tag:blogger.com,1999:blog-39977975684003744682024-03-19T02:48:39.804-06:00Estadística, Matemática y ComputaciónMaterial de los cursos impartidos por el Ingeniero Luis Manfredo Reyes
Guatemala, Centro América para el mundo. Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.comBlogger239125tag:blogger.com,1999:blog-3997797568400374468.post-16457240926349378452019-06-12T17:38:00.000-06:002019-06-12T17:54:40.888-06:00GEOMETRIA ANALITICA CON R<br />
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<span lang="ES"><b>GEOMETRIA ANALITICA CON R</b><o:p></o:p></span></div>
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<span lang="ES"><br /></span></div>
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<span lang="ES">Ing. Luis Manfredo Reyes<o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";">R es actualmente el lenguaje de
análisis<span style="mso-spacerun: yes;"> </span>más utilizado en el mundo
académico y de investigación, primero por su potencia de procesamiento, y en
segundo lugar por ser un software libre.</span><span lang="ES"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";">R<span style="mso-spacerun: yes;">
</span>puede ser descargado del siguiente sitio: </span><span lang="ES"><a href="http://www.r-project.org/"><span style="background: white; color: #888888; font-family: "arial" , "sans-serif";">http://www.r-project.org/</span></a><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";">Existen versiones para Windows, Mac, Linux
y Android<o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";"><b>UN ERROR MUY COMUN POR TODOS LADOS ES: HAY
GENTE QUE CREE QUE R ES UN PAQUETE ESTADÍSTICO (COMO SAS O SPSS).<o:p></o:p></b></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";">Por ser un lenguaje y no un paquete, R
tiene capacidades para muchos procesos, pero se debe reconocer que quizá donde
más se le ha utilizado es en el campo de la estadística.<o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif";">Una de las ventajas de R, es la enorme
cantidad de librerías (o paquetes), que han sido desarrolladas para ampliar sus
capacidades.<o:p></o:p></span></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 10.5pt;"></span></div>
<a name='more'></a><br />
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif;">Sin embargo, R no fue diseñado como un
software matemático, por lo que no posee todas las capacidades de análisis que
se utilizan en geometría analítica. Para quienes deseen trabajar Geometría
Analítica con mejor potencia se recomienda el software libre <b>GEOGEBRA.</b><o:p></o:p></span></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif;">Sin embargo, para los que desean
realizar éste tipo de cálculos sin abandonar el ambiente de R, hay esperanzas.<span style="font-size: 10.5pt;"><o:p></o:p></span></span></div>
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<span lang="ES"><b>CASO 1. LÍNEAS RECTAS </b><o:p></o:p></span></div>
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<span lang="ES">Ejemplo: y=3x-1, entre x=-5, x=5<o:p></o:p></span></div>
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<span lang="ES">curve(3*x-1, from=-5, to=5)<o:p></o:p></span></div>
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<span lang="ES"><b>CASO 2: CIRCUNFERENCIAS</b><o:p></o:p></span></div>
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<span lang="ES">Se puede realizar directamente con las funciones nativas, o con
paquetes adicionales.<o:p></o:p></span></div>
<pre><span lang="ES"><a href="https://stackoverflow.com/questions/22265704/drawing-circle-in-r">https://stackoverflow.com/questions/22265704/drawing-circle-in-r</a><span style="color: black;"><o:p></o:p></span></span></pre>
<pre><span lang="ES" style="color: black;"><o:p> </o:p></span></pre>
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<span style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">usando plotrix: el paquete plotrix debe estar
instalado<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">library(plotrix)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">plot(</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">1<span style="background: #EBF4F7;">:</span>5<span style="background: #EBF4F7;">,seq(</span>1<span style="background: #EBF4F7;">,</span>10<span style="background: #EBF4F7;">,length=</span>5<span style="background: #EBF4F7;">),type=</span></span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"n"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,xlab=</span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">""</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,ylab=</span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">""</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,main=</span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"Test draw.circle"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;"><span style="mso-spacerun: yes;"> </span>draw.circle(</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">2<span style="background: #EBF4F7;">,</span>4<span style="background: #EBF4F7;">,c(</span>1<span style="background: #EBF4F7;">,</span>0.66<span style="background: #EBF4F7;">,</span>0.33<span style="background: #EBF4F7;">),border=</span></span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"purple"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;"><span style="mso-spacerun: yes;"> </span>col=c(</span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"#ff00ff"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,</span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"#ff77ff"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,</span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"#ffccff"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">),lty=</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">1<span style="background: #EBF4F7;">,lwd=</span>1<span style="background: #EBF4F7;">)<o:p></o:p></span></span></div>
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<span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;"><span style="mso-spacerun: yes;"> </span>draw.circle(</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">2.5<span style="background: #EBF4F7;">,</span>8<span style="background: #EBF4F7;">,</span>0.6<span style="background: #EBF4F7;">,border=</span></span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"red"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,lty=</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">3<span style="background: #EBF4F7;">,lwd=</span>3<span style="background: #EBF4F7;">)<o:p></o:p></span></span></div>
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<span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;"><span style="mso-spacerun: yes;"> </span>draw.circle(</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">4<span style="background: #EBF4F7;">,</span>3<span style="background: #EBF4F7;">,</span>0.7<span style="background: #EBF4F7;">,border=</span></span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"green"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,col=</span><span lang="EN-US" style="color: #0048ab; font-family: "courier new"; font-size: 9.0pt;">"yellow"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;">,lty=</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">1<span style="background: #EBF4F7;">,<o:p></o:p></span></span></div>
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<span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-family: "courier new"; font-size: 9.0pt;"><span style="mso-spacerun: yes;"> </span>density=</span><span lang="EN-US" style="color: #00193a; font-family: "courier new"; font-size: 9.0pt;">5<span style="background: #EBF4F7;">,angle=</span>30<span style="background: #EBF4F7;">,lwd=</span>10<span style="background: #EBF4F7;">)<o:p></o:p></span></span></div>
<pre><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-size: 9.0pt;"><span style="mso-spacerun: yes;"> </span>draw.circle(</span><span lang="EN-US" style="color: #00193a; font-size: 9.0pt;">3.5<span style="background: #EBF4F7;">,</span>8<span style="background: #EBF4F7;">,</span>0.8<span style="background: #EBF4F7;">,border=</span></span><span lang="EN-US" style="color: #0048ab; font-size: 9.0pt;">"blue"</span><span lang="EN-US" style="background: #ebf4f7; color: #00193a; font-size: 9.0pt;">,lty=</span><span lang="EN-US" style="color: #00193a; font-size: 9.0pt;">2<span style="background: #EBF4F7;">,lwd=</span>2<span style="background: #EBF4F7;">)</span></span><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
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<span style="mso-ansi-language: ES-GT;">Otro ejemplo: aplicación del método Monte Carlo
para encontrar el área de un círculo<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">n <- </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1000</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">set.seed(</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">x <- runif(n, </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">-1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">y <- runif(n, </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">-1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">#x and y have the same scale -> Circle</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">plot(x, y, asp=</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">symbols(x=</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, y=</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, circles=</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, inches=F, add=T)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">#In case x and y don't have the same scale
-> Ellipse</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">#Use Gregor's Answer</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">plot(x,y)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">radius <- </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">theta <- seq(</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">2</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"> * pi, length = </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">200</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">lines(x = radius * cos(theta), y = radius *
sin(theta))<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">#Using plotrix</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">library(</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">"plotrix"</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">plot(x, y, asp=</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">draw.circle(x=</span><span style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, y=</span><span style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, radius=</span><span style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)<o:p></o:p></span></div>
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<span style="mso-ansi-language: ES-GT; mso-fareast-language: ES-GT; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape
id="Imagen_x0020_5" o:spid="_x0000_i1030" type="#_x0000_t75" style='width:249.75pt;
height:222pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\LUISMA~1\AppData\Local\Temp\msohtmlclip1\01\clip_image005.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="mso-ansi-language: ES-GT;"><o:p></o:p></span></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgd9AGLn_p2CQOU7Hd5Y-rLzezIQQu6Yj84fkzL2Jpp9ykSq50RKOao2S51geRIK9k8wm7kRmNKVshIWcHfI7p8DRCD3_MUJRWRE0poHghyCxcoM09kGvqrF0HTHqAlT6V4WOx1XRpxLbE/s1600/circle2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="301" data-original-width="333" height="289" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgd9AGLn_p2CQOU7Hd5Y-rLzezIQQu6Yj84fkzL2Jpp9ykSq50RKOao2S51geRIK9k8wm7kRmNKVshIWcHfI7p8DRCD3_MUJRWRE0poHghyCxcoM09kGvqrF0HTHqAlT6V4WOx1XRpxLbE/s320/circle2.png" width="320" /></a></div>
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<span lang="ES"><b>CASO 3: ELIPSES</b><o:p></o:p></span></div>
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<span lang="ES">Se puede realizar directamente con las funciones nativas, o con
paquetes adicionales.<o:p></o:p></span></div>
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<span lang="ES">Ejempo tomado de sitio: <o:p></o:p></span></div>
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<span lang="ES"><a href="https://stat.ethz.ch/pipermail/r-help/2005-August/077600.html">https://stat.ethz.ch/pipermail/r-help/2005-August/077600.html</a></span><i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;"><o:p></o:p></span></i></div>
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<br /></div>
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<span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;">Grafique
la elipse con centro en el origen, distancia del vértice al foco de 3 unidades,
excentricidad 1.5<i> <o:p></o:p></i></span></div>
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<br /></div>
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<i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;">theta=seq(0,2*pi,len=500)<o:p></o:p></span></i></div>
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<i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;">e=1.5<o:p></o:p></span></i></div>
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<i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;">r=3<o:p></o:p></span></i></div>
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<i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;">x=e*r*cos(theta)<o:p></o:p></span></i></div>
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<i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;">y=r*sin(theta)<o:p></o:p></span></i></div>
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<i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;">plot(x,y,asp=1)</span></i><span lang="ES" style="color: black; font-family: "courier new"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="mso-ansi-language: ES-GT; mso-fareast-language: ES-GT; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="_x0000_i1029" type="#_x0000_t75"
style='width:255.75pt;height:231.75pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\LUISMA~1\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="ES"><o:p></o:p></span></div>
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<span lang="ES"><br /></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi88epNRTffXKalPpt2rKlecoYYyeXdjIb7oR92XYMi39BzD2UZXl5W2H2LCdh278N4hidlPMMhiHKA381nsCyNSxiaVUZGooACBQxdIBjmB5nZUynJkiNsudwj6T2FtWkr9M4e9r5hmTg/s1600/elipse1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="480" data-original-width="531" height="289" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi88epNRTffXKalPpt2rKlecoYYyeXdjIb7oR92XYMi39BzD2UZXl5W2H2LCdh278N4hidlPMMhiHKA381nsCyNSxiaVUZGooACBQxdIBjmB5nZUynJkiNsudwj6T2FtWkr9M4e9r5hmTg/s320/elipse1.png" width="320" /></a></div>
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<span lang="ES"><br /></span></div>
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<span lang="ES"><br /></span></div>
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<span lang="ES">Grafique la misma elipse del ejemplo anterior,
rotada 45 gradoss<o:p></o:p></span></div>
<pre><i><span lang="ES" style="color: black;">phi=pi/4<o:p></o:p></span></i></pre>
<pre><i><span lang="ES" style="color: black;">xr=x*sin(phi)+y*cos(phi)<o:p></o:p></span></i></pre>
<pre><i><span lang="ES" style="color: black;">yr=-x*cos(phi)+y*sin(phi)<o:p></o:p></span></i></pre>
<pre><i><span lang="ES" style="color: black;">plot(xr,yr,asp=1)<o:p></o:p></span></i></pre>
<pre><span lang="ES" style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT; mso-fareast-language: ES-GT; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="_x0000_i1028" type="#_x0000_t75"
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<v:imagedata src="file:///C:\Users\LUISMA~1\AppData\Local\Temp\msohtmlclip1\01\clip_image009.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="ES" style="color: black;"><o:p></o:p></span></pre>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyrf-0_PSm47fbFpeJ_k6d3bYk94fb1C2mLkJFIKQDJ2LKdxbSvs0B96UAY59YF3xLOkngclSYnNjtLZHu_Bb_K_e29GOTnQfM0We5BqVBaLv9JqW6hDWJnQXJXHSOA2zCfIWkQRM_K8k/s1600/ELIPSE2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="465" data-original-width="525" height="283" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyrf-0_PSm47fbFpeJ_k6d3bYk94fb1C2mLkJFIKQDJ2LKdxbSvs0B96UAY59YF3xLOkngclSYnNjtLZHu_Bb_K_e29GOTnQfM0We5BqVBaLv9JqW6hDWJnQXJXHSOA2zCfIWkQRM_K8k/s320/ELIPSE2.png" width="320" /></a></div>
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<span style="mso-ansi-language: ES-GT;"><br /></span></div>
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<span style="mso-ansi-language: ES-GT;">Otra forma, tomada del
sitio:<o:p></o:p></span></div>
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<span lang="ES"><a href="https://stackoverflow.com/questions/41820683/how-to-plot-ellipse-given-a-general-equation-in-r"><span lang="ES-GT" style="mso-ansi-language: ES-GT;">https://stackoverflow.com/questions/41820683/how-to-plot-ellipse-given-a-general-equation-in-r</span></a><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">xc <- </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"> </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"># center x_c or h</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">yc <- </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">2</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"> </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"># y_c or k</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">a <- </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">5</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"> </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"># major axis length</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">b <- </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">2</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"> </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"># minor axis length</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">phi <- pi/</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">3</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"> </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #858c93; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"># angle of major axis with x axis phi or tau</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;"><o:p></o:p></span></div>
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<br /></div>
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<span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">t <- seq(</span><span style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, </span><span style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">2</span><span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">*pi, </span><span style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0.01</span><span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">) <o:p></o:p></span></div>
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<span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">x <- xc + a*cos(t)*cos(phi) -
b*sin(t)*sin(phi)<o:p></o:p></span></div>
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<span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">y <- yc + a*cos(t)*cos(phi) +
b*sin(t)*cos(phi)<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">plot(x,y,pch=</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">19</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">, col=</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">'blue'</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)</span><span lang="EN-US" style="color: #393318; font-family: "consolas"; font-size: 10.0pt;"><o:p></o:p></span></div>
<pre><span style="color: black; mso-ansi-language: ES-GT; mso-fareast-language: ES-GT; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="_x0030__x0020_Imagen"
o:spid="_x0000_i1027" type="#_x0000_t75" style='width:279.75pt;height:258pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\LUISMA~1\AppData\Local\Temp\msohtmlclip1\01\clip_image011.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p>
</o:p></span></pre>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfYt6j0iSEghOLtv0ypYmOAE6FxpBv5BWKh2HuTLXKrBDPqLkQebJVgxTZdLHjbJ1dM69iV6nLGOcjw9gyLKD5u_y_R99QegbIuPmzLRlcDQ45ePFBv47JrEVCl2kCRe_AFILI2acjbnE/s1600/elipse3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="489" data-original-width="530" height="295" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfYt6j0iSEghOLtv0ypYmOAE6FxpBv5BWKh2HuTLXKrBDPqLkQebJVgxTZdLHjbJ1dM69iV6nLGOcjw9gyLKD5u_y_R99QegbIuPmzLRlcDQ45ePFBv47JrEVCl2kCRe_AFILI2acjbnE/s320/elipse3.png" width="320" /></a></div>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT;"><b>CASO 4: PARÁBOLAS VERTICALES</b><o:p></o:p></span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT;">Ejemplo: y=x<sup>2</sup>, entre x=-5, x=5<o:p></o:p></span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT;"><o:p> </o:p></span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT;">curve(x^2, from = -5, to = 5)<o:p></o:p></span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT;"><o:p> </o:p></span></pre>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia2K4D1kE8vg6pVSw_Rz4g5oY_gTUzpUvKuZqzOVULcwEl82N4bUfD03Db9koTtbwDL5TdyCSIFNPJhczXrkNulZw2nM9kah_Gw2EcbGLc_gmyw__PvMVNNCypEjCj8s_u7Rn_62zR850/s1600/parabola1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="495" data-original-width="534" height="296" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia2K4D1kE8vg6pVSw_Rz4g5oY_gTUzpUvKuZqzOVULcwEl82N4bUfD03Db9koTtbwDL5TdyCSIFNPJhczXrkNulZw2nM9kah_Gw2EcbGLc_gmyw__PvMVNNCypEjCj8s_u7Rn_62zR850/s320/parabola1.png" width="320" /></a></div>
<pre><span style="color: black; mso-ansi-language: ES-GT; mso-fareast-language: ES-GT; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Imagen_x0020_7" o:spid="_x0000_i1026"
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<pre><span style="color: black; mso-ansi-language: ES-GT;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><b>CASO 5: PARÁBOLAS HORIZONTALES<o:p></o:p></b></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p><b> </b></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT;">La ecuación de la parabola debe ser reescrita para dejar y en función de x, queda una</span></pre>
<pre><span style="color: black; mso-ansi-language: ES-GT;"> ecuación para el brazo positivo y una para el brazo negativo.<o:p></o:p></span></pre>
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<span lang="ES">Ejemplo: x=(-y<sup>2</sup>-2y+3599)/3600<o:p></o:p></span></div>
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<span lang="ES">Al despejar y de la<span style="mso-spacerun: yes;"> </span>ecuación queda y=+/ - 60*raíz(1-x)-1<o:p></o:p></span></div>
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<span lang="ES">Fuente:<o:p></o:p></span></div>
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<span lang="ES"><a href="https://stackoverflow.com/questions/42279878/how-to-plot-a-horizontal-quadratic-function">https://stackoverflow.com/questions/42279878/how-to-plot-a-horizontal-quadratic-function</a><o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">eq1 = </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #101094; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">function</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">(x){ </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">-60</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">*(sqrt(</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">-x)</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">-1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">) }<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">eq2 = </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #101094; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">function</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">(x){ </span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">60</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">*(sqrt(</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">-x)</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">+1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">) }<o:p></o:p></span></div>
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<span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">plot(eq1, xlim=c(</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">,</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">1</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">), ylim=c(</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">0</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">,</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">100</span><span lang="EN-US" style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">))<o:p></o:p></span></div>
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<span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">plot(eq2, add=</span><span style="background: #eff0f1; border: none 1.0pt; color: #7d2727; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">TRUE</span><span style="background: #eff0f1; border: none 1.0pt; color: #303336; font-family: "inherit" , "serif"; font-size: 10.0pt; padding: 0cm;">)</span><span style="color: #393318; font-family: "consolas"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXJ6skMZrtLlT7zv_N2jBQF7v3iTRqJqGv-b_c2XpuQiI3ovCAbw8vWiqwdTojtUgGiJnvYFzOOGxOKNsx3cgtFpkdZST-Sp-EtQ5j1m9DxnibxbqrAGLd-GOh-xVgqNW06fDgJW_Msoo/s1600/parabola2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="484" data-original-width="548" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXJ6skMZrtLlT7zv_N2jBQF7v3iTRqJqGv-b_c2XpuQiI3ovCAbw8vWiqwdTojtUgGiJnvYFzOOGxOKNsx3cgtFpkdZST-Sp-EtQ5j1m9DxnibxbqrAGLd-GOh-xVgqNW06fDgJW_Msoo/s320/parabola2.png" width="320" /></a></div>
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type="#_x0000_t75" style='width:259.5pt;height:228.75pt;visibility:visible;
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<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-68191000360860489812019-06-05T07:27:00.003-06:002019-06-05T07:27:47.615-06:00Computación Simbólica con R<br />
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<span lang="EN-US" style="color: #333333; font-family: "inherit","serif"; font-size: 28.5pt; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-font-kerning: 18.0pt;">Computación
Simbólica con R<o:p></o:p></span></div>
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<i><span style="color: #333333; font-family: "inherit","serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Ing. Luis Manfredo Reyes<o:p></o:p></span></i></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 7.5pt; mso-outline-level: 4;">
<span style="color: #333333; font-family: "inherit","serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-style: italic; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Traducción y adaptación del
documento:<o:p></o:p></span></div>
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<a href="http://www.di.fc.ul.pt/~jpn/r/symbolic/">http://www.di.fc.ul.pt/~jpn/r/symbolic/</a><span style="color: #333333; font-family: "inherit","serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-style: italic; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">R es actualmente el lenguaje de análisis<span style="mso-spacerun: yes;"> </span>más utilizado en el mundo académico y de
investigación, primero por su potencia de procesamiento, y en segundo lugar por
ser un software libre.</span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
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<o:p></o:p><span style="color: #222222; font-family: "Arial","sans-serif";">R<span style="mso-spacerun: yes;"> </span>puede ser descargado
del siguiente sitio: </span><a href="http://www.r-project.org/"><span style="background: white; color: #888888; font-family: "Arial","sans-serif";">http://www.r-project.org/</span></a><o:p></o:p></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "Arial","sans-serif";">Existen versiones para Windows, Mac, Linux y Android<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">UN ERROR MUY COMUN POR TODOS LADOS ES: HAY GENTE QUE CREE QUE R
ES UN PAQUETE ESTADÍSTICO (COMO SAS O SPSS).<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">Por ser un lenguaje y no un paquete, R tiene capacidades para
muchos procesos, pero se debe reconocer que quizá donde más se le ha utilizado
es en el campo de la estadística.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">Una de las ventajas de R, es la enorme cantidad de librerías (o
paquetes), que han sido desarrolladas para ampliar sus capacidades.<o:p></o:p></span></div>
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<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><br /></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt;">
<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">R no fue diseñado como un software
matemático, por lo que no posee muchas capacidades de cálculo simbólico (Por ejemplo
(x-1)<sup>7 </sup>). Sin embargo, para los que desean realizar éste tipo de
cálculos sin abandonar el ambiente de R, hay esperanzas.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt;">
<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Existen varios paquetes desarrollados
para “salir” momentáneamente de R a realizar cálculos en programas
especializados y “retornar” con el resultado<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt;">
<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"></span></div>
<a name='more'></a><br /><br />
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt;">
<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Para éste fin, los paquetes más
utilizados son: rSymPy y Yacas.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt;">
<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">rSympy permite realizar cálculos
simbólicos usando el software Python<o:p></o:p></span></div>
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<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Yacas permite realizar cálculos
simbólicos usando el software Yacas<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt;">
<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">En éste documento se discute el uso de
rSymPy, debido a que posee más funciones disponibles.<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">REQUISITOS
PREVIOS:<o:p></o:p></span></b></div>
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<span style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Debe estar instalado el software Python,
su agregado Symphy y el Java<o:p></o:p></span></div>
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<span lang="EN-US" style="color: #333333; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Referencias:<o:p></o:p></span></div>
<ul type="disc">
<li class="MsoNormal" style="background: white; color: #333333; line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span style="font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Manual de Referencia (muy corto)<o:p></o:p></span></li>
<li class="MsoNormal" style="background: white; color: #333333; line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span style="font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: windowtext;"><a href="http://cran.r-project.org/web/packages/rSymPy/rSymPy.pdf"><span style="color: #337ab7; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; text-decoration: none; text-underline: none;">http://cran.r-project.org/web/packages/rSymPy/rSymPy.pdf</span></a></span><span style="font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></li>
<li class="MsoNormal" style="background: white; color: #333333; line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span style="font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Tutorial de SymPy: </span><span style="color: windowtext;"><a href="http://docs.sympy.org/latest/tutorial/"><span style="color: #337ab7; font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; text-decoration: none; text-underline: none;">http://docs.sympy.org/latest/tutorial/</span></a></span><span style="font-family: "Helvetica","sans-serif"; font-size: 10.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></li>
</ul>
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<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<b><span style="color: #990000; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Cargar la Librería:<o:p></o:p></span></b></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<b><span style="color: #990000; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">library</span></b><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">rSymPy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Definir x como variable simbólica<o:p></o:p></b></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"x"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Suma de dos veces x</b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">+</span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "2*x"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Simplificación sencilla</b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">*</span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">/</span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
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<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "x"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Potencias</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">y</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"x**3"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">/</span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">y</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "x**(-2)"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Expresiones<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">z</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"2.5*x**2"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">z</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">+</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">y</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"2.5*x**2 + x**3"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Cálculos numéricos</b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Raíz cuadrada de 8</b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"sqrt(8).evalf()"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"2.82842712474619"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Raíz cuadrada de 8
con 50 decimales<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"sqrt(8).evalf(50)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"2.8284271247461900976033774484193961571393437507539"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Pi con 120
decimales</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"pi.evalf(120)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
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<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230665"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Trigonométricas</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"one = cos(1)**2 + sin(1)**2"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "cos(1)**2 + sin(1)**2"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Errores de redondeo</b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"(one - 1).evalf()"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-.0e-124"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Redondeo </b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"(one - 1).evalf(chop=True)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"0"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Crear una ecuación </b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"Eq(x**2+2*x+1,(x+1)**2)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "1 +
2*x + x**2 == (1 + x)**2"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Asignar expression
a variable<o:p></o:p></b></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"a = x**2+2*x+1"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "1 +
2*x + x**2"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"b = (x+1)**2"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "(1
+ x)**2"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Simplificar</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"0"</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">==</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"simplify(a-b)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] TRUE<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<i><span style="color: #999988; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Otras simplificaciones</b><o:p></o:p></span></i></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"simplify((x**3 + x**2 - x - 1)/(x**2 + 2*x +
1))"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-1
+ x"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"(x + 2)*(x - 3)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-(2
+ x)*(3 - x)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Desarrollo de
productos</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"expand((x + 2)*(x - 3))"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-6
- x + x**2"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Factorización</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"factor(x**3 - x**2 + x - 1)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-(1
+ x**2)*(1 - x)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Simplificaciones</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">y</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"y"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">z</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"z"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"collect(x*y + x - 3 + 2*x**2 - z*x**2 + x**3,
x)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-3 + x*(1 + y) + x**2*(2 -
z) + x**3"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"(x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z +
z**2)/(x**2 - 1)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-(-2*y*z - 2*x*y*z + y**2 +
z**2 + x*y**2 + x*z**2)/(1 - x**2)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"cancel((x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z
+ z**2)/(x**2 - 1))"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"(2*y*z - y**2 - z**2)/(1 - x)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"expand_func(gamma(x + 3))"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"x*(1 + x)*(2 + x)*gamma(x)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Crear una variable “persistente” <o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"y = x*x"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "x**2"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Matrices</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"A = Matrix([[1,x], [y,1]])"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"[<span style="mso-spacerun: yes;"> </span>1, x]\n[x**2, 1]"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"A**2"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "[1
+ x**3,<span style="mso-spacerun: yes;"> </span>2*x]\n[<span style="mso-spacerun: yes;"> </span>2*x**2, 1 + x**3]"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"B = A.subs(x,1.1)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"[<span style="mso-spacerun: yes;"> </span>1, 1.1]\n[1.21,<span style="mso-spacerun: yes;"> </span>1]"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"B**2"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
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<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"[2.331,<span style="mso-spacerun: yes;"> </span>2.2]\n[ 2.42, 2.331]"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"expr = sin(2*x) + cos(2*x)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
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<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "cos(2*x) + sin(2*x)"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"expr.subs(sin(2*x), 2*sin(x)*cos(x))"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "2*cos(x)*sin(x) +
cos(2*x)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"expr.subs(x,pi/2)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"-1"<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Courier New"; font-size: 10pt;">a1</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10pt;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;"><-</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10pt;"> </span><span lang="EN-US" style="font-family: "Courier New"; font-size: 10pt;">Var</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10pt;">"a1"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;">)</span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">a2</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"a2"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">a3</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"a3"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">a4</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"a4"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Inversa y
Determinante de matrices</b><o:p></o:p></span></div>
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<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">A</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Matrix</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">List</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">a1</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">a2</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">List</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">a3</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">a4</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">))</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Courier New"; font-size: 10pt;">Inv</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10pt;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;"><-</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10pt;"> </span><b><span lang="EN-US" style="color: #990000; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">function</span></b><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;">(</span><span lang="EN-US" style="font-family: "Courier New"; font-size: 10pt;">x</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10pt;"> </span><span lang="EN-US" style="font-family: "Courier New"; font-size: 10pt;">Sym</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10pt;">"("</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10pt;">, </span><span lang="EN-US" style="font-family: "Courier New"; font-size: 10pt;">x</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10pt;">, </span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10pt;">").inv()"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10pt;">)</span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Det</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><b><span lang="EN-US" style="color: #990000; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">function</span></b><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Sym</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"("</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">").det()"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="font-family: "Courier New"; font-size: 10pt;">A</span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"[a1, a2]\n[a3, a4]"<o:p></o:p></span></div>
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<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">cat</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">A</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">,</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"\n"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## ( Matrix( ( [
a1,a2 ] ),( [ a3,a4 ] ) ) )<o:p></o:p></span></div>
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<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Inv</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">A</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"[1/a1 + a2*a3/(a1**2*(a4 - a2*a3/a1)), -a2/(a1*(a4 - a2*a3/a1))]\n[<span style="mso-spacerun: yes;"> </span>-a3/(a1*(a4 - a2*a3/a1)),<span style="mso-spacerun: yes;"> </span>1/(a4 - a2*a3/a1)]"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Det</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">A</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"a1*a4 - a2*a3"<o:p></o:p></span></div>
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<i><span style="color: #999988; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Crear función exponencial</b><o:p></o:p></span></i></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Exp</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><b><span style="color: #990000; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">function</span></b><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Sym</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"exp("</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">")"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Exp</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(-</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">*</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Exp</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"1"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">y</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><-</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Var</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"y"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"sqrt(8)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"2*2**(1/2)"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Resolver ecuación </b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"solve(x**2 - 2, x)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"[2**(1/2), -2**(1/2)]"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Límites </b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"limit(1/x, x, oo)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"0"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"limit(1/x, x, 0)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"oo"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Integración </b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"integrate(exp(-x))"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"-exp(-x)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"integrate(exp(-x*y),x)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-exp(-x*y)/y"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Integral definida<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"integrate(exp(-x), (x, 0, oo))"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"1"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Integración definida con función nativa de
R<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">integrate</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><b><span lang="EN-US" style="color: #990000; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">function</span></b><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">exp</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(-</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: #009999; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">0</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: #990073; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Inf</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## 1 with absolute error < 5.7e-05<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"integrate(x**2 - y, (x, -5, 5), (y, -pi,
pi))"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"500*pi/3"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Derivada</b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"diff(sin(2*x), x, 1)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"2*cos(2*x)"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Función nativa de R para derivada</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">D</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">expression</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sin</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #009999; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">2</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">*</span><span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">))</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">, </span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"x"</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> </span><i><span lang="EN-US" style="color: #999988; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"># also possible in base R</span></i><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## cos(2 * x) *
2<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Segunda derivada<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"diff(sin(2*x), x, 2)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"-4*sin(2*x)"<o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Tercera derivada </b><o:p></o:p></span></div>
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<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"diff(sin(2*x), x, 3)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "-8*cos(2*x)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Derivadas parciales</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"diff(exp(x*y*z), x, y, y)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "z*exp(x*y*z) +
x*y*z**2*exp(x*y*z)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"diff(exp(x*y*z), x, z, 3)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "y*exp(x*y*z) +
x*z*y**2*exp(x*y*z)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Serie de Taylor <o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"(1/cos(x)).series(x, 0, 10)"</span><span style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> <o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "1 + x**2/2 + 5*x**4/24 +
61*x**6/720 + 277*x**8/8064 + O(x**10)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"exp(x).series(x, 0, 5)"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "1 +
x + x**2/2 + x**3/6 + x**4/24 + O(x**5)"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"exp(x).series(x, 0, 5).removeO()"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1] "1 +
x + x**2/2 + x**3/6 + x**4/24"<o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>Calcular
eigenvalores de una matriz</b><o:p></o:p></span></div>
<div class="MsoNormal" style="background: whitesmoke; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: black; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">sympy</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">(</span><span lang="EN-US" style="color: #dd1144; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">"Matrix([[1, 2], [2, 2]]).eigenvals()"</span><span lang="EN-US" style="color: #687687; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">)</span><span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "Courier New"; font-size: 10.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">## [1]
"{3/2 - 17**(1/2)/2: 1, 3/2 + 17**(1/2)/2: 1}"<o:p></o:p></span></div>
</div>
<div class="MsoNormal">
<br /></div>
<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-83477632572699911052019-06-04T07:25:00.002-06:002019-06-04T07:27:09.602-06:00ALGEBRA LINEAL CON R<br />
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">ALGEBRA LINEAL CON R<o:p></o:p></b></div>
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<b style="mso-bidi-font-weight: normal;"><br /></b></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , "sans-serif";">Ing. Luis Manfredo Reyes Chávez<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif";"><br /></span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , "sans-serif";">R es actualmente el lenguaje de análisis <span style="mso-spacerun: yes;"> </span>más utilizado en el mundo académico y de
investigación, primero por su potencia de procesamiento, y en segundo lugar por
ser un software libre.</span><o:p></o:p></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , "sans-serif";">R<span style="mso-spacerun: yes;"> </span>puede ser descargado
del siguiente sitio: </span><a href="http://www.r-project.org/"><span style="background: white; color: #888888; font-family: "arial" , "sans-serif";">http://www.r-project.org/</span></a><o:p></o:p></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif";">Existen versiones para Windows, Mac, Linux y Android<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif";">UN ERROR MUY COMUN POR TODOS LADOS ES: HAY GENTE QUE CREE QUE R
ES UN PAQUETE ESTADÍSTICO (COMO SAS O SPSS).<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , "sans-serif";">Por ser un lenguaje y no un paquete, R tiene capacidades para
muchos procesos, pero se debe reconocer que quizá donde más se le ha utilizado
es en el campo de la estadística.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , "sans-serif";">Una de las ventajas de R, es la enorme cantidad de librerías (o
paquetes), que han sido desarrolladas para ampliar sus capacidades.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif";">Hay muchos
procesos estadísticos que requieren operaciones con matrices, de modo que no es
extraño que R posea una vasta cantidad de funciones para trabajar con ellas. Y
para casos especiales, hay un paquete que permite emular a MATLAB, y otros para
trabajar con casos más complejos (Matrix, LAPACK, TAUC)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif";"></span></div>
<a name='more'></a><br />
<br />
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif";">Traducción y
adaptación del documento:<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 14.0pt; line-height: 115%;"><a href="https://www.statmethods.net/advstats/matrix.html">https://www.statmethods.net/advstats/matrix.html</a><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 24.0pt; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 16.5pt; mso-outline-level: 2;">
<span lang="EN-US" style="color: #3d4251; font-family: "arial" , "sans-serif"; font-size: 18.0pt;">Funciones disponibles en R<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 24.0pt; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 16.5pt; mso-outline-level: 2;">
<span style="color: #3d4251; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">En la siguiente lista,
A y B son matrices, x y b son vectores.<o:p></o:p></span></div>
<table border="0" cellpadding="0" class="MsoNormalTable" style="background: white; margin-left: 30.0pt; mso-cellspacing: 1.5pt; mso-padding-alt: 0cm 0cm 0cm 0cm; mso-yfti-tbllook: 1184;">
<tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">Operador o Función</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">Descripción</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">A * B</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Producto por elementos<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">A %*% B</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Pproducto de Matrices<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">A %o% B</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Producto externo.<b> AB'</b><o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";">crossprod(A,B)<br />
crossprod(A)</span></b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">A'B</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"> and <b>A'A </b>respectivamente.<o:p></o:p></span></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 5;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">t(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Traspuesta<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 6;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">diag(x)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Crea una matriz diagonal con los elementos de <b>x</b> en la
diagonal principal<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 7;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">diag(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Retorna un vector que contiene los elementos de la diagonal principal<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 8;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">diag(k)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Si k es un escalar, crea una matriz identidad <span style="mso-spacerun: yes;"> </span>k x k <o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 9;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">solve(A, b)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Retorna el vector <b>x</b> en la ecuación <b>b = Ax </b>(i.e., <b>A<sup>-1</sup>b</b>)<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 10;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">solve(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Inversa de <b>A</b> si A es una matriz cuadrada (no todas las
matrices cuadradas tienen inversa).<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 11;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">ginv(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Inversa generalizada de Moore-Penrose de <b>A</b>.<br />
ginv(A) utiliza el paquete <b> MASS</b> <o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 12;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">y<-eigen(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">y$val </span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">son los<b> </b></span><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><span style="mso-spacerun: yes;"> </span>eigenvalores de <b>A</b><br />
<b>y$vec</b> son los <span style="mso-spacerun: yes;"> </span>eigenvectores
of <b>A</b><o:p></o:p></span></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 13;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">y<-svd(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Descomposición de valores simples de A .<br />
<b>y$d</b> = vector con los valores singulares de A <b><br />
y$u</b> = matriz con columnas que <span style="mso-spacerun: yes;"> </span>contienen los vectores singulares izquierdos
de <b>A</b><br />
<b>y$v </b>= matriz con columnas que contienen los vectores singulares
derechos de <b>A</b><o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 14;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">R <- chol(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Factorización de Choleski para <b>A</b>. Retorna el factor
triangular superopr, de modo que <b>R'R = A</b>.<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 15;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">y <- qr(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";">Descomposición QR de <b>A</b>.<br />
</span><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">y$qr </span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">contiene el triangulo superior que contiene la
descomposición, y un triangulo inferior que contiene información de la
descomposición <b><o:p></o:p></b></span></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">Qy$rank</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"> es el rango de A.<br />
<b>y$qraux</b> un vector que contiene información adicional de Q <br />
<b>y$pivot</b> contiene informacion de la estrategia de pivoteo
utilizada <o:p></o:p></span></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 16;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">cbind(A,B,...)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><span style="color: #4d5356; font-family: "arial" , "sans-serif";">Combina matrices (o vectores) horizontalmente. </span><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";">Retorna <span style="mso-spacerun: yes;"> </span>una matriz.<o:p></o:p></span></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 17;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">rbind(A,B,...)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Combina matrices (o vectores) verticalmente. Retorna
una matriz.<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 18;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";">rowMeans(A)</span></b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Retorna un vector de medias por filas.<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 19;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";">rowSums(A)</span></b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Retorna un vector de sumas por filas <o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 20;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";">colMeans(A)</span></b><span lang="EN-US" style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Retorna un <span style="mso-spacerun: yes;"> </span>vector de medias por
columnas<o:p></o:p></span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 21; mso-yfti-lastrow: yes;">
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><b><span style="color: #4d5356; font-family: "arial" , "sans-serif";">colSums(A)</span></b><span style="color: #4d5356; font-family: "arial" , "sans-serif";"><o:p></o:p></span></span></div>
</td>
<td style="background: #F2F2F2; padding: 2.25pt 2.25pt 2.25pt 2.25pt;" valign="top"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #4d5356; font-family: "arial" , "sans-serif"; font-size: x-small;">Retorna un <span style="mso-spacerun: yes;"> </span>vector de sumas por
columnas<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-size: x-small;"><br /></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Y ahora…. Los ejemplos, traducidos y adaptados del
documento:<o:p></o:p></div>
<div class="MsoNormal">
<span style="font-size: 14.0pt; line-height: 115%;"><a href="http://www.philender.com/courses/multivariate/notes/matr.html">http://www.philender.com/courses/multivariate/notes/matr.html</a><o:p></o:p></span></div>
<h4>
<span style="color: black; font-size: 13.5pt; line-height: 115%;"><o:p> </o:p></span></h4>
<h4>
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Crear una <span style="mso-spacerun: yes;"> </span>Matriz<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;"># La función matrix<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;"># Los datos se ingresan por columnas, indicando luego cuántas filas y cuántas columnas se tienen<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> A <- matrix(c(2,3,-2,1,2,2),3,2)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> A</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Determinar si un objeto es una <span style="mso-spacerun: yes;"> </span>Matriz<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> is.matrix(A)</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] TRUE<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">…. O si es un vector<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> is.vector(A)</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] FALSE<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Multiplicación de una matriz por un escalar<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> c <- 3<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> c*A</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>6<span style="mso-spacerun: yes;"> </span>3<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>9<span style="mso-spacerun: yes;"> </span>6<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-6<span style="mso-spacerun: yes;"> </span>6<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; font-size: 13.5pt; line-height: 115%;">Adición
y sustracción de matrices<o:p></o:p></span></h4>
<div class="MsoNormal">
<br /></div>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> B <- matrix(c(1,4,-2,1,2,1),3,2)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> B</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- A + B<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"> <o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>7<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-4<span style="mso-spacerun: yes;"> </span>3<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D <- A - B<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>-1<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; font-size: 13.5pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>Multiplicación de matrices<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D <- matrix(c(2,-2,1,2,3,1),2,3)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>3<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- D %*% A<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>10<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- A %*% D<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>7<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>7<span style="mso-spacerun: yes;"> </span>11<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-8<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>-4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D <- matrix(c(2,1,3),1,3)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>3<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- D %*% A<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>10<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- A %*% D</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">Error in A %*% D : non-conformable arguments<o:p></o:p></span></pre>
<pre><b style="mso-bidi-font-weight: normal;"><span style="color: black;">(En este caso no se puede realizar la operación por no ser conformables, es decir que el número de columnas de la primera debe ser igual al número de filas de la segunda)<o:p></o:p></span></b></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; font-size: 13.5pt; line-height: 115%;">Transpuesta
de una matriz<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> AT <- t(A)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> AT</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>-2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> ATT <- t(AT)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">>ATT</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">Vectores comunmente usados<o:p></o:p></span></h2>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">Vector unitario<o:p></o:p></span></h2>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> U <- matrix(1,3,1)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> U</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black;">Vector
nulo o cero<o:p></o:p></span></h2>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;">> Z <- matrix(0,3,1)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> Z</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black;">Matrices
comúnmente usadas<o:p></o:p></span></h2>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black;">Matriz
unitaria<o:p></o:p></span></h2>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;">> U <- matrix(1,3,2)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> U</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black;">Matriz
nula o cero<o:p></o:p></span></h2>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;">> Z <- matrix(0,3,2)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> Z</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black;">Matriz
diagonal<o:p></o:p></span></h2>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;">> S <- matrix(c(2,3,-2,1,2,2,4,2,3),3,3)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> S</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>3<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D <- diag(S)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 2 2 3<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D <- diag(diag(S))<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> D</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>3<o:p></o:p></span></pre>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">Matriz identidad<o:p></o:p></span></h2>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> I <- diag(c(1,1,1))<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> I</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<h2 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">Matriz simétrica<o:p></o:p></span></h2>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- matrix(c(2,1,5,1,3,4,5,4,-2),3,3)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>5<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>5<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>-2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> CT <- t(C)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> CT</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>5<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>5<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>-2<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Inversa de una matriz<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;">> A <- matrix(c(4,4,-2,2,6,2,2,8,4),3,3)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> A</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>6<span style="mso-spacerun: yes;"> </span>8<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><b><span style="color: black;">> AI <- solve(A)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> AI</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>1.0 -0.5<span style="mso-spacerun: yes;"> </span>0.5<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,] -4.0<span style="mso-spacerun: yes;"> </span>2.5 -3.0<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>2.5 -1.5<span style="mso-spacerun: yes;"> </span>2.0<o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><b><span style="color: black;">## el producto de una matriz por su inversa debe ser igual a la matriz identidad</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><b><span style="color: black;">> A %*% AI<span style="mso-spacerun: yes;"> </span></span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><b><span style="color: black;">> AI %*% A</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>0<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Inversa y Determinante de una<span style="mso-spacerun: yes;"> </span>Matriz<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- matrix(c(2,1,6,1,3,4,6,4,-2),3,3)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> C</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>6<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>6<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>-2<o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><b><span style="color: black;">> CI <- solve(C)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">CI</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1]<span style="mso-spacerun: yes;"> </span>[,2]<span style="mso-spacerun: yes;"> </span>[,3]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>0.2156863 -0.25490196<span style="mso-spacerun: yes;"> </span>0.13725490<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,] -0.2549020<span style="mso-spacerun: yes;"> </span>0.39215686<span style="mso-spacerun: yes;"> </span>0.01960784<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>0.1372549<span style="mso-spacerun: yes;"> </span>0.01960784 -0.04901961<o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><b><span style="color: black;">> d <- det(C)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> d</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] -102<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Rango de una matrix<o:p></o:p></span></h4>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">> A <- matrix(c(2,3,-2,1,2,2,4,7,0),3,3)<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">> A<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>7<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>0<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">> matA <- qr(A)<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">> matA$rank<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">[1] 3<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">> A <- matrix(c(2,3,-2,1,2,2,4,6,-4),3,3)<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">> A<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>6<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>-4<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span style="color: black;">> matA <- qr(A)<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> matA$rank<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 2<o:p></o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></b></pre>
<pre style="mso-outline-level: 5;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"># note column 3 is 2 times column 1<o:p></o:p></span></b></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Numero de filas y columnas <o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;">> X <- matrix(c(3,2,4,3,2,-2,6,1),4,2)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> X</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>-2<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>6<o:p></o:p></span></pre>
<pre><span style="color: black;">[4,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> dim(X)</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 4 2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> r <- nrow(X)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> r</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> c <- ncol(X)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> c</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 2<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Calcular sumas de filas y columnas <o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;"># ojo con la S en mayúsculas <o:p></o:p></span></b></pre>
<pre><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> A <- matrix(c(2,3,-2,1,2,2),3,2)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> A</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> c <- colSums(A)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> c</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 3 5<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> r <- rowSums(A)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> r</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 3 5 0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> a <- sum(A)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> a</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 8<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span style="color: black; font-size: 13.5pt; line-height: 115%;">Calcular medias por filas y columnas<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span style="color: black;"># Ojo con la M en mayúsculas<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;"><o:p> </o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> cm <- colMeans(A)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> cm</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 1.000000 1.666667<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> rm <- rowMeans(A)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> rm</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 1.5 2.5 0.0<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> m <- mean(A)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> m</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1] 1.333333<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; font-size: 13.5pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>Concatenación por columnas<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> A</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> B <- matrix(c(1,3,2,1,4,2),3,2)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> B</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- cbind(A,B)<o:p></o:p></span></b></pre>
<pre><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C</span></b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><o:p> </o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3] [,4]<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<h4 style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;">
<span lang="EN-US" style="color: black; font-size: 13.5pt; line-height: 115%;">Concatenación
por filas<o:p></o:p></span></h4>
<pre style="-webkit-text-stroke-width: 0px; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; widows: 2; word-spacing: 0px;"><b><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">> C <- rbind(A,B)<o:p></o:p></span></b></pre>
<pre><b><span style="color: black;">> C</span></b><span style="color: black;"><o:p></o:p></span></pre>
<pre><span style="color: black;"><o:p> </o:p></span></pre>
<pre><span style="color: black;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></pre>
<pre><span style="color: black;">[1,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;">[2,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span style="color: black;">[3,]<span style="mso-spacerun: yes;"> </span>-2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
<pre><span style="color: black;">[4,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;"> </span>1<o:p></o:p></span></pre>
<pre><span style="color: black;">[5,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></pre>
<pre><span style="color: black;">[6,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></pre>
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<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-50133282592137899132019-06-03T10:19:00.000-06:002019-06-03T10:19:40.751-06:00SOLUCIÓN DE ECUACIONES Y SISTEMAS DE ECUACIONES CON R<br />
<div class="MsoNormal" style="background: white;">
<b style="mso-bidi-font-weight: normal;"><span style="color: #222222; font-family: "Arial","sans-serif";">SOLUCIÓN
DE ECUACIONES Y SISTEMAS DE ECUACIONES CON R<o:p></o:p></span></b></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">Ing. Luis Manfredo Reyes Chávez<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">R es actualmente el lenguaje de análisis <span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>más
utilizado en el mundo académico y de investigación, primero por su potencia de
procesamiento, y en segundo lugar por ser un software libre.</span><o:p></o:p></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">R<span style="mso-spacerun: yes;"> </span>puede ser descargado
del siguiente sitio: </span><a href="http://www.r-project.org/"><span style="background: white; color: #888888; font-family: "Arial","sans-serif";">http://www.r-project.org/</span></a><o:p></o:p></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">Existen versiones para Windows, Mac, Linux y Android<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">UN ERROR MUY COMUN POR TODOS LADOS ES: HAY GENTE QUE CREE QUE R
ES UN PAQUETE ESTADÍSTICO (COMO SAS O SPSS).<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">Por ser un lenguaje y no un paquete, R tiene capacidades para
muchos procesos, pero se debe reconocer que quizá donde más se le ha utilizado
es en el campo de la estadística.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">Una de las ventajas de R, es la enorme cantidad de librerías (o
paquetes), que han sido desarrolladas para ampliar sus capacidades.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";"><br /></span></div>
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<span style="color: #222222; font-family: "Arial","sans-serif";">Si se necesita un poder de computación algébrica, o cálculos
demasiado complejos , es mejor usar paquetes libres que sí poseen esas
capacidades, por ejemplo MAXIMA O REDUCE, que están disponibles sin costo en el
sitio del proyecto GNU.<o:p></o:p></span></div>
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<span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Sin embargo, para situaciones comunes, es posible usar R<o:p></o:p></span></div>
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<br /></div>
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<br /></div>
<a name='more'></a><br />
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<!--[if !supportLists]--><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: Arial; mso-fareast-language: ES;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">SOLUCIÓN DE ECUACIONES
LINEALES<o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Resuelva:<span style="mso-spacerun: yes;"> </span>5x = 10<span style="mso-spacerun: yes;"> </span>(si, muy fácil)<o:p></o:p></span></div>
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<span style="color: red; font-family: "Courier New"; font-size: 16.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">>solve(5,10)<o:p></o:p></span></div>
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<span style="color: blue; font-family: "Courier New"; font-size: 16.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">[1] 2<o:p></o:p></span></div>
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<span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><br />
2. SOLUCIÓN DE SISTEMAS DE ECUACIONES LINEALES<br style="mso-special-character: line-break;" />
<!--[if !supportLineBreakNewLine]--><br style="mso-special-character: line-break;" />
<!--[endif]--><o:p></o:p></span></div>
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<span style="color: black; font-family: "Arial","sans-serif"; font-size: 16.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">3x + 2y = 8<br />
x + y =2<br />
</span><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><br />
Las ecuaciones se expresan en forma matricial Ax=bI<o:p></o:p></span></div>
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<span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">En este ejemplo se usa la función Matrix para especificar las matrices, el
ingreso se realiza por columnas<o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">La matriz A es 3 2<o:p></o:p></span></div>
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<span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span> 1
1<br />
El vector b <span style="mso-spacerun: yes;"> </span>is:<br />
<span style="mso-spacerun: yes;"> </span>8<br />
<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></div>
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<span lang="EN-US" style="color: red; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">>
a <- matrix(c(3,1,2,1),nrow=2,ncol=2)<o:p></o:p></span></div>
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<span lang="EN-US" style="color: red; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">>
a<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span>[,1] [,2]<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">[1,]<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;">
</span>2<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">[2,]<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;">
</span>1<o:p></o:p></span></div>
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<span lang="EN-US" style="color: red; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">>
b <- matrix(c(8,2),nrow=2,ncol=1)<o:p></o:p></span></div>
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<span lang="EN-US" style="color: red; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">>
b<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span>[,1]<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">[1,]<span style="mso-spacerun: yes;"> </span>8<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">[2,]<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></div>
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<span lang="EN-US" style="color: red; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">>
solve(a,b)<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><span style="mso-spacerun: yes;"> </span>[,1]<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">[1,]<span style="mso-spacerun: yes;"> </span>4<o:p></o:p></span></div>
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<span lang="EN-US" style="color: blue; font-family: "Courier New"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">[2,]<span style="mso-spacerun: yes;"> </span>-2<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; mso-ansi-language: EN-US; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><br />
Solución: <span style="mso-spacerun: yes;"> </span>x = 4, y = -2.<br />
<br style="mso-special-character: line-break;" />
<!--[if !supportLineBreakNewLine]--><br style="mso-special-character: line-break;" />
<!--[endif]--></span><span style="color: blue; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: blue; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Otro ejemplo:<o:p></o:p></span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="background: white; border-collapse: collapse; mso-padding-alt: 0cm 0cm 0cm 0cm; mso-yfti-tbllook: 1184; width: 189px;">
<tbody>
<tr style="height: 15.0pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="height: 15.0pt; padding: 0cm 0cm 0cm 0cm; width: 5.0cm;" width="189">
<div class="MsoNormal" style="line-height: normal; margin: 6.0pt;">
<b><span style="color: #0b5394; font-family: "inherit","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">5x + 3y + 4z = 30<o:p></o:p></span></b></div>
<div class="MsoNormal" style="line-height: normal; margin: 6.0pt;">
<b><span style="color: #0b5394; font-family: "inherit","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">2x + 3y + 3z = 20</span></b><span style="color: #5d5d5d; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
</td>
</tr>
<tr style="height: 18.75pt; mso-yfti-irow: 1;">
<td style="height: 18.75pt; padding: 0cm 0cm 0cm 0cm; width: 5.0cm;" width="189">
<div class="MsoNormal" style="line-height: normal; margin: 6.0pt;">
<b><span style="color: #0b5394; font-family: "inherit","serif"; font-size: 12.0pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"> x + 4y + 3z = 15</span></b><span style="color: #5d5d5d; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
</td>
</tr>
<tr style="height: 15.0pt; mso-yfti-irow: 2; mso-yfti-lastrow: yes;">
<td style="height: 15.0pt; padding: 0cm 0cm 0cm 0cm; width: 5.0cm;" width="189"></td>
</tr>
</tbody></table>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-outline-level: 3;">
<b><span style="color: #5d5d5d; font-family: "Arial","sans-serif"; font-size: 17.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Solución<o:p></o:p></span></b></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-outline-level: 3;">
<span style="color: #5d5d5d; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-bidi-font-weight: bold; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">En lugar de
Matrix se puede usar rbind para construir las matrices<o:p></o:p></span></div>
<div style="background: white; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 0cm 0cm 0cm 0cm;">
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="color: blue; font-size: large;"><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">a</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"> </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"><-</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"> </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">rbind</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">(</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">c</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">(</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">5</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">3</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, 4</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">)</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, <o:p></o:p></span></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="color: blue; font-size: large;"><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"><span style="mso-spacerun: yes;">
</span></span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">c</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">(2</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, 3, </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">3</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">)</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, <o:p></o:p></span></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="color: blue; font-size: large;"><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"><span style="mso-spacerun: yes;">
</span></span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">c</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">(1</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, 4, 3</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">))</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="color: blue; font-size: large;"><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">b</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"> </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"><-</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";"> </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">c</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">(</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">20</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">15</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, </span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">30</span><span lang="EN-US" style="background: rgb(248, 248, 248); font-family: "Lucida Console";">)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<br /></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="color: blue; font-size: large;"><span style="background: rgb(248, 248, 248); font-family: "Lucida Console";">solve</span><span style="background: rgb(248, 248, 248); font-family: "Lucida Console";">(</span><span style="background: rgb(248, 248, 248); font-family: "Lucida Console";">a</span><span style="background: rgb(248, 248, 248); font-family: "Lucida Console";">, </span><span style="background: rgb(248, 248, 248); font-family: "Lucida Console";">b</span><span style="background: rgb(248, 248, 248); font-family: "Lucida Console";">)</span><span style="background: rgb(248, 248, 248); font-family: "Lucida Console";"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<br /></div>
<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 0cm 0cm 0cm 0cm; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: "Lucida Console";"><span style="color: blue; font-size: large;">[1] -1.25 -6.25 13.75</span><span style="color: #5d5d5d; font-size: 10pt;"><o:p></o:p></span></span></div>
</div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<br /></div>
<div class="MsoListParagraphCxSpFirst" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo2; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Courier New"; mso-fareast-language: ES;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";"> </span><span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-style: normal; font-variant: normal; line-height: normal;"><b> </b></span></span></span><!--[endif]--><span style="font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><b>SOLUCIÓN DE ECUACIONES CUADRÁTICAS</b><o:p></o:p></span></div>
<div class="MsoListParagraphCxSpLast" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-add-space: auto; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Por ejemplo: </span><span style="font-family: "Courier New"; font-size: 18.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">x<sup>2</sup>+5x-6=0</span><span style="font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span style="font-family: "Courier New";"><span style="color: blue; font-size: 10pt;"> </span><span style="font-size: 10pt;"><span style="color: blue;"> </span><span style="color: blue; font-size: large;"> p<-</span></span></span><span style="font-family: Arial, sans-serif;"><span style="color: blue; font-size: large;"> polynomial(c(-6,-5,1))<span style="mso-spacerun: yes;"> </span>## ojo con el orden<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpFirst" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-add-space: auto; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: "Courier New";"><span style="color: blue; font-size: large;"> p<o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-add-space: auto; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: "Courier New";"><span style="color: blue; font-size: large;"> solve(p)<o:p></o:p></span></span></div>
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<br /></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-add-space: auto; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: "Courier New";"><span style="color: blue; font-size: large;"> [1]
-1 6</span><span style="color: blue; font-size: 10pt;"><o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpMiddle" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-add-space: auto; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<br /></div>
<div class="MsoListParagraphCxSpLast" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo2; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: Arial, sans-serif; font-size: 10pt;"><span style="color: blue;">4.</span><span style="font: 7pt "Times New Roman";"><span style="color: blue;"> </span> </span><span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-style: normal; font-variant: normal; line-height: normal;"><b>
</b></span></span><!--[endif]--><span style="font-family: Arial, sans-serif; font-size: 10pt;"><b>SOLUCION DE ECUACIONES NO LINEALES</b><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;">En este caso no existe una función que resuelva directamente. Se requiere
implementar métodos numéricos (bisección, Newton Rapson, etc).<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"><br /></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<span style="font-family: "Courier New"; font-size: 10pt;">Se recomienda el uso del paquete rootSolve, el cual se puede instalar desde
el sitio CRAN<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt;">
<br /></div>
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<b style="mso-bidi-font-weight: normal;"><span style="color: blue; font-family: "Courier New"; font-size: 10.0pt; mso-effects-shadow-align: topleft; mso-effects-shadow-alpha: 100.0%; mso-effects-shadow-angledirection: 0; mso-effects-shadow-anglekx: 0; mso-effects-shadow-angleky: 0; mso-effects-shadow-color: black; mso-effects-shadow-dpidistance: 0pt; mso-effects-shadow-dpiradius: 4.0pt; mso-effects-shadow-pctsx: 100.0%; mso-effects-shadow-pctsy: 100.0%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-style-textfill-fill-gradientfill-shade-linearshade-angle: 5400000; mso-style-textfill-fill-gradientfill-shade-linearshade-fscaled: no; mso-style-textfill-fill-gradientfill-shadetype: linear; mso-style-textfill-fill-gradientfill-stoplist: "0 \#505050 -1 100000 tint=92000 shade=100000 satm=150000\,49000 \#595959 -1 100000 tint=89000 shade=90000 satm=150000\,50000 \#000000 -1 100000 tint=100000 shade=75000 satm=150000\,95000 \#000000 -1 100000 shade=47000 satm=150000\,100000 \#000000 -1 100000 shade=39000 satm=150000"; mso-style-textfill-type: gradient; mso-style-textoutline-outlinestyle-align: center; mso-style-textoutline-outlinestyle-compound: simple; mso-style-textoutline-outlinestyle-dash: solid; mso-style-textoutline-outlinestyle-dpiwidth: .7pt; mso-style-textoutline-outlinestyle-join: miter; mso-style-textoutline-outlinestyle-linecap: flat; mso-style-textoutline-outlinestyle-pctmiterlimit: 0%; mso-style-textoutline-type: none;">Ejemplo:<o:p></o:p></span></b></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilKczB8-lCIqnM0jee8ZRIu8HqqZeYsUEomHqqOTnETpiZYVI6xQ3UzZVPuenpN9EEVjAoPkFurJl9-AkGybv6BrEXB_bt3L5bI5TDv7HSlWfM9X-7-1bvNpQioLAEmqVV5vxJ12oYBA4/s1600/ecua2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="32" data-original-width="404" height="31" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilKczB8-lCIqnM0jee8ZRIu8HqqZeYsUEomHqqOTnETpiZYVI6xQ3UzZVPuenpN9EEVjAoPkFurJl9-AkGybv6BrEXB_bt3L5bI5TDv7HSlWfM9X-7-1bvNpQioLAEmqVV5vxJ12oYBA4/s400/ecua2.png" width="400" /></a></div>
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<span style="background: white; color: black; font-size: 10.5pt;">Definir función:<o:p></o:p></span></div>
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<code><span style="color: black; font-size: 10.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">library(rootSolve)<o:p></o:p></span></code></div>
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<code><span style="color: black; font-size: 10.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">f<-</span></code><code><span style="font-size: 10.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">function<span style="color: black;">(x) </span>cos<span style="color: black;">(x)^4 - 4*</span>cos<span style="color: black;">(x)^3 + 8*</span>cos<span style="color: black;">(x)^2 - 5*</span>cos<span style="color: black;">(x) + 1/2</span></span></code><b style="mso-bidi-font-weight: normal;"><span style="color: blue; font-family: "Courier New"; font-size: 10.0pt; mso-effects-shadow-align: topleft; mso-effects-shadow-alpha: 100.0%; mso-effects-shadow-angledirection: 0; mso-effects-shadow-anglekx: 0; mso-effects-shadow-angleky: 0; mso-effects-shadow-color: black; mso-effects-shadow-dpidistance: 0pt; mso-effects-shadow-dpiradius: 4.0pt; mso-effects-shadow-pctsx: 100.0%; mso-effects-shadow-pctsy: 100.0%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-style-textfill-fill-gradientfill-shade-linearshade-angle: 5400000; mso-style-textfill-fill-gradientfill-shade-linearshade-fscaled: no; mso-style-textfill-fill-gradientfill-shadetype: linear; mso-style-textfill-fill-gradientfill-stoplist: "0 \#505050 -1 100000 tint=92000 shade=100000 satm=150000\,49000 \#595959 -1 100000 tint=89000 shade=90000 satm=150000\,50000 \#000000 -1 100000 tint=100000 shade=75000 satm=150000\,95000 \#000000 -1 100000 shade=47000 satm=150000\,100000 \#000000 -1 100000 shade=39000 satm=150000"; mso-style-textfill-type: gradient; mso-style-textoutline-outlinestyle-align: center; mso-style-textoutline-outlinestyle-compound: simple; mso-style-textoutline-outlinestyle-dash: solid; mso-style-textoutline-outlinestyle-dpiwidth: .7pt; mso-style-textoutline-outlinestyle-join: miter; mso-style-textoutline-outlinestyle-linecap: flat; mso-style-textoutline-outlinestyle-pctmiterlimit: 0%; mso-style-textoutline-type: none;"><o:p></o:p></span></b></div>
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<span style="font-family: "Courier New"; font-size: 10.0pt; mso-effects-shadow-align: topleft; mso-effects-shadow-alpha: 100.0%; mso-effects-shadow-angledirection: 0; mso-effects-shadow-anglekx: 0; mso-effects-shadow-angleky: 0; mso-effects-shadow-color: black; mso-effects-shadow-dpidistance: 0pt; mso-effects-shadow-dpiradius: 4.0pt; mso-effects-shadow-pctsx: 100.0%; mso-effects-shadow-pctsy: 100.0%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-style-textoutline-outlinestyle-align: center; mso-style-textoutline-outlinestyle-compound: simple; mso-style-textoutline-outlinestyle-dash: solid; mso-style-textoutline-outlinestyle-dpiwidth: .7pt; mso-style-textoutline-outlinestyle-join: miter; mso-style-textoutline-outlinestyle-linecap: flat; mso-style-textoutline-outlinestyle-pctmiterlimit: 0%; mso-style-textoutline-type: none;">Graficar la función
para determinar el intervalo donde hay solución:<o:p></o:p></span></div>
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<span style="font-family: "Courier New"; font-size: 10.0pt; mso-effects-shadow-align: topleft; mso-effects-shadow-alpha: 100.0%; mso-effects-shadow-angledirection: 0; mso-effects-shadow-anglekx: 0; mso-effects-shadow-angleky: 0; mso-effects-shadow-color: black; mso-effects-shadow-dpidistance: 0pt; mso-effects-shadow-dpiradius: 4.0pt; mso-effects-shadow-pctsx: 100.0%; mso-effects-shadow-pctsy: 100.0%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-style-textoutline-outlinestyle-align: center; mso-style-textoutline-outlinestyle-compound: simple; mso-style-textoutline-outlinestyle-dash: solid; mso-style-textoutline-outlinestyle-dpiwidth: .7pt; mso-style-textoutline-outlinestyle-join: miter; mso-style-textoutline-outlinestyle-linecap: flat; mso-style-textoutline-outlinestyle-pctmiterlimit: 0%; mso-style-textoutline-type: none;">plot(f)<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEvnEW5_LTllth12CX4Li_L8wr1ioZfpjMNBm4x4yaQrk4flPxnKRQohCAZCNf8kDSQC9edYTV6tHDiO5_CNdMrmJbhh_Gk9gXjk4eUhvbxUB2BPb4ZUJXmevbo0Xdr0BYb3U2QH7gEqw/s1600/root.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="565" data-original-width="631" height="286" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEvnEW5_LTllth12CX4Li_L8wr1ioZfpjMNBm4x4yaQrk4flPxnKRQohCAZCNf8kDSQC9edYTV6tHDiO5_CNdMrmJbhh_Gk9gXjk4eUhvbxUB2BPb4ZUJXmevbo0Xdr0BYb3U2QH7gEqw/s320/root.png" width="320" /></a></div>
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<b style="mso-bidi-font-weight: normal;"><span style="color: blue; font-family: "Courier New"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape id="Imagen_x0020_19" o:spid="_x0000_i1025"
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o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span></b><b style="mso-bidi-font-weight: normal;"><span style="color: blue; font-family: "Courier New"; font-size: 10.0pt; mso-effects-shadow-align: topleft; mso-effects-shadow-alpha: 100.0%; mso-effects-shadow-angledirection: 0; mso-effects-shadow-anglekx: 0; mso-effects-shadow-angleky: 0; mso-effects-shadow-color: black; mso-effects-shadow-dpidistance: 0pt; mso-effects-shadow-dpiradius: 4.0pt; mso-effects-shadow-pctsx: 100.0%; mso-effects-shadow-pctsy: 100.0%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-style-textfill-fill-gradientfill-shade-linearshade-angle: 5400000; mso-style-textfill-fill-gradientfill-shade-linearshade-fscaled: no; mso-style-textfill-fill-gradientfill-shadetype: linear; mso-style-textfill-fill-gradientfill-stoplist: "0 \#505050 -1 100000 tint=92000 shade=100000 satm=150000\,49000 \#595959 -1 100000 tint=89000 shade=90000 satm=150000\,50000 \#000000 -1 100000 tint=100000 shade=75000 satm=150000\,95000 \#000000 -1 100000 shade=47000 satm=150000\,100000 \#000000 -1 100000 shade=39000 satm=150000"; mso-style-textfill-type: gradient; mso-style-textoutline-outlinestyle-align: center; mso-style-textoutline-outlinestyle-compound: simple; mso-style-textoutline-outlinestyle-dash: solid; mso-style-textoutline-outlinestyle-dpiwidth: .7pt; mso-style-textoutline-outlinestyle-join: miter; mso-style-textoutline-outlinestyle-linecap: flat; mso-style-textoutline-outlinestyle-pctmiterlimit: 0%; mso-style-textoutline-type: none;"><o:p></o:p></span></b></div>
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<span style="font-family: "Courier New"; font-size: 10pt;">Una raíz está entre 0 y 1, pero pueden haber otras en otro intervalo<o:p></o:p></span></div>
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<span style="font-family: "Courier New"; font-size: 10pt;">Invocar a la función <span style="mso-spacerun: yes;"> </span>en el intervalo
de cero a dos<o:p></o:p></span></div>
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<code><span style="font-size: 16.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">uniroot.all<span style="color: black;">(f, </span>c<span style="color: black;">(0,2))<o:p></o:p></span></span></code></div>
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<span style="color: blue; font-family: "Courier New"; font-size: 16.0pt; mso-effects-shadow-align: topleft; mso-effects-shadow-alpha: 100.0%; mso-effects-shadow-angledirection: 0; mso-effects-shadow-anglekx: 0; mso-effects-shadow-angleky: 0; mso-effects-shadow-color: black; mso-effects-shadow-dpidistance: 0pt; mso-effects-shadow-dpiradius: 4.0pt; mso-effects-shadow-pctsx: 100.0%; mso-effects-shadow-pctsy: 100.0%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-style-textfill-fill-gradientfill-shade-linearshade-angle: 5400000; mso-style-textfill-fill-gradientfill-shade-linearshade-fscaled: no; mso-style-textfill-fill-gradientfill-shadetype: linear; mso-style-textfill-fill-gradientfill-stoplist: "0 \#505050 -1 100000 tint=92000 shade=100000 satm=150000\,49000 \#595959 -1 100000 tint=89000 shade=90000 satm=150000\,50000 \#000000 -1 100000 tint=100000 shade=75000 satm=150000\,95000 \#000000 -1 100000 shade=47000 satm=150000\,100000 \#000000 -1 100000 shade=39000 satm=150000"; mso-style-textfill-type: gradient; mso-style-textoutline-outlinestyle-align: center; mso-style-textoutline-outlinestyle-compound: simple; mso-style-textoutline-outlinestyle-dash: solid; mso-style-textoutline-outlinestyle-dpiwidth: .7pt; mso-style-textoutline-outlinestyle-join: miter; mso-style-textoutline-outlinestyle-linecap: flat; mso-style-textoutline-outlinestyle-pctmiterlimit: 0%; mso-style-textoutline-type: none;">[1] 0.6293342 1.4478556<o:p></o:p></span></div>
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Mientras
mayor sea el intervalo, mayor es la posibilidad de encontrar más raíces<o:p></o:p></div>
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Más
detalles: <o:p></o:p></div>
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<a href="https://www.r-bloggers.com/finding-multiple-roots-of-univariate-functions-in-r/">https://www.r-bloggers.com/finding-multiple-roots-of-univariate-functions-in-r/</a><b style="mso-bidi-font-weight: normal;"><span style="color: blue; font-family: "Courier New"; font-size: 10.0pt; mso-effects-shadow-align: topleft; mso-effects-shadow-alpha: 100.0%; mso-effects-shadow-angledirection: 0; mso-effects-shadow-anglekx: 0; mso-effects-shadow-angleky: 0; mso-effects-shadow-color: black; mso-effects-shadow-dpidistance: 0pt; mso-effects-shadow-dpiradius: 4.0pt; mso-effects-shadow-pctsx: 100.0%; mso-effects-shadow-pctsy: 100.0%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES; mso-style-textfill-fill-gradientfill-shade-linearshade-angle: 5400000; mso-style-textfill-fill-gradientfill-shade-linearshade-fscaled: no; mso-style-textfill-fill-gradientfill-shadetype: linear; mso-style-textfill-fill-gradientfill-stoplist: "0 \#505050 -1 100000 tint=92000 shade=100000 satm=150000\,49000 \#595959 -1 100000 tint=89000 shade=90000 satm=150000\,50000 \#000000 -1 100000 tint=100000 shade=75000 satm=150000\,95000 \#000000 -1 100000 shade=47000 satm=150000\,100000 \#000000 -1 100000 shade=39000 satm=150000"; mso-style-textfill-type: gradient; mso-style-textoutline-outlinestyle-align: center; mso-style-textoutline-outlinestyle-compound: simple; mso-style-textoutline-outlinestyle-dash: solid; mso-style-textoutline-outlinestyle-dpiwidth: .7pt; mso-style-textoutline-outlinestyle-join: miter; mso-style-textoutline-outlinestyle-linecap: flat; mso-style-textoutline-outlinestyle-pctmiterlimit: 0%; mso-style-textoutline-type: none;"><o:p></o:p></span></b></div>
<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-75758447952126450392019-06-03T07:49:00.004-06:002019-06-03T07:56:00.898-06:00OPERACIONES CON POLINOMIOS EN R<br />
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<a href="https://www.blogger.com/null" name="page1"></a><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">OPERACIONES CON POLINOMIOS CON R<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><br /></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Ing. Luis Manfredo Reyes<o:p></o:p></span></div>
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<br /></div>
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<br /></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES-GT" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">R es actualmente el lenguaje estadístico más
utilizado en el mundo académico y de investigación, primero por su potencia de
procesamiento, y en segundo lugar por ser un software libre.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES-GT" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">R puede ser descargado del siguiente sitio: <a href="http://www.r-project.org/"><span style="background: white; color: #888888; text-decoration: none;">http://www.r-project.org/</span></a><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES-GT" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Existen versiones para Windows, Mac, Linux y Android<o:p></o:p></span></div>
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<br /></div>
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<span lang="ES-GT" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">UN ERROR MUY COMUN POR TODOS LADOS ES: HAY GENTE QUE CREE QUE R
ES UN PAQUETE ESTADÍSTICO (COMO SAS O SPSS).<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES-GT" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Por ser un lenguaje y no un paquete, R tiene capacidades para
muchos procesos, pero se debe reconocer que quizá donde más se le ha utilizado
es en el campo de la estadística.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES-GT" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Una de las ventajas de R, es la enorme cantidad de librerías (o
paquetes), que han sido desarrolladas para ampliar sus capacidades.<o:p></o:p></span></div>
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<br /></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Para el manejo de operaciones con polinomios con R, se
recomienda el paquete polynom.<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"></span><br />
<a name='more'></a><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><br /></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">El manual del paquete se puede encontrar en:<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><a href="https://cran.r-project.org/web/packages/polynom/polynom.pdf"><span style="font-size: large;">https://cran.r-project.org/web/packages/polynom/polynom.pdf</span></a></span><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<br /></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Este paquete fue desarrollado por Bill Venables, e
incorporado a R por Kurt Hornik y Martin Maechler, publicado el 22 de marzo de
2019 bajo la licencia GPL-2.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: 10.0pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<b style="mso-bidi-font-weight: normal;"><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Instalar el paquete:<o:p></o:p></span></b></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Se realiza desde cualquiera de los
espejos CRAN disponibles por todo el mundo.<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Por ejemplo, en el espejo CRAN
principal, la ubicación del paquete es:<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNormal" style="line-height: 10.0pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">https://cloud.r-project.org/bin/windows/contrib/3.5/polynom_1.4-0.zip
<o:p></o:p></span></div>
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<br /></div>
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<br /></div>
<div class="MsoNormal" style="line-height: 10.0pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<b style="mso-bidi-font-weight: normal;"><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Cargar la librería:<span style="mso-spacerun: yes;"> </span><o:p></o:p></span></b></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Library(polynom)<o:p></o:p></span></div>
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<br /></div>
<ol start="1" style="margin-top: 0cm;" type="1">
<li class="MsoNormal" style="line-height: 10.0pt; mso-line-height-rule: exactly; mso-list: l0 level1 lfo1;"><b style="mso-bidi-font-weight: normal;"><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Crear el objeto “Polinomio”<o:p></o:p></span></b></li>
</ol>
<div class="MsoNormal" style="line-height: 10.0pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Se usa la función concatenar, los
coeficientes se ingresan en orden ascendente de potencias.<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNormal" style="line-height: 10.0pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">El polinomio se puede asignar a un
objeto, y su tipo será polynomial.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 1.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 1.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Ejemplos: <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , sans-serif;"><span style="font-size: large;">polynomial(c(1, 2, 1))<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , sans-serif;"><span style="font-size: large;">1 + 2*x + x^2<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , sans-serif;"><span style="font-size: large;">> polynomial(c(-2,18,6))<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">-2 + 18*x + 6*x^2<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">p<-polynomial(c(1,1,1,1,1,1,1,1,1))<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">p<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">1 + x + x^2 + x^3 + x^4 + x^5 + x^6
+ x^7 + x^8<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">polynomial(1:4)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">1 + 2*x + 3*x^2 + 4*x^3 <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">> p <- as.polynomial(c(1,0,3,0))<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">> p<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">1 + 3*x^2.</span><span style="font-size: 12pt;"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 28.0pt; mso-line-height-alt: 0pt;">
<br /></div>
<ol start="2" style="margin-top: 0cm;" type="1">
<li class="MsoNormal" style="line-height: 9.5pt; mso-line-height-rule: exactly; mso-list: l0 level1 lfo1;"><b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Calcular un polinomio a partir de
sus ceros (raíces</span></b><b><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">)</span></b></li>
</ol>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">poly.calc(rep(1,3))<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">-1 + 3*x -
3*x^2 + x^3 <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , sans-serif;"><span style="font-size: large;">>
poly.calc(0:4, (0:4)^2 + 1)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">1 + x^2 <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">>
poly.calc(0:4, cbind(0:4, (0:4)^2 + 1), lab = letters[1:2])<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">$`a`<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">x <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , sans-serif; font-size: large;">$b</span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">1 + x^2 <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 9.5pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="margin-left: 46.0pt; mso-line-height-alt: 0pt;">
<br /></div>
<ol start="3" style="margin-top: 0cm;" type="1">
<li class="MsoNormal" style="mso-line-height-alt: 0pt; mso-list: l0 level1 lfo1;"><b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Hallar los ceros de un polinomio<o:p></o:p></span></b></li>
</ol>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">R puede calcular ceros reales o complejos<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;">p <- polynomial(6:1)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>p<o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>6 + 5*x + 4*x^2 + 3*x^3 + 2*x^4 +
x^5<o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>pz <- solve(p)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>pz <o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>-1.49180+0.0000i -0.80579-1.2229i
-0.80579+1.2229i<o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>0.55169-1.2533i 0.55169+1.2533i<o:p></o:p></span></span></div>
<div class="MsoNormal" style="margin-left: 46.0pt; mso-line-height-alt: 0pt;">
<br /></div>
<div class="MsoNormal" style="line-height: 19.5pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><span style="mso-spacerun: yes;">
</span>p<-polynomial(c(1,1,1,1,1,1,1,1,1))<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 46.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">z<-solve(p)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 46.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">z<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 46.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">[1] -0.9396926-0.3420201i -0.9396926+0.3420201i
-0.5000000-0.8660254i -0.5000000+0.8660254i<span style="mso-spacerun: yes;">
</span>0.1736482-0.9848078i<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 46.0pt; mso-line-height-alt: 0pt;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">[6]<span style="mso-spacerun: yes;">
</span>0.1736482+0.9848078i<span style="mso-spacerun: yes;">
</span>0.7660444-0.6427876i<span style="mso-spacerun: yes;">
</span>0.7660444+0.6427876i<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 46.0pt; mso-line-height-alt: 0pt;">
<br /></div>
<ol start="4" style="margin-top: 0cm;" type="1">
<li class="MsoNormal" style="line-height: 10.0pt; mso-line-height-rule: exactly; mso-list: l0 level1 lfo1;"><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Derivada
de un polinomio<o:p></o:p></span></b></li>
</ol>
<div class="MsoNormal" style="line-height: 10.0pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">R Calcula la derivada de un polinomio
aplicando la regla de la derivada de una potencia<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">pr <- poly.calc(1:5) <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">-120 + 274*x - 225*x^2 + 85*x^3 - 15*x^4 +
x^5 <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">deriv(pr)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span style="font-size: large;"><span lang="ES-GT"><span style="mso-spacerun: yes;"><br /></span></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span style="font-size: large;"><span lang="ES-GT"><span style="mso-spacerun: yes;"> </span>274
- 450*x + 255*x^2 - 60*x^3 + 5*x^4</span><span style="font-family: "arial" , sans-serif;"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<br /></div>
<ol start="5" style="margin-top: 0cm;" type="1">
<li class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly; mso-list: l0 level1 lfo1;"><b style="mso-bidi-font-weight: normal;"><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Integral
de un polinomio<o:p></o:p></span></b></li>
</ol>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">La
integral de un polinomio de calcula usando la regla de la antiderivada de una
potencia<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">En
el ejemplo se crea un polinomio, se deriva y luego se integra. El resultado
debe ser el mismo que al inicio<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">p <- poly.calc(1:5)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">p<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">-120 + 274*x - 225*x^2 +
85*x^3 - 15*x^4 + x^5<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><span style="mso-spacerun: yes;"><br /></span></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>deriv(p)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">274 - 450*x + 255*x^2 -
60*x^3 + 5*x^4<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">integral(deriv(p)) – 120<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><span style="font-size: large;">-120 + 274*x - 225*x^2 +
85*x^3 - 15*x^4 + x^5</span><span style="font-size: 12pt;"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<ol start="6" style="margin-top: 0cm;" type="1">
<li class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly; mso-list: l0 level1 lfo1;"><b style="mso-bidi-font-weight: normal;"><span lang="ES-GT" style="font-size: 12.0pt;">Operaciones aritméticas con polinomios<o:p></o:p></span></b></li>
</ol>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="ES-GT" style="font-size: 12.0pt;">Los objetos “polynomial” se
pueden operar aritméticamente sin problema<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="font-size: large;">p <-
polynomial(c(1, 2, 1))<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="mso-ansi-language: ES;"><span style="font-size: large;">p<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="mso-ansi-language: ES;"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="mso-ansi-language: ES;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>1 + 2*x + x^2<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="mso-ansi-language: ES;"><span style="font-size: large;"><span style="mso-spacerun: yes;"><br /></span></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="mso-ansi-language: ES;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>r <- poly.calc(-1 : 1)<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="mso-ansi-language: ES;"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="mso-ansi-language: ES;"><span style="font-size: large;">r <o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="font-size: large;"><span style="mso-spacerun: yes;"><br /></span></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>-1*x + x^3<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="font-size: large;"><span style="mso-spacerun: yes;"><br /></span></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="font-size: large;"><span style="mso-spacerun: yes;"> </span>(r - 2 * p)^2<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="font-size: large;"><span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"><br /></span></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="font-size: large;"><span lang="EN-US" style="mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>4 + 20*x + 33*x^2 + 16*x^3 - 6*x^4 - 4*x^5 +
x^6</span><span lang="EN-US"><o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<ol start="7" style="margin-top: 0cm;" type="1">
<li class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly; mso-list: l0 level1 lfo1;"><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Graficar un polinomio<o:p></o:p></span></b></li>
</ol>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Se usa la función plot que es parte del paquete base de R<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US"><span style="font-size: large;">plot(p
<- poly.calc(-1:5))<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US"><span style="font-size: large;"> p<o:p></o:p></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US"><span style="font-size: large;"><br /></span></span></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<span lang="EN-US"><span style="font-size: large;">-120*x+154*x^2+49*x^3-140*x^4+70*x^5-14*x^6+</span></span><span style="font-size: large;">x^7</span><br />
<span style="font-size: large;"><br /></span>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinECa0V-39F08P69nVK99cduk0K3dc0ktS7Cq_Am8YKvmrrT5CMNssFEw7wgskqYQXr4Vt7DAXMyVpscrbAht68YXHsM4KBbHrEVhWTNVA8f2sxXcZIMeLipRI44uZL5cYHEZddSaVMCw/s1600/poly.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="561" data-original-width="640" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinECa0V-39F08P69nVK99cduk0K3dc0ktS7Cq_Am8YKvmrrT5CMNssFEw7wgskqYQXr4Vt7DAXMyVpscrbAht68YXHsM4KBbHrEVhWTNVA8f2sxXcZIMeLipRI44uZL5cYHEZddSaVMCw/s320/poly.png" width="320" /></a></div>
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
<br /></div>
<div class="MsoNormal" style="line-height: 10.55pt; margin-left: 36.0pt; mso-line-height-rule: exactly;">
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<div class="MsoNormal" style="line-height: 10.55pt; mso-line-height-rule: exactly;">
<span lang="ES-GT"><a href="https://cran.r-project.org/web/packages/polynom/polynom.pdf"><span lang="EN-US" style="mso-ansi-language: EN-US;">https://cran.r-project.org/web/packages/polynom/polynom.pdf</span></a></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-5799123999717707012019-05-29T16:43:00.002-06:002019-05-29T16:44:08.923-06:00COMPUTACIÓN NUMÉRICA CON R<!--[if gte mso 9]><xml>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 5"/>
<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Closing"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Signature"/>
<w:LsdException Locked="false" Priority="1" SemiHidden="true"
UnhideWhenUsed="true" Name="Default Paragraph Font"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Message Header"/>
<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Salutation"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Date"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/>
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<br />
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">COMPUTACIÓN NUMÉRICA CON R</span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">Ing. Luis Manfredo Reyes Chávez</span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">R es actualmente el lenguaje<span style="mso-spacerun: yes;">
</span>estadístico más utilizado en el mundo académico y de investigación,
primero por su potencia de procesamiento, y en segundo lugar por ser un
software libre.</span></div>
<div class="MsoNormal" style="background: white;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">R<span style="mso-spacerun: yes;"> </span>puede ser descargado
del siguiente sitio: </span><a href="http://www.r-project.org/"><span style="background: white; color: #888888; font-family: "arial" , sans-serif;">http://www.r-project.org/</span></a></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">Existen versiones para Windows, Mac, Linux y Android</span></div>
<div class="MsoNormal" style="background: white;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;"></span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">UN ERROR MUY COMUN POR TODOS LADOS ES: HAY GENTE QUE CREE QUE R
ES UN PAQUETE ESTADÍSTICO (COMO SAS O SPSS).</span></div>
<div class="MsoNormal" style="background: white;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">Por ser un lenguaje y no un paquete, R tiene capacidades para
muchos procesos, pero se debe reconocer que quizá donde más se le ha utilizado
es en el campo de la estadística.</span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">Una de las ventajas de R, es la enorme cantidad de librerías (o
paquetes), que han sido desarrolladas para ampliar sus capacidades.</span></div>
<div class="MsoNormal" style="background: white;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">La computación simbólica, aunque se puede realizar con una
interfase a otros programas matemáticos, no es el fuerte de R.</span></div>
<div class="MsoNormal" style="background: white;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">Si se necesita un poder de computación simbólica, es mejor usar
paquetes libres que sí poseen esas capacidades, por ejemplo MAXIMA O REDUCE,
que están disponibles sin costo en el sitio del proyecto GNU.</span></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;"></span></div>
<a name='more'></a><br />
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">Tampoco es el objeto del presente documento profundizar en cada
una de las funciones existentes en R para cómputo numérico, ya sean nativas o
incorporadas en librerías. Se pretende realizar un recorrido por las
capacidades disponibles, y posteriormente se profundizará en temas específicos.</span></div>
<div class="MsoNormal" style="background: white;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #222222; font-family: "arial" , sans-serif;">La siguiente discusión es una traducción y adaptación del
documento: Numerical Mathematics, disponible en: <span style="mso-spacerun: yes;"> </span></span></div>
<div class="MsoNormal" style="background: white;">
<span style="font-size: large;">https://cran.r-project.org/web/views/NumericalMathematics.html</span></div>
<br />
<b><span style="color: black; font-family: "arial" , sans-serif;">Algebra
Lineal Numérica</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Como las estadísticas se basan en gran medida en el álgebra
lineal, muchas rutinas numéricas de álgebra lineal están presentes en R, y
algunas solo de forma implícita.</span></span><span style="color: black; font-family: "arial" , sans-serif;"> </span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Ejemplos de funciones explícitamente disponibles son: operaciones
de vectores y matrices, descomposiciones matriciales (QR), resolución de
ecuaciones lineales, valores propios / vectores, descomposición de valores
singulares o aproximación de mínimos cuadrados.</span></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Paquetes contribuídos para Algebra lineal numérica</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Matrix/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjl9J6722OBzsULYAadI04w5y322A">Matrix</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
clases y métodos para matrices densas y dispersas y operaciones en ellas,
por ejemplo, descomposición de Cholesky y Schur, matriz exponencial, o
normas y números condicionales para matrices dispersas.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/MASS/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhh2QGci0QcQcUXAN99qINvcmQHN3w">MASS</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> agrega
inversos generalizados (Penrose) y espacios nulos de matrices.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/expm/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiVZGNcPNx8aSobsSbdEyocJSs4tQ">expm</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> calcula
el exponencial, el logaritmo y la raíz cuadrada de las matrices cuadradas,
pero también las potencias de las matrices o la derivada de Frechet.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">expm ()</span></tt><span class="notranslate"> debe ser
preferido a la función con el mismo nombre en <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Matrix/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjl9J6722OBzsULYAadI04w5y322A"><span style="background: white;">Matrix</span></a>.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/SparseM/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhNu20vVPaA0agjIzGUNrvitthfNg">SparseM</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
clases y métodos para matrices dispersas y para resolver problemas
lineales y de mínimos cuadrados en álgebra lineal dispersa</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/rmumps/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhT8HJxv4eWqdEXonU4HVqunoG2JA">rmumps</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
un acceso para la biblioteca MUMPS, que resuelve grandes sistemas lineales
de ecuaciones aplicando un solucionador directo paralelo disperso</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Rlinsolve/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhi8VIIuIt96ZWSx7kgeIk9mQZqy5A">Rlinsolve</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> es
una colección de solucionadores iterativos para sistemas de ecuaciones
lineales dispersos.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Se proporcionan
solucionadores iterativos estacionarios como Jacobi o Gauss-Seidel, así
como métodos no estacionarios (subespacio Krylov).</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/svd/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgEB1fjiKURclxU-Zad6QTqQzcXew">svd</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
enlaces R a implementaciones de vanguardia de la descomposición de valores
singulares (SVD) y cálculos de valores propios / vectores propios.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">El paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/ssvd/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgfYyT25BVW1xuOdh_6yLldBd7W1w"><span style="background: white;">ssvd</span></a> obtendrá SVD dispersos utilizando
un método de umbral iterativo, mientras que <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/irlba/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjUK8UN1q1yddxWOWqirlLW67Favw"><span style="background: white;">irlba</span></a> calculará valores /
vectores singulares aproximados de matrices grandes.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/PRIMME/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjxENdQtWANaJ45ojLk15b1rGql9w">PRIMME</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> interactúa
con PRIMME, una biblioteca de C para calcular valores propios y los
vectores propios correspondientes de matrices de Hermitt simétricas reales
o complejas.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Puede encontrar <span style="mso-spacerun: yes;"> </span>eigenvalores <span style="mso-spacerun: yes;"> </span>singulares más grandes, más pequeños o
interiores y aplicará un preacondicionamiento para acelerar la
convergencia.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Los
paquetes <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/geigen/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgdf5DjUOCSKdu5eDgB8wdY8J21IA"><span style="background: white;">geigen</span></a> y </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/QZ/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjcZB_0OXtCF7cz5x0AKxft1h8dew">QZ</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> calculan
eigen valores y eigen vectores para pares de matrices, y descomposiciones
QZ (Schur generalizado).</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/eigeninv/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiPYVGzDB6edgYACnkhdqDPhh5p4w">eigeninv</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> genera
matrices con un conjunto dado de valores propios ('problema de valores
propios inversos').</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/rARPACK/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiVdKiA1Hlb2X8JYF9J0uvQ8NdvHA"><span style="background: white;">rARPACK</span></a> , es <span style="mso-spacerun: yes;"> </span>un enlace para la biblioteca ARPACK, se
usa normalmente para calcular solo unos pocos valores / vectores, por
ejemplo, un pequeño número de valores propios más grandes.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/RSpectra/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhi9T3_wu4F5pJExnkJTk6emymrLCg">RSpectra</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> interactúa
con la biblioteca 'Spectra' para la descomposición de valores propios a
gran escala y los problemas de SVD.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/optR/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjrsXC6ccy0qGms_4edxBl1aA6QGQ">optR</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> usa
métodos elementales de álgebra lineal (Gauss, LU, CGM, Cholesky) para
resolver sistemas lineales.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/matrixcalc/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjA3UH2RKNnNSemfncWziBy2MA-cg">matrixcalc</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> contiene
una colección de funciones para cálculos matriciales, matrices especiales
y pruebas de propiedades matriciales, por ejemplo, definición (semi)
positiva.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/onion/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgAqWUkjviKBmdrTuzVJ25QwQ9hDQ">onion</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> contiene
rutinas para manipular cuaterniones y octonianos (álgebras de división
normalizada sobre los números reales);</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Los cuaterniones pueden ser útiles para manejar
rotaciones en el espacio tridimensional.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l1 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/RcppArmadillo/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgYm7fdCfSol_ksZEW76U0cytF4zw">RcppArmadillo</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> y <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/RcppEigen/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhg8woDJUvYFzg7WoDq_MPqSYoXqhA"><span style="background: white;">RcppEigen</span></a> permiten la integración
de las bibliotecas de plantillas de C ++ 'Resp. Armadillo'.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">'Eigen' para aplicaciones de álgebra lineal escritas en
C ++ e integradas en R usando <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Rcpp/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgICh7rQYes7gPAqDLVm8fWf3u-cw"><span style="background: white;">Rcpp</span></a> para rendimiento y facilidad
de uso.</span></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Funciones
especiales</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Muchas funciones matemáticas especiales están presentes en R,
especialmente logaritmos y exponenciales, funciones trigonométricas e
hiperbólicas, o funciones de Bessel y Gamma.</span></span><span style="color: black; font-family: "arial" , sans-serif;"> <span class="notranslate">Muchas más funciones especiales están disponibles en paquetes
contribuidos.</span></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/gsl/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgiGJdvocnolLvQ6m2xoEZroD-6eA">gsl</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
una interfaz a la 'Biblioteca Científica GNU' que contiene
implementaciones de muchas funciones especiales, por ejemplo las funciones
Airy y Bessel, las integrales elípticas y exponenciales, la función
hipergeométrica, la función W de Lambert y muchos más.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Las
funciones de Airy y Bessel, para números reales y complejos, también se
computan en el paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Bessel/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjUGPXKIIe0v0ancqpB0nlbnBCYxw"><span style="background: white;">Bessel</span></a> , con aproximaciones para
argumentos grandes.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/pracma/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhsRijdxWO_eTd2TGAnnCVU_K0T9g">pracma</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> incluye
funciones especiales, tales como funciones de error e inversas, función
gamma incompleta y compleja, integrales exponenciales y logarítmicas,
integrales de Fresnel, el poligamma y las funciones zeta de Dirichlet y
Riemann.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/appell/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjQjdHqp9FfkfzabQCUP8G8wybOBg">appell</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> calcula
las funciones hipergeométricas de Gauss '2F1 y Appell F1 para parámetros y
argumentos complejos con bastante precisión.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">La
función hipergeométrica (y hipergeométrica generalizada) se calcula
en <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/hypergeo/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhhlIdc0JbDufbiRNgjVylFuaqtug"><span style="background: white;">hipergeo</span></a> , incluidas las fórmulas
de transformación y los valores especiales de los parámetros.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Las
funciones elípticas y modulares están disponibles en el paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/elliptic/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhoJlymD9yfdbuLf73GX3D-rMWrcw">elíptico</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> ,
incluida la función Weierstrass P y las funciones theta de Jacobi.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Existen herramientas para visualizar funciones
complejas.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/expint/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjdwOT46HeGrugiqTAc_GodH_uENg"><span style="background: white;">expint</span></a> envuelve las funciones C
de la Biblioteca Científica GNU para calcular las integrales exponenciales
y la función Gamma incompleta, incluidos los valores negativos para su
primer argumento.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/fourierin/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhi_fgwli7Mlavb4BRgX41KPq-zdpg">fourierin</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> calcula
las integrales de Fourier de funciones de una y dos variables utilizando
la transformada rápida de Fourier.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/logOfGamma/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiDZHak_ydsy3aLB2DIyMgYLAdp_w">logOfGamma</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> utiliza
aproximaciones para calcular los logaritmos naturales de la función Gamma
para valores grandes.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l10 level1 lfo2; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Package <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/lamW/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjxQlEq6HdlIi7djE_sFDrXCSqOzg"><span style="background: white;">lamW</span></a> implementa ambas ramas de
valor real de la función Lambert W (usando Rcpp).</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Polinomios</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">La función polyroot () en la base R determina todos los ceros de
un polinomio, según el algoritmo de Jenkins-Traub.</span></span><span style="color: black; font-family: "arial" , sans-serif;"> <span class="notranslate">La función de regresión lineal lm () puede realizar un ajuste
polinomial cuando se usa </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">poly ()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> en
la fórmula del modelo (con la opción </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">raw = TRUE</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> ).</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l9 level1 lfo3; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/polynom/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjfNue3aU2GqJ_a19BAX97aIHWn-Q">polinom</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> y <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/PolynomF/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgpWRzT4KyBUggDLDYCiX2MI44iVA"><span style="background: white;">Polinom</span></a> proporcionan una
funcionalidad similar para manipular polinomios univariados, como evaluar
polinomios (esquema de Horner), diferenciarlos o integrarlos, o resolver
polinomios, es decir, encontrar todas las raíces (basadas en un cálculo de
valores propios).</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l9 level1 lfo3; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/MonoPoly/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgvE1sI_ZjH4zOBroxeEM11u3Lwmg">MonoPoly</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> ajusta
polinomios univariados a los datos dados, aplicando diferentes algoritmos.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l9 level1 lfo3; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/multipol/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhxexk5gmtSOkknGInRo_AOBnr6Wg">multipol</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
varias herramientas para manipular y combinar estos polinomios de varias
variables.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l9 level1 lfo3; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/mpoly/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhld_cI5j0meIj3JY8CeVipppkD7A">mpoly</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> facilita
las manipulaciones simbólicas en polinomios multivariados, incluidas las
operaciones básicas de cálculo diferencial en polinomios, más algunos
cálculos de base de Groebner.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l9 level1 lfo3; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/mvp/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhZdy2jbb0rWm57JSIiziK3NZZLDQ">mvp</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> permite
la manipulación rápida de polinomios multivariados simbólicos, utilizando
métodos de impresión y coerción del paquete 'mpoly', pero ofrece mejoras
de velocidad.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l9 level1 lfo3; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/orthopolynom/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhg91WUMoCbiCzDM1ngAH5IfhZFv9w">orthopolynom</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> consiste
en una colección de funciones para construir polinomios ortogonales y sus
relaciones de recurrencia, entre ellos los polinomios de Chebyshev,
Hermite y Legendre, así como los polinomios esféricos y ultrasónicos.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Hay funciones para operar en estos polinomios.</span></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Diferenciación
e integración</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">D
()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> y </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">deriv ()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> en
base R calculan derivadas de expresiones simples simbólicamente.</span></span><span style="color: black; font-family: "arial" , sans-serif;"> <span class="notranslate">La función </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">integra ()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> implementa
un enfoque para integrar numéricamente funciones univariadas en R. Aplica la
cuadratura de Gauss-Kronrod adaptativa y puede manejar singularidades y
dominios ilimitados hasta cierto punto.</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Deriv/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiqRicc2Z8SpIczKE1ZoN2XqU3e2Q">Deriv</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
una solución extendida para la diferenciación simbólica en R;</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">el usuario puede agregar reglas derivadas
personalizadas, y la salida para una función será nuevamente una función
ejecutable.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/numDeriv/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhg3ygbdR6vSXzY-FpVFr3YG1crUjA">numDeriv</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> establece
el estándar para la diferenciación numérica en R, que proporciona
gradientes numéricos, jacobianos y hessianos, calculados por diferencias
finitas simples, extrapolación de Richardson o el enfoque de pasos
complejos de gran precisión.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="github"><u><span style="background: white; color: #9f7be1; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://github.com/Non-Contradiction/autodiffr/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjuZ0Xv7b37zsK7PWmzkpAAES5naw"><span style="color: #9f7be1;">autodiffr</span></a></span></u></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> (en
Github) proporciona un acceso a R para los paquetes Julia ForwardDiff.jl y
ReverseDiff.jl para hacer una diferenciación automática para las funciones
R nativas.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">(Funciona solo con Julia v0.6 por el momento)</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/pracma/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhsRijdxWO_eTd2TGAnnCVU_K0T9g">pracma</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> contiene
funciones para calcular derivadas numéricas, incluida la extrapolación de
Richardson o el paso complejo.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">fderiv
()</span></tt><span class="notranslate"> calcula derivadas numéricas de
órdenes superiores.</span> <span class="notranslate"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/pracma/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhsRijdxWO_eTd2TGAnnCVU_K0T9g"><span style="background: white;">pracma</span></a> tiene varias rutinas para
la integración numérica: cuadratura adaptativa de Lobatto, integración de
Romberg, fórmulas de Newton-Cotes, reglas de cuadratura de
Clenshaw-Curtis.</span> <tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">integral2 ()</span></tt><span class="notranslate"> integra funciones en dos dimensiones, también para
dominios caracterizados por coordenadas polares o con límites de intervalo
variable.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/gaussquad/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhVbMQawLitRjo4vNBU116rgZ8IKQ">gaussquad</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> contiene
una colección de funciones para realizar la cuadratura gaussiana, entre
ellas las reglas de cuadratura de Chebyshev, Hermite, Laguerre y Legendre,
que devuelven explícitamente los nodos y los pesos en cada caso.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">La función </span><tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">gaussquad
()</span></tt><span class="notranslate"> en el paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/statmod/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiWai7cKSoBAn-t3CmfvQp0CS-Gjg"><span style="background: white;">statmod</span></a> hace un trabajo similar.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/fastGHQuad/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhguEaems9FmB-A1z-qMr0E1aD2thw">fastGHQuad</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
una implementación rápida basada en <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Rcpp/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgICh7rQYes7gPAqDLVm8fWf3u-cw"><span style="background: white;">Rcpp</span></a> de la cuadratura de
Gauss-Hermite (adaptativa).</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/mvQuad/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgwhHmwPpC60KIjkCyHACiHZZcL5w">mvQuad</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
métodos para generar cuadrículas multivariables que pueden utilizarse para
la integración multivariable.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Estas
cuadrículas se basarán en diferentes reglas de cuadratura, como las
fórmulas de cuadratura de Newton-Cotes o Gauss.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/SparseGrid/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhha2GpRumwNI43RY4IwTbH5vN3I5w">SparseGrid</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
otro enfoque para la integración multivariable en espacios de alta
dimensión.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Crea grillas n-dimensionales dispersas que se pueden
usar como con las reglas de cuadratura.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/SphericalCubature/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhi9755ZxRLihddmfeT8sbG3Yt0vwg">SphericalCubature</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> emplea <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/cubature/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhid92w-n9zYnVib9-2GGrsKFA-DXw"><span style="background: white;">cubature</span></a> para integrar funciones
sobre esferas y bolas unitarias en el espacio n-dimensional;</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/SimplicialCubature/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjcLD3lzdFsmoI3nOjgqsjKXbQHJA"><span style="background: white;">SimplicialCubature</span></a> proporciona
métodos para integrar funciones sobre <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/SimplicialCubature/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjcLD3lzdFsmoI3nOjgqsjKXbQHJA"><span style="background: white;">simplicidades</span></a> m dimensionales en el
espacio n-dimensional.</span> <span class="notranslate">Ambos paquetes
comprenden métodos exactos para los polinomios.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/polyCub/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiGOCSxx22tcMysSVY4ajQ07m6hZQ"><span style="background: white;">polyCub</span></a> contiene algunas rutinas
para la integración numérica sobre dominios poligonales en dos
dimensiones.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Pade/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjip2TNCbPO6cCmnpWFinOf4U1uNg">Pade</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> calcula
los coeficientes de numerador y denominador de la aproximación de Pade,
dados los coeficientes de la serie de Taylor de longitud suficiente.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l7 level1 lfo4; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/features/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhVUtDEK4hC_dzgW-_WG2ytA3qH3Q"><span style="background: white;">características</span></a> extrae
características de datos funcionales, como primera y segunda derivadas, o
curvatura en puntos críticos, mientras que <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/RootsExtremaInflections/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiBu3UJ-tZRO3IrEHzaK20aTe9NNg"><span style="background: white;">RootsExtremaInflections</span></a> encuentra
raíces, extremos y puntos de inflexión de curvas definidas por puntos
discretos.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Interpolación
y aproximación</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">La Base de R proporciona funciones </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">aprox()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> para
la interpolación constante y lineal, y </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">spline()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> para
la interpolación de spline cúbica (Hermite), mientras que </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">smooth.spline
()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> realiza una aproximación de spline cúbica.Las splines del
paquete base crean splines de interpolación periódica en la función </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">periodicSpline
()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> .</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">La
interpolación de datos espaciados irregularmente es posible con el
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/akima/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgNjbTWEgPGs2NTogvcbr-vrFeYsg"><span style="background: white;">akima</span></a> : </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">aspline ()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
datos univariados, </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">bicúbico ()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> o </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">interp ()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
datos en un dominio rectangular 2D.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">(Este
paquete se distribuye bajo la licencia ACM y no está disponible para uso
comercial).</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/signal/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhfuCCD4fK4PcYj4wrtRYSn_Dz0nw">signal
</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><span style="mso-spacerun: yes;"> </span>contiene varios </span></span><i><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">filtros</span></i><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
suavizar datos discretos, en particular </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">interp1()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
interpolación lineal, spline y cúbica, </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">pchip()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
interpolación de Hermite cúbica por partes, y </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">sgolay ()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
el suavizado de Savitzky-Golay.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/pracma/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhsRijdxWO_eTd2TGAnnCVU_K0T9g"><span style="background: white;">pracma</span></a> proporciona interpolación
de Lagrange baricéntrica (en 1 y 2 dimensiones) en </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">barylag()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> resp.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">barylag2d()</span></tt><span class="notranslate"> , 1-dim.</span> <span class="notranslate">akima en </span><tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">akimaInterp()</span></tt><span class="notranslate"> , e interpolación y aproximación de datos con
funciones racionales, es decir, en presencia de singularidades, en </span><tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">ratinterp ()</span></tt><span class="notranslate"> y </span><tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">rationalfit()</span></tt><span class="notranslate"> .</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/interp/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhg8GyTx3SgEXjG6U7B4iUm3BGKWqw"><span style="background: white;">interp</span></a> proporciona interpolación
de datos bivariados en cuadrículas regulares e irregulares, ya sea lineal
o usando splines.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Actualmente se
implementa la parte de interpolación lineal por tramos.</span> <span class="notranslate">(Está destinado a proporcionar un reemplazo gratuito
para las </span><tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">funciones akima :: interp</span></tt><span class="notranslate"> y </span><tt><span style="font-family: "arial" , sans-serif; mso-ansi-font-size: 12.0pt; mso-bidi-font-size: 12.0pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin;">tripack
:: tri.mesh con</span></tt><span class="notranslate"> licencia de ACM).</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/chebpol/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhZsxV9e0TBKLAJ24Bmf1tLtHdO_w"><span style="background: white;">chebpol</span></a> contiene métodos para
crear polinomios de <span style="mso-spacerun: yes;"> </span>Chebyshev
multivariable y la interpolación multilineal en cuadrículas regulares, por
ejemplo, el método de baricentro Floater-Hormann, o splines poliarmónicos
para datos dispersos.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/tripack/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgzZWjDfF1MXxJklZLCSKTTBRmqsQ">tripack</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
triangulación de datos espaciados irregularmente es un paquete
bidimensional de triangulación de Delaunay restringido que proporciona
triangulación y generación de mosaicos Voronoi de datos espaciados
irregulares.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">sinterp()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> en
el paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/stinepack/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhi3X88dYc1SpFUjcPPVt201HFsCNw"><span style="background: white;">stinepack</span></a> realiza la
interpolación basada en funciones racionales por partes aplicando el
algoritmo de Stineman.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">La función
de interpolación será monótona en regiones donde los puntos especificados
cambien monótonamente.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Schumaker()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> en
el paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/schumaker/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgKRB9rEWMYerH6yCSO6M2gr247Qg"><span style="background: white;">schumaker</span></a> implementa splines que
preservan la forma, con garantía de ser monótona cóncavo o convexo si los
datos son monotónicos, cóncavos o convexos.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/ADPF/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgadAR5fpAQICCJ4sCoLjwohSIM_Q">ADPF</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> utiliza
regresión polinomial de mínimos cuadrados y pruebas estadísticas para mejorar
el suavizado de Savitzky-Golay.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l0 level1 lfo5; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/conicfit/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhg7OxKS7cq2O8GpRlKMeSRejZtgyw"><span style="background: white;">conicfit</span></a> proporciona varios
algoritmos (geométricos y algebraicos) para ajustar círculos, elipses y
cónicas en general.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Búsqueda de
raíces y puntos fijos</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">uniroot()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> ,
que implementa el algoritmo Brent-Decker, es la rutina básica en R para
encontrar raíces de funciones univariadas.</span></span><span style="color: black; font-family: "arial" , sans-serif;"> <span class="notranslate">Hay
implementaciones del algoritmo de bisección en varios paquetes contribuidos.</span> <span class="notranslate">Para encontrar la raíz con mayor precisión, hay una función </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">unirootR()</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> en
el paquete de precisión múltiple <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Rmpfr/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhj0J28BmorPSCUAXOmKz9H8SYLvng"><span style="background: white;">Rmpfr</span></a> .</span></span><span style="color: black; font-family: "arial" , sans-serif;"> <span class="notranslate">Y para encontrar raíces de funciones multivariadas, vea los
siguientes dos paquetes:</span></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l12 level1 lfo6; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Para
resolver sistemas de ecuaciones no lineales, el paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/BB/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhg5g3-_nMUrRHrJjBjxS2aDKufEwg">BB</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
métodos espectrales de Barzilai-Borwein (no monótonos) en </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">sane()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> ,
incluida una variante sin derivados en </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">dfsane()</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> ,
y características de inicio múltiple con análisis de sensibilidad.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l12 level1 lfo6; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/nleqslv/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhmFvltqyWBEyHE4w_jy0cndIPG_w"><span style="background: white;">nleqslv</span></a> resuelve sistemas no
lineales de ecuaciones usando alternativamente el método de Broyden o
Newton, apoyado por estrategias como búsquedas de líneas o regiones de
confianza.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l12 level1 lfo6; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/ktsolve/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjVx2OSlgyT32jy8gEasfYaZ4HS6Q">ktsolve</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> define
una interfaz común para resolver un conjunto de ecuaciones con </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">BB</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> o </span></span><tt><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">nleqslv</span></tt><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> .</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l12 level1 lfo6; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/FixedPoint/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjhpto08snd1Se0ac8MsscjecGXdg"><span style="background: white;">FixedPoint</span></a> proporciona algoritmos
para encontrar vectores de puntos fijos.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Estos algoritmos incluyen la aceleración de Anderson,
los métodos de extrapolación de épsilon y los métodos polinomiales
mínimos.</span></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Matemática
discreta y teoría de números</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">No hay tantas funciones disponibles para la teoría numérica
computacional.</span></span><span style="color: black; font-family: "arial" , sans-serif;"> Hay que tener <span class="notranslate">en cuenta que los
enteros en doble precisión se pueden representar exactamente hasta </span></span><tt><span style="color: black; font-family: "arial" , sans-serif; font-size: 10.0pt;">2 ^ 53 - 1</span></tt><span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;"> ,
por encima de ese límite se necesita un paquete de precisión múltiple
como <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/gmp/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiN5q_OH-o7IXHiHFcF1Afe6MV9CA"><span style="background: white;">gmp</span></a> , consulte a continuación.</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/numbers/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjzgLS7gEeAu86alc2dt1uGBZH5OQ">n</a>umbers</span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">
proporciona funciones para la factorización, números primos, primos
gemelos, raíces primitivas, inversos modulares, MCD extendido, etc. Se
incluyen algunas funciones de teoría de números como funciones de divisor
o función Phi de Euler.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/contfrac/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhg0dRXDlaTK9GX1cl71G1Z0oR7RAA">Contfrac</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> contiene
varias utilidades para evaluar fracciones continuas y convergentes
parciales.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/magic/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgGTJ34Tj96OK5FbQRyJRrzSzUaDw">magic</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> crea
e investiga cuadrados mágicos e hipercubos, incluidas funciones para la
manipulación y el análisis de matrices dimensionadas arbitrariamente.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/freegroup/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgxcutsH8wO_nYE9XkOu8_xQYRLdA"><span style="background: white;">freegroup</span></a> proporciona
funcionalidad para manipular elementos de un grupo libre, incluidas la
yuxtaposición, la inversión, la multiplicación por un escalar, las
operaciones de potencia y las formas de Tietze.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/partitions/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgNcSo8KNlAsyRmZitJdH5KYgRNhg">partitions</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> enumera
particiones aditivas de enteros, incluyendo particiones restringidas y
desiguales.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/permutations/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhh_vk_oFcriTN91VbJkDP9tykfAaw">permutations</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> tratan
las permutaciones como funciones invertibles de conjuntos finitos e
incluyen varias operaciones matemáticas en ellas.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/combinat/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgYt-_J7YgjW4Kvet-erGpcLJ_V2Q"><span style="background: white;">combinat</span></a> genera todas las
permutaciones o todas las combinaciones de una cierta longitud de un
conjunto de elementos (es decir, un vector);</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">también calcula los coeficientes binomiales.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/arrangements/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhj63UfdyWHClWwAhkm9TfOJ3huARA">arrangements</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">
proporciona generadores e iteradores para permutaciones, combinaciones y
particiones.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Los iteradores permiten a los usuarios generar arreglos
de una manera rápida y eficiente en memoria.</span> <span class="notranslate">Las permutaciones y combinaciones se pueden dibujar con
/ sin reemplazo y son compatibles con multisets.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/RcppAlgos/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjJaiyJ-raI6TxR-lt7WRRrISw3dQ">RcppAlgos</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
funciones flexibles para generar combinaciones o permutaciones de un
vector con o sin restricciones.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">El paquete
de extensión <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/bigIntegerAlgos/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgUSurzGL3wsn7kX9zRGMlNdWo_yw"><span style="background: white;">bigIntegerAlgos</span></a> presenta un
algoritmo de tamiz cuadrático para factorizar completamente enteros
grandes.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l4 level1 lfo7; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Zseq/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiWuFcvgBOzE6h953WbJqMkzJxhqQ"><span style="background: white;">Zseq</span></a> genera secuencias enteras
conocidas;</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">El paquete 'gmp' es adoptado para computar con números
arbitrariamente grandes.</span> <span class="notranslate">Cada función
tiene en su página de ayuda un hipervínculo a la entrada correspondiente
en la Enciclopedia en línea de secuencias de números enteros ( </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://oeis.org/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiKepOvIw97U1jMfWBynPbXxGi3YA">OEIS</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> ).</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Matemática
aritmética y simbólica de precisión múltiple</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l5 level1 lfo8; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">La
aritmética de precisión múltiple está disponible en R a través del
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/gmp/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiN5q_OH-o7IXHiHFcF1Afe6MV9CA"><span style="background: white;">gmp</span></a> que se interconecta con la
biblioteca GMP C.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Algunos ejemplos son la
factorización de enteros, una prueba de número primo probabilístico u
operaciones en grandes racionales, para los cuales se pueden resolver
sistemas de ecuaciones lineales.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l5 level1 lfo8; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Las
operaciones y funciones de punto flotante de precisión múltiple se
proporcionan a través del paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Rmpfr/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhj0J28BmorPSCUAXOmKz9H8SYLvng"><span style="background: white;">Rmpfr</span></a> utilizando las bibliotecas
MPFR y GMP.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Se incluyen números especiales y algunas funciones
especiales, así como rutinas para la búsqueda, integración y optimización
de raíz con precisión arbitraria.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l5 level1 lfo8; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Brobdingnag/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhihQ2uWjqJ5l9rC72O1K5fT4qZ5Qw">Brobdingnag</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> maneja
números muy grandes manteniendo su logaritmo más una bandera que indica su
signo.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">(Una excelente viñeta explica cómo se hace esto usando
los métodos S4).</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l5 level1 lfo8; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/VeryLargeIntegers/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiEnDVGYWMdauq7mbaR4FkVcFSDwg">VeryLargeIntegers</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> implementa
una biblioteca de precisión múltiple que permite almacenar y administrar
enteros arbitrariamente grandes;</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Incluye
pruebas de primalidad probabilística y algoritmos de factorización.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l5 level1 lfo8; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Ryacas/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhKu68KJ2kNBin_CWk1GNmLu6JJjg"><span style="background: white;">Ryacas</span></a> interconecta el sistema de
álgebra computacional 'Yacas'.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Admite cálculos
de precisión simbólicos y arbitrarios en cálculo y álgebra lineal.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l5 level1 lfo8; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/rSymPy/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhBbHhByuGQk0_cwMJCSCyM-UZpag"><span style="background: white;">rSymPy</span></a> accede al sistema de
álgebra simbólica 'SymPy' (escrito en Python) desde R. Admite cálculos de
precisión arbitrarios, álgebra lineal y cálculo, resolución de ecuaciones,
matemáticas discretas y mucho más.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">Interfaces
Python</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Python, a través de sus módulos 'NumPy', 'SciPy', 'Matplotlib',
'SymPy' y 'pandas', tiene disponibles herramientas numéricas y gráficas
eficientes y elaboradas.</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l8 level1 lfo9; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/reticulate/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhtAzMuUgWwCp71eTZq-3DP7KwXcw">reticulate</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> es
una interfaz R para los módulos, clases y funciones de Python.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Cuando se llama a Python en R, los tipos de datos se
convierten automáticamente a sus tipos de Python equivalentes;</span> <span class="notranslate">cuando los valores se devuelven de Python a R, se
vuelven a convertir en los tipos R.</span> <span class="notranslate">Este
paquete del equipo de RStudio es un tipo de estándar para llamar a Python
desde R.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l8 level1 lfo9; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete R <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/rPython/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhYEl5R8YjWtxDywswbexKX-TQpsg"><span style="background: white;">rPython</span></a> permite llamadas de R a
Python, mientras que <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://sourceforge.net/projects/rpy/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhISw6CX90_ffsOXLwvJkc4wmd20Q"><span style="background: white;">RPy</span></a> (con el módulo de Python
'rpy2') interconecta R de Python.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/SnakeCharmR/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhh7QPdUWHQ5o-kw3LqS7DpOSkFaQQ"><span style="background: white;">SnakeCharmR</span></a> es una bifurcación de
'rPython' con varias correcciones y mejoras.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l8 level1 lfo9; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/PythonInR/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhGL5VCHHexPGLtJMHMuTp_K7ziJw">PythonInR</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> es
otro paquete para interactuar con Python desde dentro R. Proporciona
clases de Python para vectores, matrices y marcos de datos que permiten
una conversión fácil de R a Python y viceversa.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l8 level1 lfo9; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/feather/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhDPBpM6bOVHdNqhjov9xeEgnvc7g">feather</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
enlaces para leer y escribir archivos feather, un ligero almacén de datos
binarios diseñado para la máxima velocidad.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">También se puede acceder a este formato de
almacenamiento en Python, Julia o Scala.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l8 level1 lfo9; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/findpython/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhZqK_QA5KmHStMTbQcFceq9zUJkg">findpython</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> es
un paquete diseñado para encontrar un binario Python aceptable en la ruta,
incl.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Versión mínima o módulos requeridos.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l8 level1 lfo9; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">'pyRserve'
es un módulo de Python para conectar Python a un proceso R que
ejecuta <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/Rserve/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhp82srpiZqNAGa_ldwAeRjLrfFHA"><span style="background: white;">Rserve</span></a> como una puerta de enlace
RPC.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Este proceso R se puede ejecutar en una máquina remota,
el acceso variable y las llamadas de función se delegarán a través de la
red.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l8 level1 lfo9; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/XRPython/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhj7IKS6mIkfgAdqZC_0GKJEYvwzqQ">XRPython</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> (y
'XRJulia') se basan en el paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/XR/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhi0K7MzbYOI_d6uLekJg5BacmOeqQ">XR
de</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> John Chambers y su libro
"Extending R" y permiten una integración muy estructurada de R
con Python resp.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Julia</span></span></li>
</ul>
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l11 level1 lfo10; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Tenga
en cuenta que <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://www.sagemath.org/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhEka6BEMAJrurlt7jZ6wei34LFjQ"><span style="background: white;">SageMath</span></a> es un sistema gratuito
de matemáticas de código abierto basado en Python, que permite ejecutar
funciones R, pero también brinda acceso a los programas de matemáticas
Maxima, GAP, FLINT y muchos más.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">SageMath se
puede descargar o utilizar a través de una interfaz web en <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cocalc.com/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhg0sKBLfc5HmxuuktpL2woP6_A1Q"><span style="background: white;">CoCalc</span></a> .</span></span></li>
</ul>
<b><span style="color: black; font-family: "arial" , sans-serif;">MATLAB,
Octave, Julia y otras interfaces</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Las interfaces para software de computación numérica como MATLAB
(comercial) u Octave (gratis) serán importantes cuando se resuelven problemas
numéricos difíciles.</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l2 level1 lfo11; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete de emulación <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/matlab/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhiaHHBCo4zbtX-f72I3n0sTL6FfYA"><span style="background: white;">matlab</span></a> contiene aproximadamente
30 funciones simples, replicando funciones MATLAB, usando los nombres
respectivos de MATLAB y implementándose en R. pura (vea también la </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/pracma/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhsRijdxWO_eTd2TGAnnCVU_K0T9g">práctica</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> para
muchas más funciones matemáticas diseñadas con MATLAB en mente).</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l2 level1 lfo11; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/R.matlab/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhV8qtkd8SBbx8UhrCNneL6NX-IEg"><span style="background: white;">R.matlab</span></a> proporciona herramientas
para leer y escribir archivos MAT, que es el formato de datos MATLAB.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">También permite una interfaz unidireccional con un
proceso MATLAB, enviando y recuperando objetos a través de una conexión
TCP / IP.</span></span></li>
</ul>
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Julia es "un lenguaje de programación dinámico de alto nivel
y alto rendimiento para computación numérica", lo que lo hace interesante
para problemas de optimización y otros cálculos científicos exigentes en R.</span></span><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l13 level1 lfo12; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">La
interfaz Julia del paquete <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/XRJulia/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhitudKEwmmCSxTrLB1lmyIuU8KuUg"><span style="background: white;">XRJulia</span></a> de John Chambers
proporciona análogos directos a las llamadas de función de Julia.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Un paquete 'juliaExamples' está disponible en Github.</span></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l13 level1 lfo12; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/JuliaCall/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgIdrBavKas_bcuUC6mqfkSZ0K3qA">JuliaCall</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
una integración perfecta entre R y Julia.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Usando la interfaz de alto nivel, el usuario puede
llamar a cualquier función Julia como una función R con conversión
automática de tipos.</span></span></li>
</ul>
<span class="notranslate"><span style="color: black; font-family: "arial" , sans-serif;">Los programas comerciales SAS y Mathematica tienen facilidades
para llamar a las funciones R.</span></span><span style="color: black; font-family: "arial" , sans-serif;"> <span class="notranslate">Aquí hay otro Sistema de Álgebra
Computacional (CAS) en Matemáticas Puras que se puede llamar desde R.</span></span><br />
<ul type="disc">
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l6 level1 lfo13; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">El
paquete </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/packages/m2r/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhglw8F_hMTKf7QgrjWfLAzt2-PU_w">m2r</a></span></span><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> proporciona
una interfaz persistente para Macauley2, un programa de software extendido
que apoya la investigación en geometría algebraica y álgebra conmutativa.</span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> <span class="notranslate">Macauley2 debe instalarse de forma independiente, de lo
contrario, se instanciará un proceso Macauley2 en la nube.</span></span></li>
</ul>
<h3 style="background: white;">
<span class="notranslate"><span style="color: #666666; font-family: "arial" , sans-serif; font-size: 12.0pt;">Enlaces relacionados:</span></span><span style="color: #666666; font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></h3>
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<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Vista
de Tarea CRAN: <a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/views/DifferentialEquations.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjI7mT2pI6H0lRWxSCfpeVSxsJfBA"><span style="background: white;">Ecuaciones Diferenciales</span></a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Vista
de tareas CRAN: </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://cran.r-project.org/web/views/Optimization.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhilKlq523t7jNNbM4xL-WCe-JSUbw">Optimización</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
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de alto rendimiento</span></a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Libro
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de matriz práctica con R</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Libro
de texto: </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://www.ms.unimelb.edu.au/~apro%40unimelb/spuRs/index.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhdxGUcyF_G6IObQJ2Dc2Z4VhCMsg">Introducción
a la programación científica y simulación usando R</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Libro
de texto: </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://www.crcpress.com/product/isbn/9781439884485&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhgmQpE7aZKe6oDjAlX-iNfjBYrfIw">Métodos
numéricos en ciencia e ingeniería usando R</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="notranslate"><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Libro
de texto: </span></span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=https://www.crcpress.com/Computational-Methods-for-Numerical-Analysis-with-R/II/p/book/9781498723633&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjp8b6xPBpNFJPiFw6ATJrBqTGU4A">Métodos
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<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://www.math.umaine.edu/~hiebeler/comp/matlabR.html&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhh8IxS7LlKg32JxKSnO4KNB8mj4ZQ">R
y MATLAB</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://www.nr.com/aands/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhwtVZ6G1VlDUR2rgAGDU1X-yJ_YQ">Abramowitz
y Stegun.</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> </span><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://www.nr.com/aands/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhhwtVZ6G1VlDUR2rgAGDU1X-yJ_YQ">Manual
de funciones matematicas</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://www.nrbook.com/a/bookcpdf.php&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjZQy38lE7uWtxKImvayavu_kjHeg">Recetas
Numéricas: El Arte de la Computación Numérica</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
<li class="MsoNormal" style="color: black; line-height: normal; mso-list: l3 level1 lfo14; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list 36.0pt;"><span class="MsoHyperlink"><span style="background: white; font-family: "arial" , sans-serif; font-size: 12.0pt;"><a href="https://translate.googleusercontent.com/translate_c?depth=1&hl=es&prev=search&rurl=translate.google.com&sl=en&sp=nmt4&u=http://mathworld.wolfram.com/&xid=17259,15700021,15700043,15700186,15700191,15700256,15700259&usg=ALkJrhjNB2OHrFoT5MJLW7UUFwCubc4GpA">Wolfram
Mathworld de E. Weisstein</a></span></span><span style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></li>
</ul>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-21873192293416899002019-03-18T11:41:00.001-06:002019-03-18T11:52:58.015-06:00COMPROBACIÓN DE LOS SUPUESTOS DEL ANÁLISIS DE VARIANZA CON INFOSTAT<br />
<div class="MsoNormal">
<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">COMPROBACIÓN DE LOS SUPUESTOS DEL
ANÁLISIS DE VARIANZA CON INFOSTAT<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">Ing. Luis Manfredo Reyes<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">La técnica conocida como Análisis
de Varianza (ANDEVA, ANOVA o ANVA), se utiliza intensamente en la
experimentación para comparar varios grupos, llamados tratamientos.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">La hipótesis nula que se prueba con
el ANDEVA, es que “todos los tratamientos son iguales”,<span style="mso-spacerun: yes;"> </span>contra la hipótesis alterna que “al menos uno
de los tratamientos es distinto a los demás”. <o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">El análisis de varianza, está
construido sobre supuestos teóricos, que en la mayoría de casos nunca son
comprobados.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">Una de las excusas más utilizadas
es “SI LA VARIABLE DE RESPUESTA ES CONTÍNUA, SEGURAMENTE ES NORMAL”. <o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">También anteriormente, no se tenía
la tecnología informática que permitiera realizar las pruebas en forma rápida y
efectiva. <o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">Ahora sí existe la tecnología (se
acabaron las excusas). De todas las opciones disponibles, en ésta ocasión se
discute el uso de Infostat para el proceso de comprobación de supuestos.<o:p></o:p></span></div>
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<span lang="ES" style="background: white; color: #222222; font-family: "verdana" , sans-serif;"></span><br />
<a name='more'></a><span lang="ES" style="background: white; color: #222222; font-family: "verdana" , sans-serif;"><br /></span></div>
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<span lang="ES" style="background: white; color: #222222; font-family: "verdana" , sans-serif;">Infostat
es un paquete estadístico producido en la Universidad de Córdoba, Argentina,
que tiene la ventaja de estar escrito en español, con versión comercial y
libre. Se puede obtener en: </span><span lang="ES"><a href="http://www.infostat.com.ar/"><span style="background: white; color: #bb5521; font-family: "verdana" , sans-serif; text-decoration: none;">www.infostat.com.ar</span></a></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 13.85pt;">
<span lang="ES" style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10.0pt;"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10.0pt;">Se asume que el lector tiene instalado el paquete y conoce
el uso básico del mismo.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">SUPUESTOS DEL ANÁLISIS DE VARIANZA<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">1)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">La
variable de respuesta debe medirse en una escala por lo menos de intervalo<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">2)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">Las
observaciones son independientes<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">3)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">La
distribución de los errores debe ser normal<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">4)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">Las
varianzas deben ser <span style="mso-spacerun: yes;"> </span>iguales
(Homocedasticidad)<o:p></o:p></span></div>
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<div style="background: #E0DFD8;">
<b><span lang="ES" style="color: blue; font-family: "arial" , sans-serif;">COMPROBACION DEL SUPUESTO DE Homogeneidad de Varianzas</span></b></div>
<div style="background: #E0DFD8;">
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">Para verificar el supuesto de homogeneidad de varianzas de los
grupos comparados, de aplica la prueba de Levene.<o:p></o:p></span><br />
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"><br /></span>
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">El test de Levene consiste en realizar un análisis de varianza, usando como variable dependiente el valor absoluto de los errores.</span><br />
<br /></div>
<div style="background: #E0DFD8;">
<b><span lang="ES" style="color: blue; font-family: "arial" , sans-serif;">COMPROBACION DEL SUPUESTO DE NORMALIDAD DE LOS ERRORES</span></b><span lang="ES" style="color: black; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span style="color: #333333; font-family: "arial" , sans-serif; mso-ansi-language: ES-GT;">Métodos </span><span lang="ES" style="color: #333333; font-family: "arial" , sans-serif;">gráficos</span><span lang="ES" style="color: black; font-family: "arial" , sans-serif;">: Una primera
posibilidad consiste en obtener histogramas con curvas superpuestas de la
distribución Normal. <o:p></o:p></span><br />
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"><br /></span></div>
<div style="background: #E0DFD8;">
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">Otra posibilidad son las gráficas QQ que sitúan los cuantiles de
la distribución de la muestra respecto de los cuantiles del modelo de
distribución normal. <o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"><br /></span>
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">Cuando ambas distribuciones coinciden los puntos se sitúan sobre
una recta, y en la medida que la distribución de la muestra difiera de la
distribución normal se sitúan fuera de la recta.<o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"><br /></span>
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"> </span><b><span lang="ES" style="color: blue; font-family: "arial" , sans-serif;">Pruebas de bondad de ajuste</span></b><span lang="ES" style="color: black; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">Procedimientos analíticos más precisos se basan en pruebas de
bondad de ajuste que someten a prueba la hipótesis de que la distribución de la
muestra se ajusta al modelo Normal. <o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">La prueba de Kolmogorov-Smirnov o la de Shapiro-Wilk pueden ser
utilizadas a este propósito.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></b>
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">COMPROBACION DE LA INDEPENDENCIA</span></b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Los datos de cada tratamiento deben provenir
de observaciones no correlacionadas, es decir al azar. Esto en parte se logra
al inicio del experimento por medio del proceso de aleatorización .</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<br /></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Sin embargo es posible que aún se viole el
supuesto.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Pruebas para detectar falta de independencia</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Prueba de rachas (aleatoridad)</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Prueba de Durbin Watson</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Pruebas de chi cuadrado de bondad de ajuste</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="background: white;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Consecuencias de la violación de las
hipótesis del modelo</span></b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">La pregunta que ahora surge
es: qué efectos genera la violación de los supuestos del análisis de
varianza :</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">i) Independencia de los errores.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">ii) Normalidad de los errores.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">iii) Igualdad de varianzas de tratamientos.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></b></div>
<div class="MsoNormal" style="background: white;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Dependencia entre los términos de error</span></b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">La no independencia entre los términos de
error puede producir graves efectos en la inferencia. Debido a que este
incumplimiento es a veces difícil de corregir, es importante
evitarlo cuando sea factible. Una forma de conseguirlo es mediante el uso de la
aleatorización y otra forma es modificando el modelo.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Desviación de los errores de la normalidad</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; text-indent: -18.0pt;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">a)</span><span lang="ES" style="color: #222222; font-size: 7.0pt; line-height: 115%;"> </span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Inferencias sobre
las medias: se refiere únicamente a los efectos del factor (estimación puntual
o por intervalos para medias, contraste de igualdad de medias, contrastes
múltiples de medias, etc.)</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">b) Inferencias sobre las varianzas: incluye
la estimación puntual o por intervalos de la varianza o de componentes de la
varianza e inferencia sobre la razón de varianzas.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">El efecto de la falta de
normalidad afecta de manera desigual a estos tipos de inferencias. En general,
si la desviación de la normalidad no es muy grave, es poco importante en la
inferencia sobre medias y más grave en la inferencia sobre varianzas.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Además, en los dos casos, las estimaciones
puntuales siguen siendo insesgadas y en los contrastes de hipótesis se
modifican el error de tipo I y su potencia. Generalmente, dicho
error es ligeramente mayor que el nominal y la potencia menor que la teórica.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></b></div>
<div class="MsoNormal" style="background: white;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Heterocedasticidad </span></b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">del término error</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Cuando las varianzas de los términos de error
de cada nivel no son iguales,lal prueba F para la igualdad de medias
de los tratamientos está poco afectado si todas las repeticiones de los
tratamientos son iguales o difieren muy poco, pero no ocurre lo mismo cuando
hay grandes diferencias entre dichas repeticiones o cuando una varianza es
mucho mayor que las otras.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">En el caso de dos grupos de igual tamaño, la
violación de la hipótesis de homocedasticidad para n grande es aún menos
importante que en el caso general, así, por ejemplo, la potencia del test F
será la teórica si y sólo si los tamaños de cada grupo son iguales.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; text-indent: 35.4pt;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></b></div>
<div class="MsoNormal" style="background: white; text-indent: 35.4pt;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">EJEMPLO DE
APLICACIÓN USANDO INFOSTAT <span style="mso-spacerun: yes;"> </span>(DATOS
REALES)</span></b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">En 1996 se realizó un diagnóstico de los
conocimientos con los que ingresaban los alumnos de nuevo ingreso a la Facultad
de Ciencias Químicas y Farmacia de la Universidad de San Carlos de Guatemala (Universidad Pública). Se les aplicó un test con temas de álgebra,
geometría, física y trigonometría y se calculó una nota final (entre cero y 100)</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">La pregunta de investigación que se desea
responder es:</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">“¿HAY DIFERENCIAS ENTRE LAS NOTAS OBTENIDAS,
DEPENDIENDO DEL TITULO DE SECUNDARIA CON EL QUE INGRESARON ¿”</span></b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Los datos están contenidos en un archivo tipo
CSV, que se puede obtener en el sitio: </span><span lang="ES"><a href="http://www.mediafire.com/file/54xb7t6rygxy5qb/DIAGNOSTICO.csv/file"><span style="color: #888888; font-family: "arial" , sans-serif; text-decoration: none;">http://www.mediafire.com/file/54xb7t6rygxy5qb/DIAGNOSTICO.csv/file</span></a></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; line-height: 115%;">Suponiendo
que los datos están en la carpeta: documentos:<span style="font-size: 10pt;"><o:p></o:p></span></span></div>
<div class="MsoListParagraphCxSpFirst" style="background: white; mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Importar los datos
a Infostat<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Se ingresa a Archivo,
luego a Abrir y en la ventana que aparece seleccionar la carpeta Documentos (o
donde esté ubicado el archivo)<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB-l56rzoM88fxI2rzaTrgaXOYozmziIEwlYRKznVLBx5tzKPV1Js2n7cMdu2u6thvq7BckOhEaetrzLnfaEgK_1pwu_6XPh3BeD6m1sjVkaf7bn_XJavoK2yOOUGs8HEYVVQQcBEQ8vI/s1600/supaovi1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="423" data-original-width="568" height="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB-l56rzoM88fxI2rzaTrgaXOYozmziIEwlYRKznVLBx5tzKPV1Js2n7cMdu2u6thvq7BckOhEaetrzLnfaEgK_1pwu_6XPh3BeD6m1sjVkaf7bn_XJavoK2yOOUGs8HEYVVQQcBEQ8vI/s320/supaovi1.png" width="320" /></a></div>
<br /></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; mso-fareast-language: ES; mso-no-proof: yes;"><v:shape id="_x0000_i1044" style="height: 243.75pt; mso-wrap-style: square; visibility: visible; width: 327.75pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:/Users/Lreyes1/AppData/Local/Temp/msohtmlclip1/01/clip_image003.png">
</v:imagedata></v:shape></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Luego se especifican los
detalles del archivo:<o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; mso-fareast-language: ES; mso-no-proof: yes;"><v:shape id="_x0000_i1043" style="height: 208.5pt; mso-wrap-style: square; visibility: visible; width: 276pt;" type="#_x0000_t75">
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</v:imagedata></v:shape></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
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<br /></div>
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</v:imagedata></v:shape></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_zuh7FCRGEqd8CDWiBwoz4Lv3z0vuL6mDd8_kFkXeU3pza_YdeuB_2cjHhapoFPlq5Kga-DxVw-UqClXGxW_uOfWPT7z_NzAeYPwwH3N5mTcTMP6FKaMuztpMydDR2TzeNDQvOXiy5pc/s1600/supaovi2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="397" data-original-width="526" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_zuh7FCRGEqd8CDWiBwoz4Lv3z0vuL6mDd8_kFkXeU3pza_YdeuB_2cjHhapoFPlq5Kga-DxVw-UqClXGxW_uOfWPT7z_NzAeYPwwH3N5mTcTMP6FKaMuztpMydDR2TzeNDQvOXiy5pc/s320/supaovi2.png" width="320" /></a></div>
<br />
<span style="color: #222222; font-family: "arial" , sans-serif;">Ya importados los datos a Infostat, aparecen así:</span><br />
<span style="color: #222222; font-family: "arial" , sans-serif;"><br /></span>
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4xc70nFkt1QvUVzczqG79eEHio3Lwzzcd3IprlMz7C5rg_oSpMsKXI_p_Zjyo9ZlXVn6HNJ54sdqJt8bZ_4PdfTuWJfp0gDOlL0Q2Irc2DKvK3KxFVqZRK9qC3JlsDpgz77Qp860-2Ng/s1600/supaovi3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="505" data-original-width="415" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4xc70nFkt1QvUVzczqG79eEHio3Lwzzcd3IprlMz7C5rg_oSpMsKXI_p_Zjyo9ZlXVn6HNJ54sdqJt8bZ_4PdfTuWJfp0gDOlL0Q2Irc2DKvK3KxFVqZRK9qC3JlsDpgz77Qp860-2Ng/s320/supaovi3.png" width="262" /></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Comparación gráfica de los datos</span><br />
Se realiza mediante la gráfica de Cajitas de Tukey: (Box Plot)<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4r4Cqsf1R7O_9HgbA4nih4uIlCEvk5cUzFqU5jLb6_cZ_3M7Ryq00uxDbLuyfdOi9cpz2zGGErBjjPh5zyvQw03vs2-Vr03zEUJ8spGkek-nQOcVfLf4HH3rKi3tUQTUtVfuVHNlYwb8/s1600/SUPAOVI90.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="253" data-original-width="310" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4r4Cqsf1R7O_9HgbA4nih4uIlCEvk5cUzFqU5jLb6_cZ_3M7Ryq00uxDbLuyfdOi9cpz2zGGErBjjPh5zyvQw03vs2-Vr03zEUJ8spGkek-nQOcVfLf4HH3rKi3tUQTUtVfuVHNlYwb8/s1600/SUPAOVI90.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrXKIphjPShAvXE7mSE8SQZHkEf3OVZQ3E-LV9wzk7mTwO1KLEV-4VRsqN3ox0XzlREXRgbn3A0tjXcKzC5PHbGQjWh1cNOxnvRAQ06v-A0UV36hR2DYYxt4ejLSWQGdtntjePlgUnSUE/s1600/SUPAOVI91.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="520" data-original-width="497" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrXKIphjPShAvXE7mSE8SQZHkEf3OVZQ3E-LV9wzk7mTwO1KLEV-4VRsqN3ox0XzlREXRgbn3A0tjXcKzC5PHbGQjWh1cNOxnvRAQ06v-A0UV36hR2DYYxt4ejLSWQGdtntjePlgUnSUE/s320/SUPAOVI91.png" width="305" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjX1NP0U3GwReJTxiMEfrVtMx52dCBFCG6PDSzB94hnjFwKBdzRXf4i8v73zMOlDe0bqohwd_lPxA99jGfFr4T0YfVfBLIPL-YIRLtotZMVH1Kc3lgv0Yc4eGnxIxb2tjModkdrJGXckQ4/s1600/supaovi92.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="545" data-original-width="582" height="299" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjX1NP0U3GwReJTxiMEfrVtMx52dCBFCG6PDSzB94hnjFwKBdzRXf4i8v73zMOlDe0bqohwd_lPxA99jGfFr4T0YfVfBLIPL-YIRLtotZMVH1Kc3lgv0Yc4eGnxIxb2tjModkdrJGXckQ4/s320/supaovi92.png" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">La gráfica indica que sí existen diferencias entre las notas obtenidas dependiendo del título con el que ingresaron. </span><br />
<br />
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; text-indent: -18pt;"><span style="mso-list: Ignore;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; text-indent: -18pt;">Realizar el
análisis de varianza, conservando los residuos</span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Se ingresa a Estadísticas
y luego a análisis de la varianza.<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">La variable dependiente en
este caso es la nota, y la independiente el título, los que se ingresan a las
cajas respectivas y luego se indica aceptar<o:p></o:p></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
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<v:imagedata o:title="" src="file:///C:/Users/Lreyes1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png">
</v:imagedata></v:shape></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; mso-fareast-language: ES; mso-no-proof: yes;"><v:shape id="_x0000_i1040" style="height: 248.25pt; mso-wrap-style: square; visibility: visible; width: 234.75pt;" type="#_x0000_t75">
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</v:imagedata></v:shape></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0wAC3q1asVaL_FoolS-j1rAW0cpuFBAA76Xp5fUszNGj1kq6iRCmtgRlYm6tuopJnx2S1lFcQRa6zaWE23T-mU4ngwJcsgP7SLv6_4RJxC7PVzZlb1jVPWDNq0-didErS5xUw9hMEl40/s1600/supaovi5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="522" data-original-width="494" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0wAC3q1asVaL_FoolS-j1rAW0cpuFBAA76Xp5fUszNGj1kq6iRCmtgRlYm6tuopJnx2S1lFcQRa6zaWE23T-mU4ngwJcsgP7SLv6_4RJxC7PVzZlb1jVPWDNq0-didErS5xUw9hMEl40/s320/supaovi5.png" width="302" /></a></div>
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">En la siguiente ventana se
seleccionan los residuos, el valor absoluto de los residuos, los residuos
estandarizados <span style="mso-spacerun: yes;"> </span>y los valores predichos.<o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; mso-fareast-language: ES; mso-no-proof: yes;"><v:shape id="Imagen_x0020_23" o:spid="_x0000_i1039" style="height: 218.25pt; mso-wrap-style: square; visibility: visible; width: 270pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:/Users/Lreyes1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png">
</v:imagedata></v:shape></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYbYj75IaVbHV7TOhCZa46CII5qLuXe8OZhjGJtGBkYYZju2etlmlEAk8r-CSxc5HY4leGECiG4kq6L5ecUJAiqgUORKI4KNO14gKVTGuGUXW-njRfD-oNp2EWEKltq6zORTe0gZ13Eck/s1600/supaovi6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="401" data-original-width="484" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYbYj75IaVbHV7TOhCZa46CII5qLuXe8OZhjGJtGBkYYZju2etlmlEAk8r-CSxc5HY4leGECiG4kq6L5ecUJAiqgUORKI4KNO14gKVTGuGUXW-njRfD-oNp2EWEKltq6zORTe0gZ13Eck/s320/supaovi6.png" width="320" /></a></div>
<br /></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Los resultados del
Análisis de varianza son los siguientes:<o:p></o:p></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><br /></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9s9tvZlPMndWr-lnWsm5i5T2w_MixGVvy9LBKNcq-PpIN-8F2qub9Iy7M7p3wifHJXvYiS9qdQl7-d2bm0T4HJRLi7-Cq8E2B4BF4CSZBz9c1I00MPZ-7LdXcIgNmSeIsZFjgww36mIQ/s1600/supaovi7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="340" data-original-width="435" height="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9s9tvZlPMndWr-lnWsm5i5T2w_MixGVvy9LBKNcq-PpIN-8F2qub9Iy7M7p3wifHJXvYiS9qdQl7-d2bm0T4HJRLi7-Cq8E2B4BF4CSZBz9c1I00MPZ-7LdXcIgNmSeIsZFjgww36mIQ/s320/supaovi7.png" width="320" /></a></div>
<br /></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white; mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;"><b><span style="font-size: large;">Verificación de la normalidad de los Errores</span></b><span style="font: 7.0pt "Times New Roman";"> </span></span></span><br />
<!--[endif]--><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Pruebas de
normalidad gráfica y analítica<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Se procede a realizar un
gráfico QQ Plot.<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Se ingresa a la pestaña
Gráficos y luego se selecciona QQ Plot. Se define como variable de interés el
residuo de la nota y en la siguiente ventana se especifica la distribución
normal.<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white;">
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</v:imagedata></v:shape></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">.</span><span style="color: #222222; font-family: "arial" , sans-serif;">La</span><span style="color: #222222; font-family: "arial" , sans-serif;"> </span><span style="color: #222222; font-family: "arial" , sans-serif;">Gráfica indica que en general los datos se
aproximan a una distribución normal, pero se desvían en los extremos.</span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">Para confirmar
analíticamente ésta conclusión, se procede a realizar una prueba de Shapiro y
Wilks:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjRJNX-yDIVYvta4THr7IrmzSG0KtPdRKdCs9v5mXB5WeaeH5xXbHWD7UVJ88O9V8RSbIWkKN-jpcQIwsUOlIj3u7pI0ZLJ3L79kPhxuGUECE_suE1_y0y2aRmGsnpSeDaAMMwYq05tKE/s1600/supaovi14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="209" data-original-width="413" height="161" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjRJNX-yDIVYvta4THr7IrmzSG0KtPdRKdCs9v5mXB5WeaeH5xXbHWD7UVJ88O9V8RSbIWkKN-jpcQIwsUOlIj3u7pI0ZLJ3L79kPhxuGUECE_suE1_y0y2aRmGsnpSeDaAMMwYq05tKE/s320/supaovi14.png" width="320" /></a></div>
<span style="color: #222222; font-family: "arial" , sans-serif;">De acuerdo a la prueba se concluye
que los errores no tienen distribución normal</span></div>
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<span style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 18.0pt;">Verificación de la Homogeneidad de varianzas.</span><b><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 13.5pt;"><o:p></o:p></span></b></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 10.5pt;">Este es el supuesto más importante que los residuos deben cumplir para que
el modelo empleado sea válido.</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 10.5pt;">Para corroborar esto se realiza una prueba de Cochran o una de Levine</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 10.5pt;">La prueba de C de Cocharn, es una prueba unilateral del límite superior
atípico de la varianza. La prueba C se utiliza para determinar si una sola
estimación de una varianza es significativamente más grande que un grupo de
varianzas, se presenta el resultado de la prueba de C de Cochran para la
variable porcentaje de parasitismo. En este caso no es posible usarla porque el
modelo es desbalanceado (diferentes repeticiones por tratamiento)</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 10.5pt;">Una alternativa a la prueba C de Cochran <b>es la prueba de </b>Levene**</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 10.5pt;">La prueba de <b>Levene</b> consiste en realizar un análisis de la
varianza usando como variable dependiente el valor absoluto de los residuos.<o:p></o:p></span></div>
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<span lang="ES" style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 10.5pt;">La variable dependiente es el valor absoluto de los residuos y la
independiente el titulo<o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; mso-fareast-language: ES; mso-no-proof: yes;"><v:shape id="Imagen_x0020_20" o:spid="_x0000_i1030" style="height: 219pt; mso-wrap-style: square; visibility: visible; width: 267.75pt;" type="#_x0000_t75">
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;">El análisis indica que no
hay diferencia significativa entre los tratamientos, por lo cual se concluye
que las varianzas son homogéneas.<o:p></o:p></span></div>
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<span style="color: #333333; font-family: "times new roman" , serif; font-size: 7.0pt;"> </span><span style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 18.0pt;">Verificación de Independencia
de los residuos</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #333333; font-family: "helvetica" , sans-serif; font-size: 12.0pt;">Este supuesto necesita que la probabilidad
de que el residuo de una observación cualquiera tenga un determinado valor, no
debe depender de los valores de los otros residuos. Al momento de aleatorizar,
debió controlarse. En Infostat solamente se puede comprobar gráficamente,
realizando un diagrama de dispersión de los residuos contra los predichos. <o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">De acuerdo a la gráfica, se nota que no hay evidencia de correlación en los
datos, por lo cual se concluye que sí se cumple el supuesto (LOS RESIDUOS SON
INDEPENDIENTES).<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;"><b>CONCLUSIONES:</b></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">1.</span><span style="color: #222222; font-family: "times new roman" , serif; font-size: 7.0pt;"> </span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">Los residuos son
independientes</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span><br />
<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">2.</span><span style="color: #222222; font-family: "times new roman" , serif; font-size: 7.0pt;"> </span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">Los residuos no tienen
distribución normal, pero tomando en cuenta que lo que interesa es comparar las
medias, la violación del supuesto no se considera crítica para seguir con el
análisis</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span><br />
<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">3.</span><span style="color: #222222; font-family: "times new roman" , serif; font-size: 7.0pt;"> </span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">Las varianzas son homogéneas</span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt;">4. <b>EL ANÁLISIS
DE VARIANZA POR LA VÍA ORDINARIA ES VÁLIDO, LO MISMO QUE LAS CONCLUSIONES</b></span><span lang="ES" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt;"><o:p></o:p></span></div>
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<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-53734000996597298542019-01-24T10:37:00.000-06:002019-03-18T11:55:19.588-06:00COMPROBACIÓN DE LOS SUPUESTOS DEL ANÁLISIS DE VARIANZA CON R<br />
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<b style="mso-bidi-font-weight: normal;"><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">COMPROBACIÓN DE LOS SUPUESTOS DEL ANÁLISIS DE VARIANZA
CON R<o:p></o:p></span></b></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Ing. Luis Manfredo
Reyes<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">La técnica conocida
como Análisis de Varianza (ANDEVA, ANOVA o ANVA), se utiliza intensamente en la
experimentación para comparar varios grupos, llamados tratamientos.<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">La hipótesis nula que
se prueba con el ANDEVA, es que “todos los tratamientos son iguales”,<span style="mso-spacerun: yes;"> </span>contra la hipótesis alterna que “al menos uno
de los tratamientos es distinto a los demás”. <o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span>
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">El análisis de
varianza, está construido sobre supuestos teóricos, que en la mayoría de casos
nunca son comprobados.<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"></span></div>
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<a name='more'></a><br />
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Una de las excusas
más utilizadas es “<b style="mso-bidi-font-weight: normal;">SI LA VARIABLE DE
RESPUESTA ES CONTÍNUA, SEGURAMENTE ES NORMAL</b>”. <o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">También
anteriormente, no se tenía la tecnología informática que permitiera realizar
las pruebas en forma rápida y efectiva. <o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Ahora sí existe la
tecnología (se acabaron las excusas). De todas las opciones disponibles, tanto
comerciales como gratuitas y libres, destaca el lenguaje R.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">R es actualmente el lenguaje
<span style="mso-spacerun: yes;"> </span>estadístico más utilizado en el mundo
académico y de investigación, primero por su potencia de procesamiento, y en
segundo lugar por ser un software libre.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">R<span style="mso-spacerun: yes;"> </span>puede ser descargado del siguiente
sitio: </span><a href="http://www.r-project.org/"><span style="background: white; color: #888888; font-family: "arial" , "sans-serif"; font-size: 12.0pt; text-decoration: none;">http://www.r-project.org/</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Existen versiones para Windows, Mac, Linux y Android<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Se asume que los lectores tienen conocimientos básicos <span style="mso-spacerun: yes;"> </span>de cómo se utiliza el programa R<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">SUPUESTOS
DEL ANÁLISIS DE VARIANZA<o:p></o:p></span></b></div>
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<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">1)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">La
variable de respuesta debe medirse en una escala por lo menos de intervalo<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">2)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Las
observaciones son independientes<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">3)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">La
distribución de los errores debe ser normal<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">4)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Las
varianzas deben ser <span style="mso-spacerun: yes;"> </span>iguales
(Homocedasticidad)<o:p></o:p></span></div>
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<b><span style="color: blue; font-family: "arial" , "sans-serif";">Prueba de Homogeneidad de Varianza</span></b><span style="color: black; font-family: "arial" , "sans-serif";"><o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span style="color: black; font-family: "arial" , "sans-serif";">Para verificar el supuesto de homogeneidad de varianzas de los
grupos comparados, de aplica la prueba de Levene , la prueba Q de Cochran, o la
prueba de Bartlett<o:p></o:p></span></div>
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<b><span style="color: blue; font-family: "arial" , "sans-serif";"><br /></span></b></div>
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<b><span style="color: blue; font-family: "arial" , "sans-serif";">COMPROBACION DEL SUPUESTO DE NORMALIDAD DE LOS ERRORES</span></b><span style="color: black; font-family: "arial" , "sans-serif";"><o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span lang="ES-GT" style="color: #333333; font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Métodos </span><span style="color: #333333; font-family: "arial" , "sans-serif";">gráficos</span><span style="color: black; font-family: "arial" , "sans-serif";">: Una primera posibilidad consiste en
obtener histogramas con curvas superpuestas de la distribución Normal para cada
tratamiento.<o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span style="color: black; font-family: "arial" , "sans-serif";">Otra posibilidad son las gráficas QQ que sitúan los cuantiles de
la distribución de la muestra respecto de los cuantiles del modelo de
distribución normal. <o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span style="color: black; font-family: "arial" , "sans-serif";">Cuando ambas distribuciones coinciden los puntos se sitúan sobre
una recta, y en la medida que la distribución de la muestra difiera de la
distribución normal se sitúan fuera de la recta.<o:p></o:p></span></div>
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<b><span style="color: blue; font-family: "arial" , "sans-serif";">Pruebas de bondad de ajuste</span></b><span style="color: black; font-family: "arial" , "sans-serif";"><o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span style="color: black; font-family: "arial" , "sans-serif";">Los procedimientos analíticos más precisos se basan en pruebas de
bondad de ajuste que someten a prueba la hipótesis de que la distribución de la
muestra se ajusta al modelo Normal. <o:p></o:p></span></div>
<div style="background: #E0DFD8;">
<span style="color: black; font-family: "arial" , "sans-serif";">La prueba de Kolmogorov-Smirnov o la de Shapiro-Wilk pueden ser
utilizadas a este propósito.<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></b></div>
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<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">COMPROBACION
DE LA INDEPENDENCIA<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Los datos de cada tratamiento deben provenir de
observaciones no correlacionadas, es decir al azar. Esto en parte se logra al
inicio del experimento por medio del proceso de aleatorización . <o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Sin embargo es posible que aún se viole el supuesto.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Pruebas para detectar falta de independencia<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Prueba de rachas (aleatoridad)<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Prueba de Durbin Watson<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Pruebas de chi cuadrado de bondad de ajuste<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><br /></span></b></div>
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<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Consecuencias de la violación de las hipótesis del modelo<o:p></o:p></span></b></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">La pregunta que
ahora surge es: <span style="mso-spacerun: yes;"> </span>qué efectos genera la
violación de los supuestos del análisis de varianza :<o:p></o:p></span></div>
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<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">i)
Independencia de los errores.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">ii)
Normalidad de los errores.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">iii)
Igualdad de varianzas de tratamientos.<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Dependencia entre los términos de error<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">La no
independencia entre los términos de error puede producir graves efectos en la
inferencia. Debido a que este incumplimiento es a veces <span style="mso-spacerun: yes;"> </span>difícil de corregir, es importante evitarlo
cuando sea factible. Una forma de conseguirlo es mediante el uso de la
aleatorización y otra forma es modificando el modelo.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Desviación
de los errores de la normalidad<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpFirst" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">a)<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Inferencias sobre las medias: se
refiere únicamente a los efectos del factor (estimación puntual o por
intervalos para medias, contraste de igualdad de medias, contrastes múltiples
de medias, etc.)<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpLast" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-add-space: auto;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">b) Inferencias
sobre las varianzas: incluye la estimación puntual o por intervalos de la
varianza o de componentes de la varianza e inferencia sobre la razón <span style="mso-spacerun: yes;"> </span>de varianzas.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">El efecto de
la falta <span style="mso-spacerun: yes;"> </span>de normalidad afecta de manera
desigual a estos tipos de inferencias. En general, si la desviación de la
normalidad no es muy grave, es poco importante en la inferencia sobre medias y
más grave en la inferencia sobre varianzas.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Además, en los
dos casos, las estimaciones puntuales siguen siendo insesgadas y en los
contrastes de hipótesis se modifican <span style="mso-spacerun: yes;"> </span>el
error de tipo I y su potencia. Generalmente, dicho error es ligeramente mayor
que el nominal y la potencia menor que la teórica.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Heterocedasticidad </span></b><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">del término error<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">Cuando las
varianzas de los términos de error de cada nivel no son iguales,lal prueba <span style="mso-spacerun: yes;"> </span>F para la igualdad de medias de los
tratamientos está poco afectado si todas las repeticiones de los tratamientos
son iguales o difieren muy poco, pero no ocurre lo mismo cuando hay grandes
diferencias entre dichas repeticiones o cuando una varianza es mucho mayor que
las otras.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">En el caso
de dos grupos de igual tamaño, la violación de la hipótesis de homocedasticidad
para n grande es aún menos importante que en el caso general, así, por ejemplo,
la potencia del test F será la teórica si y sólo si los tamaños de cada grupo
son iguales. <o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">EJEMPLO
DE APLICACIÓN USANDO R (DATOS REALES)<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">En 1996 se realizó un diagnóstico de los conocimientos
con los que ingresaban los alumnos de nuevo ingreso a la Facultad de Ciencias
Químicas y Farmacia de la Universidad de San Carlos de Guatemala . Se les aplicó un test con temas de álgebra, geometría,
física y trigonometría y se calculó una nota final (entre cero y 100)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">La pregunta de investigación que se desea responder es:<o:p></o:p></span><br />
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">“HAY
DIFERENCIAS ENTRE LAS NOTAS OBTENIDAS, DEPENDIENDO DEL TITULO DE
SECUNDARIA<span style="mso-spacerun: yes;"> </span>CON EL QUE INGRESARON ¿”<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Los datos están contenidos en un archivo tipo CSV, que se
puede obtener en el sitio: </span><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><a href="http://www.mediafire.com/file/54xb7t6rygxy5qb/DIAGNOSTICO.csv/file">http://www.mediafire.com/file/54xb7t6rygxy5qb/DIAGNOSTICO.csv/file</a></span><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span><br />
<br /></div>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l4 level1 lfo3; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Importar
el archivo a R (suponiendo que los datos ya fueron descargados y están en la unidad E)<o:p></o:p></span></div>
<div style="background: white; border: solid #CCCCCC 1.0pt; margin-left: 36.0pt; margin-right: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<div class="MsoListParagraphCxSpMiddle" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">Datos<-read.table(“E/DIAGNOSTICO.csv,
header=TRUE, sep=”,”,<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">na.strings=”NA”)<o:p></o:p></span></div>
</div>
<div class="MsoListParagraphCxSpMiddle">
<br /></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l4 level1 lfo3; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Realizar
un análisis visual de las variables independientes (tratamientos)<o:p></o:p></span></div>
<div style="background: white; border: solid #CCCCCC 1.0pt; margin-left: 36.0pt; margin-right: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<div class="MsoListParagraphCxSpMiddle" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">Boxplot(NOTA~TITULO,data=Datos,id=(list(method=”y”))<o:p></o:p></span></div>
</div>
<div class="MsoListParagraphCxSpMiddle">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><v:shapetype coordsize="21600,21600" filled="f" id="_x0000_t75" o:preferrelative="t" o:spt="75" path="m@4@5l@4@11@9@11@9@5xe" stroked="f">
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<v:formulas>
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<v:f eqn="sum @8 21600 0">
<v:f eqn="prod @7 21600 pixelHeight">
<v:f eqn="sum @10 21600 0">
</v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:formulas>
<v:path gradientshapeok="t" o:connecttype="rect" o:extrusionok="f">
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</v:imagedata></v:shape></span><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfX9yOdYY_8NwlrrLVwXKetMqqbO6dXKkPzDxh19Du8J0yEzZY5nNrFH3gJyyl1cxu2BZ4oac4B8HTUp7S48tDaLAqB9Q9XmQr-bm9uLmdojkLnYJ2D7IkBnTjRJTqCmbLt77hMLW8sPk/s1600/supaov1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="257" data-original-width="566" height="145" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfX9yOdYY_8NwlrrLVwXKetMqqbO6dXKkPzDxh19Du8J0yEzZY5nNrFH3gJyyl1cxu2BZ4oac4B8HTUp7S48tDaLAqB9Q9XmQr-bm9uLmdojkLnYJ2D7IkBnTjRJTqCmbLt77hMLW8sPk/s320/supaov1.png" width="320" /></a></div>
<br />
<div class="MsoListParagraphCxSpMiddle">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><o:p><br /></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><o:p><br /></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Se observan diferencias de promedios
entre los títulos, y la variabilidad es aparentemente distinta . El mayor
promedio lo presentan los agrónomos y el menor los maestros.<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<br /></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l4 level1 lfo3; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Realizar
el andeva preliminar, presentar resultados <o:p></o:p></span></div>
<div style="background: white; border: solid #CCCCCC 1.0pt; margin-left: 36.0pt; margin-right: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<div class="MsoListParagraphCxSpMiddle" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">Andeva1<-aov(NOTA~TITULO,data=Datos)<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpLast" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">Summary(Andeva1)<o:p></o:p></span></div>
</div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>Df <span style="mso-spacerun: yes;"> </span>Sum Sq <span style="mso-spacerun: yes;"> </span>Mean Sq F value<span style="mso-spacerun: yes;"> </span>Pr(>F)<span style="mso-spacerun: yes;">
</span><o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">TITULO<span style="mso-spacerun: yes;">
</span>5<span style="mso-spacerun: yes;"> </span>5842<span style="mso-spacerun: yes;"> </span>1168.3<span style="mso-spacerun: yes;"> </span>4.139 0.00155 **<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Residuals<span style="mso-spacerun: yes;">
</span>139<span style="mso-spacerun: yes;"> </span>39240<span style="mso-spacerun: yes;"> </span>282.3<span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">-Signif. codes:<span style="mso-spacerun: yes;"> </span>0 '***' 0.001 '**' 0.01 '*' 0.05 '.' </span><span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">0.1
' ' 1<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">4. Presentar promedios por tratamiento<o:p></o:p></span></div>
<div style="background: white; border: solid #CCCCCC 1.0pt; margin-left: 36.0pt; margin-right: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<div class="MsoListParagraph" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">With(Datos,
numSummary(NOTA, groups=TITULO, statistics=c(“mean”,”sd”)))<o:p></o:p></span></div>
</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>mean<span style="mso-spacerun: yes;">
</span>sd data<span style="mso-spacerun: yes;"> </span>:n<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">AGRONOMO<span style="mso-spacerun: yes;"> </span>52.62500 15.65647<span style="mso-spacerun: yes;"> </span>8<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">BACHILLER<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>41.37805
17.41196<span style="mso-spacerun: yes;"> </span>82<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">COMPUTO<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>36.00000
22.62742<span style="mso-spacerun: yes;"> </span>2<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">CONTADOR<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>30.69231
17.13371<span style="mso-spacerun: yes;"> </span>13<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">MAESTRO<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>28.95455
12.98892<span style="mso-spacerun: yes;"> </span>22<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">SECRETARIA <span style="mso-spacerun: yes;"> </span>45.72222 17.80550<span style="mso-spacerun: yes;"> </span>18<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l2 level1 lfo4; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">5.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Cargar paquetes para pruebas de validación<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Si
no están instaladas es necesario instalarlas antes<o:p></o:p></span></div>
<div style="background: white; border: solid #CCCCCC 1.0pt; margin-left: 36.0pt; margin-right: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<div class="MsoListParagraphCxSpLast" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">library(agricolae)<o:p></o:p></span></div>
</div>
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<div class="MsoNormal" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">Loading
required package: agricolae<o:p></o:p></span></div>
</div>
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<div class="MsoListParagraph" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">library(car)<o:p></o:p></span></div>
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<pre style="background: white; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">Loading required package: car</span></code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
</div>
<div style="background: white; border: solid #CCCCCC 1.0pt; margin-left: 36.0pt; margin-right: 0cm; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<div class="MsoListParagraph" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">library(ggplot2)<o:p></o:p></span></div>
</div>
<div style="background: white; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<pre style="background: white; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">Loading required package: ggplot2</span></code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
</div>
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<div class="MsoListParagraph" style="background: white; border: none; line-height: normal; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; word-break: break-all;">
<span lang="EN-US" style="color: #333333; font-family: "courier new"; font-size: 10.0pt;">library(outliers)<o:p></o:p></span></div>
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<pre style="background: white; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">Loading required package: outliers</span></code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
</div>
<div class="MsoListParagraphCxSpFirst">
<br /></div>
<div class="MsoListParagraphCxSpLast" style="mso-list: l2 level1 lfo4; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">6.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Extraer y mostrar <span style="mso-spacerun: yes;"> </span>los residuos <span style="mso-spacerun: yes;"> </span>del objeto Andeva1<o:p></o:p></span></div>
<div style="background: whitesmoke; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">Residuos</span></span><span class="operator"><span lang="ES-GT" style="color: #687687;"><-</span></span><span class="identifier"><span lang="ES-GT" style="color: black;">residuals</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(Andeva1)</span></span><code><span lang="ES-GT" style="color: #333333;"> <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">Residuos</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span lang="ES-GT" style="color: #333333;"><o:p></o:p></span></pre>
</div>
<div class="MsoListParagraphCxSpFirst">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;">1<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>5<span style="mso-spacerun: yes;"> </span>6<span style="mso-spacerun: yes;"> </span>7 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>6.6219512 -17.3780488<span style="mso-spacerun: yes;"> </span>6.6219512<span style="mso-spacerun: yes;">
</span>2.6219512<span style="mso-spacerun: yes;"> </span>2.6219512<span style="mso-spacerun: yes;"> </span>18.6219512 -16.9545455 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>8<span style="mso-spacerun: yes;"> </span>9<span style="mso-spacerun: yes;"> </span>10<span style="mso-spacerun: yes;"> </span>11<span style="mso-spacerun: yes;"> </span>12<span style="mso-spacerun: yes;"> </span>13<span style="mso-spacerun: yes;"> </span>14 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>-6.6923077<span style="mso-spacerun: yes;">
</span>26.6219512 -21.3780488<span style="mso-spacerun: yes;">
</span>19.0454545<span style="mso-spacerun: yes;"> </span>2.6219512 -21.7222222<span style="mso-spacerun: yes;"> </span>-1.3780488 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>15<span style="mso-spacerun: yes;"> </span>16<span style="mso-spacerun: yes;"> </span>17<span style="mso-spacerun: yes;"> </span>18<span style="mso-spacerun: yes;"> </span>19<span style="mso-spacerun: yes;"> </span>20<span style="mso-spacerun: yes;"> </span>21 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>26.6219512<span style="mso-spacerun: yes;">
</span>-9.3780488<span style="mso-spacerun: yes;"> </span>14.6219512<span style="mso-spacerun: yes;"> </span>30.6219512<span style="mso-spacerun: yes;">
</span>-1.3780488 -13.3780488<span style="mso-spacerun: yes;"> </span>18.2777778
<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>22<span style="mso-spacerun: yes;"> </span>23<span style="mso-spacerun: yes;"> </span>24<span style="mso-spacerun: yes;"> </span>25<span style="mso-spacerun: yes;"> </span>26<span style="mso-spacerun: yes;"> </span>27<span style="mso-spacerun: yes;"> </span>28 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>7.0454545<span style="mso-spacerun: yes;">
</span>-4.9545455<span style="mso-spacerun: yes;"> </span>-1.3780488<span style="mso-spacerun: yes;"> </span>-1.3780488<span style="mso-spacerun: yes;">
</span>14.6219512 -21.3780488<span style="mso-spacerun: yes;"> </span>5.3076923
<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>29<span style="mso-spacerun: yes;"> </span>30<span style="mso-spacerun: yes;"> </span>31<span style="mso-spacerun: yes;"> </span>32<span style="mso-spacerun: yes;"> </span>33<span style="mso-spacerun: yes;"> </span>34<span style="mso-spacerun: yes;"> </span>35 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;">-17.3780488<span style="mso-spacerun: yes;"> </span>-1.3780488<span style="mso-spacerun: yes;">
</span>2.6219512<span style="mso-spacerun: yes;"> </span>-9.3780488<span style="mso-spacerun: yes;"> </span>-0.9545455<span style="mso-spacerun: yes;">
</span>6.6219512 -13.3780488 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>36<span style="mso-spacerun: yes;"> </span>37<span style="mso-spacerun: yes;"> </span>38<span style="mso-spacerun: yes;"> </span>39<span style="mso-spacerun: yes;"> </span>40<span style="mso-spacerun: yes;"> </span>41<span style="mso-spacerun: yes;"> </span>42 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>10.6219512<span style="mso-spacerun: yes;">
</span>-0.9545455<span style="mso-spacerun: yes;"> </span>26.6219512<span style="mso-spacerun: yes;"> </span>-9.3780488<span style="mso-spacerun: yes;">
</span>10.6219512<span style="mso-spacerun: yes;"> </span>19.3750000<span style="mso-spacerun: yes;"> </span>6.6219512 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>43<span style="mso-spacerun: yes;"> </span>44<span style="mso-spacerun: yes;"> </span>45<span style="mso-spacerun: yes;"> </span>46<span style="mso-spacerun: yes;"> </span>47<span style="mso-spacerun: yes;"> </span>48<span style="mso-spacerun: yes;"> </span>49 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;">-25.3780488<span style="mso-spacerun: yes;"> </span>6.6219512<span style="mso-spacerun: yes;">
</span>50.6219512 -13.3780488<span style="mso-spacerun: yes;">
</span>26.2777778<span style="mso-spacerun: yes;"> </span>30.2777778<span style="mso-spacerun: yes;"> </span>14.2777778 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>50<span style="mso-spacerun: yes;">
</span>51<span style="mso-spacerun: yes;"> </span>52<span style="mso-spacerun: yes;"> </span>53<span style="mso-spacerun: yes;"> </span>54<span style="mso-spacerun: yes;"> </span>55<span style="mso-spacerun: yes;"> </span>56 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>14.2777778<span style="mso-spacerun: yes;">
</span>26.2777778<span style="mso-spacerun: yes;"> </span>14.6219512<span style="mso-spacerun: yes;"> </span>10.6219512<span style="mso-spacerun: yes;">
</span>22.6219512<span style="mso-spacerun: yes;"> </span>30.6219512<span style="mso-spacerun: yes;"> </span>14.6219512 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>57<span style="mso-spacerun: yes;"> </span>58<span style="mso-spacerun: yes;"> </span>59<span style="mso-spacerun: yes;"> </span>60<span style="mso-spacerun: yes;"> </span>61<span style="mso-spacerun: yes;"> </span>62<span style="mso-spacerun: yes;"> </span>63 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>34.6219512<span style="mso-spacerun: yes;">
</span>23.0454545<span style="mso-spacerun: yes;"> </span>3.0454545<span style="mso-spacerun: yes;"> </span>1.3076923<span style="mso-spacerun: yes;">
</span>-4.9545455 -21.7222222<span style="mso-spacerun: yes;"> </span>-5.3780488
<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>64<span style="mso-spacerun: yes;"> </span>65<span style="mso-spacerun: yes;"> </span>66<span style="mso-spacerun: yes;"> </span>67<span style="mso-spacerun: yes;"> </span>68<span style="mso-spacerun: yes;"> </span>69<span style="mso-spacerun: yes;"> </span>70 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>-5.3780488 -17.3780488 -14.6923077
-13.3780488<span style="mso-spacerun: yes;"> </span>-8.9545455 -13.3780488<span style="mso-spacerun: yes;"> </span>-5.7222222 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>71<span style="mso-spacerun: yes;">
</span>72<span style="mso-spacerun: yes;"> </span>73<span style="mso-spacerun: yes;"> </span>74<span style="mso-spacerun: yes;"> </span>75<span style="mso-spacerun: yes;"> </span>76<span style="mso-spacerun: yes;"> </span>77 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;">-10.6923077<span style="mso-spacerun: yes;"> </span>-1.3780488<span style="mso-spacerun: yes;">
</span>-1.3780488<span style="mso-spacerun: yes;"> </span>-1.3780488<span style="mso-spacerun: yes;"> </span>-1.3780488<span style="mso-spacerun: yes;">
</span>-5.3780488<span style="mso-spacerun: yes;"> </span>-5.3780488 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>78<span style="mso-spacerun: yes;"> </span>79<span style="mso-spacerun: yes;"> </span>80<span style="mso-spacerun: yes;"> </span>81<span style="mso-spacerun: yes;"> </span>82<span style="mso-spacerun: yes;"> </span>83<span style="mso-spacerun: yes;"> </span>84 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;">-29.3780488<span style="mso-spacerun: yes;"> </span>-9.3780488 -33.3780488 -13.3780488<span style="mso-spacerun: yes;"> </span>2.6219512 -21.3780488<span style="mso-spacerun: yes;"> </span>-9.3780488 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>85<span style="mso-spacerun: yes;"> </span>86<span style="mso-spacerun: yes;"> </span>87<span style="mso-spacerun: yes;"> </span>88<span style="mso-spacerun: yes;"> </span>89<span style="mso-spacerun: yes;"> </span>90<span style="mso-spacerun: yes;"> </span>91 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>-1.3780488 -16.0000000 -12.9545455 -20.6250000
-25.3780488<span style="mso-spacerun: yes;"> </span>-8.9545455<span style="mso-spacerun: yes;"> </span>7.0454545 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;">
</span>92<span style="mso-spacerun: yes;"> </span>93<span style="mso-spacerun: yes;"> </span>94<span style="mso-spacerun: yes;"> </span>95<span style="mso-spacerun: yes;"> </span>96<span style="mso-spacerun: yes;"> </span>97<span style="mso-spacerun: yes;"> </span>98 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>-4.9545455 -13.3780488 -25.7222222<span style="mso-spacerun: yes;"> </span>2.2777778<span style="mso-spacerun: yes;">
</span>-1.3780488 -16.9545455<span style="mso-spacerun: yes;"> </span>14.6219512
<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 9.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>99<span style="mso-spacerun: yes;"> </span>100<span style="mso-spacerun: yes;"> </span>101<span style="mso-spacerun: yes;"> </span>102<span style="mso-spacerun: yes;"> </span>103<span style="mso-spacerun: yes;"> </span>104<span style="mso-spacerun: yes;"> </span>105 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span></span><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;">16.0000000<span style="mso-spacerun: yes;"> </span>-5.7222222<span style="mso-spacerun: yes;">
</span>49.3076923<span style="mso-spacerun: yes;"> </span>-2.7222222
-13.7222222<span style="mso-spacerun: yes;"> </span>15.3750000 -18.7222222 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>106<span style="mso-spacerun: yes;"> </span>107<span style="mso-spacerun: yes;"> </span>108<span style="mso-spacerun: yes;"> </span>109<span style="mso-spacerun: yes;"> </span>110<span style="mso-spacerun: yes;"> </span>111<span style="mso-spacerun: yes;"> </span>112 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;">-15.9545455<span style="mso-spacerun: yes;"> </span>-8.9545455<span style="mso-spacerun: yes;">
</span>6.0454545<span style="mso-spacerun: yes;"> </span>-7.9545455<span style="mso-spacerun: yes;"> </span>9.0454545<span style="mso-spacerun: yes;">
</span>9.0454545<span style="mso-spacerun: yes;"> </span>-5.6923077 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>113<span style="mso-spacerun: yes;"> </span>114<span style="mso-spacerun: yes;"> </span>115<span style="mso-spacerun: yes;"> </span>116<span style="mso-spacerun: yes;"> </span>117<span style="mso-spacerun: yes;"> </span>118<span style="mso-spacerun: yes;"> </span>119 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>1.3076923<span style="mso-spacerun: yes;">
</span>12.3076923 -17.6923077<span style="mso-spacerun: yes;"> </span>-0.6923077
-13.6923077<span style="mso-spacerun: yes;"> </span>0.3076923<span style="mso-spacerun: yes;"> </span>-9.7222222 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>120<span style="mso-spacerun: yes;"> </span>121<span style="mso-spacerun: yes;"> </span>122<span style="mso-spacerun: yes;"> </span>123<span style="mso-spacerun: yes;"> </span>124<span style="mso-spacerun: yes;"> </span>125<span style="mso-spacerun: yes;"> </span>126 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;">-14.6250000<span style="mso-spacerun: yes;"> </span>10.3750000<span style="mso-spacerun: yes;">
</span>11.3750000<span style="mso-spacerun: yes;"> </span>-9.6250000
-11.6250000<span style="mso-spacerun: yes;"> </span>-0.7222222<span style="mso-spacerun: yes;"> </span>-5.7222222 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>127<span style="mso-spacerun: yes;"> </span>128<span style="mso-spacerun: yes;"> </span>129<span style="mso-spacerun: yes;"> </span>130<span style="mso-spacerun: yes;"> </span>131<span style="mso-spacerun: yes;"> </span>132<span style="mso-spacerun: yes;"> </span>133 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>31.0454545 -11.3780488 -11.3780488<span style="mso-spacerun: yes;"> </span>47.6219512<span style="mso-spacerun: yes;">
</span>0.6219512<span style="mso-spacerun: yes;"> </span>16.6219512<span style="mso-spacerun: yes;"> </span>-7.3780488 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>134<span style="mso-spacerun: yes;"> </span>135<span style="mso-spacerun: yes;"> </span>136<span style="mso-spacerun: yes;"> </span>137<span style="mso-spacerun: yes;"> </span>138<span style="mso-spacerun: yes;"> </span>139<span style="mso-spacerun: yes;"> </span>140 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>26.6219512<span style="mso-spacerun: yes;">
</span>5.6219512 -24.3780488<span style="mso-spacerun: yes;"> </span>3.6219512
-18.3780488<span style="mso-spacerun: yes;"> </span>26.6219512<span style="mso-spacerun: yes;"> </span>13.6219512 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>141<span style="mso-spacerun: yes;"> </span>142<span style="mso-spacerun: yes;"> </span>143<span style="mso-spacerun: yes;"> </span>144<span style="mso-spacerun: yes;"> </span>145 <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 8.0pt; line-height: 115%;">-15.3780488
-16.3780488 -18.3780488 -24.3780488<span style="mso-spacerun: yes;">
</span>-5.3780488<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<br /></div>
<div class="MsoListParagraphCxSpLast" style="background: white; line-height: normal; margin-bottom: 7.5pt; margin-left: 36.0pt; margin-right: 0cm; margin-top: 15.0pt; mso-add-space: auto; mso-list: l2 level1 lfo4; mso-outline-level: 3; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="ES-GT" style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 18.0pt;"><span style="mso-list: Ignore;">7.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES-GT" style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 18.0pt;">Verificación de Independencia de los residuos<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 7.5pt;">
<span lang="ES-GT" style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 12.0pt;">Este supuesto necesita <span style="mso-spacerun: yes;"> </span>que la probabilidad de que el residuo de una
observación cualquiera tenga un determinado valor, no debe depender de los
valores de los otros residuos. Al momento de aleatorizar, debió controlarse, pero
se realiza la prueba de Durbin Watson (DW) para tener la seguridad del
cumplimiento de este supuesto,<o:p></o:p></span></div>
<div class="MsoListParagraph">
<br /></div>
<div style="background: whitesmoke; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">durbinWatsonTest</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(Andeva1)</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
</div>
<div class="MsoListParagraphCxSpFirst">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">lag
Autocorrelation D-W Statistic p-value<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span></span><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">1<span style="mso-spacerun: yes;"> </span>0.1110966<span style="mso-spacerun: yes;"> </span>1.775952<span style="mso-spacerun: yes;">
</span>0.128<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>Alternative hypothesis: rho != 0<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<br /></div>
<div class="MsoListParagraphCxSpMiddle">
<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Debido
a que el valor es cercano a 2, y que la significancia es mayor que 0.01 y 0.05,
se concluye que los datos no están correlacionados, con lo que se verifica el
supuesto (LOS RESIDUOS SON INDEPENDIENTES)<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpLast">
<br /></div>
<h3 style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 15.0pt;">
<span lang="ES-GT" style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 18.0pt; font-weight: normal;">Distribución normal
de los residuos<o:p></o:p></span></h3>
<div style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm;">
<span style="color: #333333; font-family: "helvetica" , "sans-serif";">Este
supuesto dice que los residuos deben tener distribución normal estándar (media
cero y varianza 1). A continuación, se calcula y presenta una curva de densidad
observada de las frecuencias de los residuos y el gráfico de probabilidad
normal, tomando en cuenta que la curva de densidad observada se debería parecer
a una <span style="mso-spacerun: yes;"> </span>campana de Gauss, y que, en el
gráfico de probabilidad normal, los cuantiles observados de los residuos de la
variable nota <span style="mso-spacerun: yes;"> </span>(representados por los
puntos negros) se aproximan a la línea central que representa los cuantiles de
una distribución normal teórica. Para tener la certeza del cumplimiento de este
supuesto se realiza la prueba de Shapiro Wilk.<o:p></o:p></span></div>
<div style="background: whitesmoke; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">par</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">mfrow</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">c</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="number"><span lang="EN-US" style="color: #009999; mso-ansi-language: EN-US;">1</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">,</span></code><span class="number"><span lang="EN-US" style="color: #009999; mso-ansi-language: EN-US;">2</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">))</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">dplot</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;"><-</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">density</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">residuos</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">)</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">plot</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(</span></span><span class="identifier"><span lang="ES-GT" style="color: black;">dplot</span></span><code><span lang="ES-GT" style="color: #333333;">,<o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="ES-GT" style="color: #333333;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="ES-GT" style="color: black;">main</span></span><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><span class="string"><span lang="ES-GT" style="color: #dd1144;">"Curva de densidad observada"</span></span><code><span lang="ES-GT" style="color: #333333;">, <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="ES-GT" style="color: #333333;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="ES-GT" style="color: black;">xlab</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="string"><span lang="ES-GT" style="color: #dd1144;">"Residuos"</span></span><code><span lang="ES-GT" style="color: #333333;">, <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="ES-GT" style="color: #333333;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="ES-GT" style="color: black;">ylab</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="string"><span lang="ES-GT" style="color: #dd1144;">"Densidad"</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">)</span></span><code><span lang="ES-GT" style="color: #333333;"> <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">polygon</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(</span></span><span class="identifier"><span lang="ES-GT" style="color: black;">dplot</span></span><code><span lang="ES-GT" style="color: #333333;">, <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span style="color: #333333; mso-ansi-language: ES;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">col</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"red"</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">border</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"black"</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">)</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p> </o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">qqPlot</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">Residuos</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">,<o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="ES-GT" style="color: black;">pch</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><span class="number"><span lang="ES-GT" style="color: #009999;">20</span></span><code><span lang="ES-GT" style="color: #333333;">, <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="ES-GT" style="color: #333333;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="ES-GT" style="color: black;">main</span></span><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><span class="string"><span lang="ES-GT" style="color: #dd1144;">"Gráfica QQ-Plot de los residuos"</span></span><code><span lang="ES-GT" style="color: #333333;">, <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="ES-GT" style="color: #333333;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="ES-GT" style="color: black;">xlab</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="string"><span lang="ES-GT" style="color: #dd1144;">"Cuantiles teóricos"</span></span><code><span lang="ES-GT" style="color: #333333;">,<span style="mso-spacerun: yes;"> </span><o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="ES-GT" style="color: #333333;"><span style="mso-spacerun: yes;"> </span></span></code><span class="identifier"><span lang="ES-GT" style="color: black;">ylab</span></span><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><span class="string"><span lang="ES-GT" style="color: #dd1144;">"Cuantiles observados de los residuos"</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">)</span></span><code><span lang="ES-GT" style="color: #333333;"> <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">mtext</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">side</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="number"><span lang="EN-US" style="color: #009999; mso-ansi-language: EN-US;">3</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">at</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">par</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"usr"</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">)[</span></span><span class="number"><span lang="EN-US" style="color: #009999; mso-ansi-language: EN-US;">1</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">]</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">adj</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="number"><span lang="EN-US" style="color: #009999; mso-ansi-language: EN-US;">0.7</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">cex</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="number"><span lang="EN-US" style="color: #009999; mso-ansi-language: EN-US;">0.6</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">col</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"gray40"</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">line</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=-</span></span><span class="number"><span lang="EN-US" style="color: #009999; mso-ansi-language: EN-US;">21</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">ext</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">paste</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"Facultad de Farmcia Usac"</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">format</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">Sys.time</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">()</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, <o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span></span></code><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"%d/%m/%Y %H:%M:%S --"</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">)) </span></span><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><v:shape id="Imagen_x0020_1" o:spid="_x0000_i1026" style="height: 437.25pt; mso-wrap-style: square; visibility: visible; width: 424.5pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\MATEYF~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png">
</v:imagedata></v:shape></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoListParagraphCxSpLast">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Se observa que en los extremos (punteos
bajos y punteos altos) los puntos se apartan de la línea recta, y algunos fuera
de los límites. Para asegurarse es necesario realizar la prueba de normalidad,
en éste caso Shapiro y Wilks<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL7G8SPkhmNvnIb1IkeAsI9Pyl8L74-aoUl_QvX5ltzcJ6UxzZqSqB0stzgVBWGnJkZ5cNn8cYu4bYjtCiYgjJbOLBY07qd-PQ6G1Ty100NCCROG496vpysSx6B5RqlLtCo72kciL3VC4/s1600/supaov2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="583" data-original-width="566" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL7G8SPkhmNvnIb1IkeAsI9Pyl8L74-aoUl_QvX5ltzcJ6UxzZqSqB0stzgVBWGnJkZ5cNn8cYu4bYjtCiYgjJbOLBY07qd-PQ6G1Ty100NCCROG496vpysSx6B5RqlLtCo72kciL3VC4/s320/supaov2.png" width="310" /></a></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div style="background: whitesmoke; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">shapiro.test</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">residuals</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(Andeva1))</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-tab-count: 1;"> </span>Shapiro-Wilk normality test<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">data:<span style="mso-spacerun: yes;"> </span>residuals(Andeva1)<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">W =
0.96628, p-value = 0.001232<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpMiddle">
<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Debido a que el valor de p es menor que
0.05 y 0.01, se rechaza la hipótesis nula (los datos tienen distribución normal)
y se acepta la alterna (los datos no tienen distribución normal).<o:p></o:p></span></div>
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<br /></div>
<h3 style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 15.0pt;">
<span lang="ES-GT" style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 18.0pt; font-weight: normal;">Homogeneidad de
varianzas.<o:p></o:p></span></h3>
<div style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm;">
<span style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 10.5pt;">Este es el supuesto más importante que los residuos deben
cumplir para que el modelo empleado sea válido.<o:p></o:p></span></div>
<div style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm;">
<span style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 10.5pt;">En siguiente gráfica (graf1) se debería observar los valores
predichos por el modelo para la variable NOTA <span style="mso-spacerun: yes;"> </span>contra la raíz cuadrada de los residuos
estandarizados, donde no se observa ninguna tendencia en la distribución de
dichos valores lo cual implica en que no existen indicios de que no se cumple el
supuesto de homogeneidad de varianzas, pero para corroborar esto se realiza una
prueba de Cochran o una de Levine<o:p></o:p></span></div>
<div style="background: whitesmoke; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-element: para-border-div; padding: 7.0pt 7.0pt 7.0pt 7.0pt;">
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">graf1</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;"><-</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">ggplot</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(Andeva1</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">aes</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">.fitted</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">sqrt</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">abs</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">.stdresid</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">))))</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">+</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">geom_point</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">na.rm</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="literal"><span lang="EN-US" style="color: #990073; mso-ansi-language: EN-US;">TRUE</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">)</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">graf1</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;"><-graf1+</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">stat_smooth</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">method</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"loess"</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;">, </span></code><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">na.rm</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">=</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"> </span></code><span class="literal"><span lang="EN-US" style="color: #990073; mso-ansi-language: EN-US;">TRUE</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">)</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">+</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">xlab</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(</span></span><span class="string"><span lang="EN-US" style="color: #dd1144; mso-ansi-language: EN-US;">"Valores predichos"</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">)</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">graf1</span></span><span class="operator"><span lang="ES-GT" style="color: #687687;"><-graf1+</span></span><span class="identifier"><span lang="ES-GT" style="color: black;">ylab</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(</span></span><span class="identifier"><span lang="ES-GT" style="color: black;">expression</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(</span></span><span class="identifier"><span lang="ES-GT" style="color: black;">sqrt</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(</span></span><span class="string"><span lang="ES-GT" style="color: #dd1144;">"|Residuos estandarizados|"</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">)))</span></span><code><span lang="ES-GT" style="color: #333333;"><o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">graf1</span></span><span class="operator"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;"><-graf1+</span></span><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">theme_bw</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">()</span></span><code><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></code></pre>
<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="EN-US" style="color: black; mso-ansi-language: EN-US;">plot</span></span><span class="paren"><span lang="EN-US" style="color: #687687; mso-ansi-language: EN-US;">(graf1)</span></span><span lang="EN-US" style="color: #333333; mso-ansi-language: EN-US;"><o:p></o:p></span></pre>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif6f31TGQFDFrnHFv11cbnSu8Vn761HVf9Gn1EdJgnKI5np4aXqdR49bCjEf9MoSi2YwNQy9gIi_eq7fyYEnsvFMcc4HdUYXNIJMBqYgemdzlOaiODPXmKl307t93Nv6ETt2lwJN8tLX4/s1600/supaqov3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="456" data-original-width="467" height="312" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif6f31TGQFDFrnHFv11cbnSu8Vn761HVf9Gn1EdJgnKI5np4aXqdR49bCjEf9MoSi2YwNQy9gIi_eq7fyYEnsvFMcc4HdUYXNIJMBqYgemdzlOaiODPXmKl307t93Nv6ETt2lwJN8tLX4/s320/supaqov3.png" width="320" /></a></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><o:p><br /></o:p></span></div>
<div style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm;">
<span style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 10.5pt;">La prueba de C de Cocharn, es una prueba unilateral del límite
superior atípico de la varianza. La prueba C se utiliza para determinar si una
sola estimación de una varianza es significativamente más grande que un grupo
de varianzas, se presenta el resultado de la prueba de C de Cochran para la
variable porcentaje de parasitismo. En este caso no es posible usarla porque el
modelo es desbalanceado (diferentes repeticiones por tratamiento)<o:p></o:p></span></div>
<div style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm;">
<span style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 10.5pt;">Una alternativa a la prueba C de Cochran <strong><span style="font-family: "helvetica" , "sans-serif";">es la prueba de </span></strong>Levene**<o:p></o:p></span></div>
<div style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm;">
<span style="color: #333333; font-family: "helvetica" , "sans-serif"; font-size: 10.5pt;">La prueba de <strong><span style="font-family: "helvetica" , "sans-serif";">Levene</span></strong> consiste
en realizar un análisis de la varianza usando como variable dependiente el
valor absoluto de los residuos.<o:p></o:p></span></div>
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<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">leveneTest</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(NOTA</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span class="operator"><span lang="ES-GT" style="color: #687687;">~</span></span><code><span lang="ES-GT" style="color: #333333;"> TITULO</span></code><span class="paren"><span lang="ES-GT" style="color: #687687;">)</span></span><code><span lang="ES-GT" style="color: #333333;"> </span></code><span lang="ES-GT" style="color: #333333;"><o:p></o:p></span></pre>
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<h4 style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 7.5pt;">
<span style="color: #333333; font-family: "inherit" , "serif"; font-size: 13.5pt; font-weight: normal; line-height: 115%;">Resultados del test<o:p></o:p></span></h4>
<h4 style="background: white; margin-bottom: 7.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 7.5pt;">
<span style="color: #333333; font-family: "inherit" , "serif"; font-size: 13.5pt; font-weight: normal; line-height: 115%;">Para solicitar el coeficiente de variación<o:p></o:p></span></h4>
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<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">cv.model</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(Andeva1)</span></span><span lang="ES-GT" style="color: #333333;"><o:p></o:p></span></pre>
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<span lang="ES-GT" style="color: #333333; font-family: "inherit" , "serif"; font-size: 18.0pt; font-weight: normal;">Para solicitar la prueba de Tukey al 99% de confianza<o:p></o:p></span></h3>
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<pre style="background: whitesmoke; border: none; margin-bottom: 7.5pt; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 7.0pt 7.0pt 7.0pt 7.0pt; padding: 0cm; word-break: break-all;"><span class="identifier"><span lang="ES-GT" style="color: black;">LTukey</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">(Andeva1</span></span><code><span lang="ES-GT" style="color: #333333;">, </span></code><span class="identifier"><span lang="ES-GT" style="color: black;">which</span></span><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><span class="string"><span lang="ES-GT" style="color: #dd1144;">"tratamiento"</span></span><code><span lang="ES-GT" style="color: #333333;">, </span></code><span class="identifier"><span lang="ES-GT" style="color: black;">conf.level</span></span><span class="operator"><span lang="ES-GT" style="color: #687687;">=</span></span><span class="number"><span lang="ES-GT" style="color: #009999;">0.99</span></span><span class="paren"><span lang="ES-GT" style="color: #687687;">)</span></span><span lang="ES-GT" style="color: #333333;"><o:p></o:p></span></pre>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">CONCLUSIONES:<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Los residuos son independientes<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Los residuos no tienen distribución normal, pero
tomando en cuenta que lo que interesa es comparar las medias, la violación del supuesto
no se considera crítica para seguir con el análisis<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">Las varianzas son homogéneas<o:p></o:p></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span lang="ES-GT" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">4. EL ANÁLISIS DE VARIANZA POR LA VÍA ORDINARIA ES VÁLIDO, LO MISMO QUE LAS CONCLUSIONES</span></div>
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<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-4739943526078768112018-04-22T13:17:00.000-06:002018-04-22T13:17:04.624-06:00ANÁLISIS DE CONGLOMERADOS (CLUSTER) USANDO PAST: EL CASO DE LAS HAMBURGUESAS DE MCDONALDS®<br />
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<b style="mso-bidi-font-weight: normal;"><span style="font-size: 16.0pt; line-height: 107%;">ANÁLISIS DE CONGLOMERADOS (CLUSTER)
USANDO PAST: <o:p></o:p></span></b></div>
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<span style="font-size: 16.0pt; line-height: 107%;">EL CASO DE
LAS HAMBURGUESAS DE MCDONALDS®<o:p></o:p></span></div>
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<span style="font-size: 16.0pt; line-height: 107%;">Ing. Luis
Manfredo Reyes<o:p></o:p></span></div>
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Introducción:<o:p></o:p></div>
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En el análisis estadístico tradicional, se analizan una por
una las variables de interés, o a lo sumo se elaboran pruebas de hipótesis para
cada variable. Pero cuando se trata de más de dos variables, la situación se
hace muy difícil de manejar manualmente.-<o:p></o:p></div>
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De todos los métodos que se han desarrollado, hay uno que
permite agrupar los datos, mediante la medición de las distancias entre los
individuos y la creación de una gráfica (Dendograma)<o:p></o:p></div>
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La técnica se denomina Análisis de Conglomerados (Cluster). <o:p></o:p></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br /></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">El objetivo de éste artículo no es profundizar en los detalles del
análisis manual de la técnica, sino definir cómo se realiza en el paquete
estadístico PAST<o:p></o:p></span></div>
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<span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Past fue desarrollado por Øyvind Hammer,
de la Universidad de Oslo y el museo de historia natural. </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
<span style="background: white;"><br />
Past es un paquete gratuito (No es software libre) <span style="mso-spacerun: yes;"> </span>para análisis de datos, graficación,
manipulación de datos, análisis descriptivo, análisis inferencial, análisis
multivariado, análisis especializado para ecología, análisis especializado para
arqueología (series de tiempo, análisis espacial, morfometría y estratigrafía).</span><br />
</span></div>
<a name='more'></a><span style="background: white;"><br />
Actualmente, la última versión disponible es la 3.2, que puede ser descargada
en el siguiente enlace: </span><a href="http://folk.uio.no/ohammer/past/"><span style="background: white; color: #888888; font-family: "arial" , "sans-serif"; font-size: 10.0pt; text-decoration: none;">http://folk.uio.no/ohammer/past/</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
<span style="background: white;"><br />
Se dispone de una versión para Windows y una preliminar para Macintosh.</span><br />
<span style="background: white;"><br />
La descripción del programa es la siguiente:</span><br />
<span style="background: white;">- Es gratuito</span><br />
<span style="background: white;">-Todo el programa está contenido en un sólo
archivo (past.exe)</span><br />
<span style="background: white;">-No necesita instalador, basta copiarlo al disco
de la computadora y hacer doble click sobre el ícono de un caracol para que
funcione.</span><br />
<span style="background: white;">Se pueden importar archivos de texto y de Excel (R)</span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span><br />
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Ejemplo:<o:p></o:p></span></div>
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La siguiente tabla muestra el contenido nutricional de los
productos más conocidos de la marca de Hamburgesas McDonalds ® (fuente: <a href="https://www.zonadiet.com/tablas/mcdonalds.htm">https://www.zonadiet.com/tablas/mcdonalds.htm</a>)<o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWtxzA6ZO8sJ1oAnpEcECd3nG0Wxk0AFcKZr-N_QOihmp01Y8wKgzgQjdpv3Jp8eFLNOG6OHfxNGgQxRJK0KfRdUA2Lkjjmr5GfAvzzAuDtKTQchL7V6Zt1oeu8dnen9z519Sh1sGMYC4/s1600/clus1info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="208" data-original-width="725" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWtxzA6ZO8sJ1oAnpEcECd3nG0Wxk0AFcKZr-N_QOihmp01Y8wKgzgQjdpv3Jp8eFLNOG6OHfxNGgQxRJK0KfRdUA2Lkjjmr5GfAvzzAuDtKTQchL7V6Zt1oeu8dnen9z519Sh1sGMYC4/s640/clus1info.png" width="640" /></a></div>
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Tabla 1: Contenido nutricional de productos McDonalds®<o:p></o:p></div>
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NA significa información no disponible<o:p></o:p></div>
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<b style="mso-bidi-font-weight: normal;">McDonalds es marca
registrada de McDonalds International <o:p></o:p></b></div>
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<b style="mso-bidi-font-weight: normal;">EN ESTE EJEMPLO NO SE
DISCUTE LA CALIDAD NUTRICIONAL DE LOS PRODUCTOS, SINO EL GRADO DE PARECIDO O
DIFERENCIACION QUE PRESENTAN, DE ACUERDO A LOS DATOS DE LA TABLA<o:p></o:p></b></div>
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Una práctica usual en el trabajo con PAST, es importar los
datos de un archivo de Excel®. Sin embargo, siendo pocos datos, también es
posible crear manualmente el archivo. Los datos pueden estar en formato CsV
(separados por comas), o bien xls. (archivos de Excel 2007).<o:p></o:p></div>
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<o:p> </o:p> </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjckDlSnz1qeIxC96lmRUADx00iTczcAIoEJMTWbr1zENkFRyOEN2lVYYViEl1bOl__a0dOG05lg1jWXKKQHMFtpiK4y_0zlTvVvjKIxQuta64wmt3n9Oo8xdkFNWdpuniflvg0v9NhXIU/s1600/past1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="414" data-original-width="801" height="329" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjckDlSnz1qeIxC96lmRUADx00iTczcAIoEJMTWbr1zENkFRyOEN2lVYYViEl1bOl__a0dOG05lg1jWXKKQHMFtpiK4y_0zlTvVvjKIxQuta64wmt3n9Oo8xdkFNWdpuniflvg0v9NhXIU/s640/past1.png" width="640" /></a></div>
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<span lang="EN-US" style="mso-ansi-language: EN-US;"><o:p> Se da click en la pesta</o:p></span>ña File y se escoge Open</div>
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<span lang="EN-US" style="mso-ansi-language: EN-US;"><o:p>Luego se especifica el nombre y tipo de archivo</o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjubX3ZXcTiqmGUzimEJnoCIxfaLqg_JLtAnLfvgcMZ1dC823rNtF-gxroPvw3-mG4BJG4UJtkYxYFL-J78xzpZgQWiKKkrU8PEWF3dXhmK3mXdJ_Gi3FZoB5UemkrIcTJ454mu57ftRQ/s1600/past3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="486" data-original-width="685" height="283" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjubX3ZXcTiqmGUzimEJnoCIxfaLqg_JLtAnLfvgcMZ1dC823rNtF-gxroPvw3-mG4BJG4UJtkYxYFL-J78xzpZgQWiKKkrU8PEWF3dXhmK3mXdJ_Gi3FZoB5UemkrIcTJ454mu57ftRQ/s400/past3.png" width="400" /></a></div>
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<span lang="EN-US" style="mso-ansi-language: EN-US;"><o:p>Se deben definir las opciones de importación: cada fila contiene nombres y datos y cada columna nombres y datos . El separador es la coma</o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFOnIk1f3w-Vb9h5Wo-602LZT7mDy6nt8l9YNPqdw52xwvF5iZkc9DGnApzilGMJa1dxJIXB-gHZKT3zRuk0gluYFBfXJg-HEX0J9ef7_xe9HxuR3ToOLJHb7zRq67PBkTaZdexugl7FA/s1600/past4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="346" data-original-width="471" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFOnIk1f3w-Vb9h5Wo-602LZT7mDy6nt8l9YNPqdw52xwvF5iZkc9DGnApzilGMJa1dxJIXB-gHZKT3zRuk0gluYFBfXJg-HEX0J9ef7_xe9HxuR3ToOLJHb7zRq67PBkTaZdexugl7FA/s320/past4.png" width="320" /></a></div>
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El archivo ya importado es el siguiente:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB9XXFVLKlFFz39Ad7xu3oXNrB6F_-QhztyH3Lyja3jS1VLxKsJVG9fStRHg3Rc0CJfmbsWsR75yBjvkbg49fwgWbvmW9t1V-ebd47oDLe7QaYGKDpYiRhjeCejqx9kPgp-CjhdQVUiao/s1600/past5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="349" data-original-width="800" height="278" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB9XXFVLKlFFz39Ad7xu3oXNrB6F_-QhztyH3Lyja3jS1VLxKsJVG9fStRHg3Rc0CJfmbsWsR75yBjvkbg49fwgWbvmW9t1V-ebd47oDLe7QaYGKDpYiRhjeCejqx9kPgp-CjhdQVUiao/s640/past5.png" width="640" /></a></div>
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Nótese que en la variable colesterol hay dos datos faltantes. Para poder realizar el análisis, se deben estimar los datos faltantes.<br />
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Una forma de hacerlo es colocar la media de la variable.<br />
Se marca la columna de colesterol<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfmH9FeBKoyiWv1G5fRCaJuWI9RRZCeas_UmcilYUv-Kf-Vlm-g8mlO2Y9dnpmlUSHDVePC-x0p4Rf2a9wYfa5ibO4qA7F0t_qSeoEC4vQzeYzW6os01qlVopn83d0VjVoIkTGTEfLiCE/s1600/past6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="331" data-original-width="622" height="212" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfmH9FeBKoyiWv1G5fRCaJuWI9RRZCeas_UmcilYUv-Kf-Vlm-g8mlO2Y9dnpmlUSHDVePC-x0p4Rf2a9wYfa5ibO4qA7F0t_qSeoEC4vQzeYzW6os01qlVopn83d0VjVoIkTGTEfLiCE/s400/past6.png" width="400" /></a></div>
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Se ingresa a la pestaña Univariate y se escoge: descriptive<br />
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Los datos se ingresan en los lugares faltantes<br />
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Y ahora se procede a especificar el análisis:<br />
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El individuo que más se diferencia de los demás es La hamburguesa. El Cuarto de libra con queso, el McRoyal y el Mc Royal con queso forman un grupo que después se une al big Mac.</div>
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Los dos grandes grupos que se forman son: Los productos "mas pesados" en contenidos, y los "menos pesados". </div>
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El programa permite tres métodos de cálculo y varias formas de medición de distancias<br />
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-13935514606400918792018-04-17T11:46:00.000-06:002018-04-17T11:50:45.836-06:00<br />
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<span style="font-family: "arial" , "sans-serif"; font-size: 16.5pt;">ANÁLISIS DE CONGLOMERADOS
(CLUSTER) CON INFOSTAT: EL CASO DE LAS HAMBURGUESAS DE McDONALDS®<o:p></o:p></span></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">ANÁLISIS
DE CONGLOMERADOS (CLUSTER) CON INFOSTAT: EL CASO DE LAS HAMBURGUESAS DE
McDONALDS®</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br />
Ing. Luis Manfredo Reyes<br />
<br />
El análisis de conglomerados (cluster), es una técnica multivariada, en la cual
en base a un conjunto de características cuantitativas, se realiza una
agrupación de individuos en base a las distancias entre ellos.<br />
</span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">
Al final del análisis, se obtiene un diagrama llamado DENDROGRAMA, que resume
gráficamente el agrupamiento.<br style="mso-special-character: line-break;" />
<!--[if !supportLineBreakNewLine]--><br style="mso-special-character: line-break;" />
<!--[endif]--></span><span style="background: white; color: #222222; font-family: "verdana" , "sans-serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">No es el objetivo de éste documento profundizar en la teoría del método, sino ejemplificar su análisis en Infostat.</span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;"></span><br />
<a name='more'></a><span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
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<span style="background: white; color: #222222; font-family: "verdana" , "sans-serif"; font-size: 12.0pt;">Infostat es un
paquete estadístico producido en la Universidad de Córdoba, Argentina, que
tiene la ventaja de estar escrito en español, con versión comercial y libre. Se
puede obtener n: </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><a href="http://www.infostat.com.ar/"><span style="background: white; color: #bb5521; font-family: "verdana" , "sans-serif"; text-decoration: none;">www.infostat.com.ar</span></a><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "verdana" , "sans-serif"; font-size: 10.0pt;">Se asume que el lector tiene
instalado el paquete y conoce el uso básico del mismo.<o:p></o:p></span></div>
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<a href="https://www.blogger.com/null" name="more"></a><span style="font-family: "times new roman" , "serif"; font-size: 10.0pt;"><br />
OJO: EN LA VERSIÓN LIBRE DE INFOSTAT SÓLAMENTE SE PUEDE TRABAJAR CON 9
VARIABLES. EN LA VERSIÓN PAGADA NO HAY LÍMITE.<br style="mso-special-character: line-break;" />
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La siguiente tabla muestra el contenido nutricional de los
productos más conocidos de la marca de Hamburgesas McDonalds ® (fuente: <a href="https://www.zonadiet.com/tablas/mcdonalds.htm">https://www.zonadiet.com/tablas/mcdonalds.htm</a>)<o:p></o:p></div>
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Tabla 1: Contenido nutricional de productos McDonalds®<o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmL3l6HIDP49i8DTEyjfVUmBdNZxlYgMuR2kiIA624waStXudUgF6cji7no0f6VOAyoKtE1P-VK3uoRZfJsguD59W6e_LV5qWfLhN6ezi5swlH_ZKRdKNQpsAd8bDCiUYPfwCRcicV8dA/s1600/clus1info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="208" data-original-width="725" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmL3l6HIDP49i8DTEyjfVUmBdNZxlYgMuR2kiIA624waStXudUgF6cji7no0f6VOAyoKtE1P-VK3uoRZfJsguD59W6e_LV5qWfLhN6ezi5swlH_ZKRdKNQpsAd8bDCiUYPfwCRcicV8dA/s640/clus1info.png" width="640" /></a></div>
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UN ESPACIO EN BLANCO significa información no disponible</div>
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<o:p></o:p></div>
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<b style="mso-bidi-font-weight: normal;">McDonalds es marca
registrada de McDonalds International <o:p></o:p></b></div>
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<b style="mso-bidi-font-weight: normal;">EN ESTE EJEMPLO NO SE
DISCUTE LA CALIDAD NUTRICIONAL DE LOS PRODUCTOS, SINO EL GRADO DE PARECIDO O
DIFERENCIACION QUE PRESENTAN, DE ACUERDO A LOS DATOS DE LA TABLA<o:p></o:p></b></div>
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Una práctica usual en el trabajo con INFOSTAT, es importar los
datos de un archivo de Excel®. Sin embargo, siendo pocos datos, también es
posible crear manualmente el archivo.En este ejemplo, el archivo es de texto, separado por comas, creado en Excel, y se encuentra en una carpeta llamada "Chernoff" de la unidad E<o:p></o:p></div>
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1) Importar el archivo a Infostat</div>
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<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;">Se ingresa: Archivo-->Abrir</span><span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;"> </span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhULqgL8HzypVWfp4A-KRlRAnGsW8zzfkksmP8O6Fvt9dPPMtcU1x5sdSAymAeHc2EdqyDlohuYOx0QD0r8Avzz5DO-t25xwPL0B3bpmdMYvQJ_5Gp1ePD-V3PcJZZsJWpwOC3pzNagCAE/s1600/c1_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="231" data-original-width="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhULqgL8HzypVWfp4A-KRlRAnGsW8zzfkksmP8O6Fvt9dPPMtcU1x5sdSAymAeHc2EdqyDlohuYOx0QD0r8Avzz5DO-t25xwPL0B3bpmdMYvQJ_5Gp1ePD-V3PcJZZsJWpwOC3pzNagCAE/s1600/c1_info.png" /></a></div>
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<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;">Pregunta si la primera fila de datos contiene los nombres de las variables y se marca la casilla </span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_o-oaEOq9rscTuF2gxO5fHx8RTW-tKz1-cQk0MX9vvqJe6tova41xM63V2BsE7ELN7oF-9MZIo8jLLJhr0MP8IkXS8De3yAc2ukgwXf0nt44xoig-Dd4a9dBFzlFglPLHunxGyr7GnhU/s1600/c2_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="428" data-original-width="572" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_o-oaEOq9rscTuF2gxO5fHx8RTW-tKz1-cQk0MX9vvqJe6tova41xM63V2BsE7ELN7oF-9MZIo8jLLJhr0MP8IkXS8De3yAc2ukgwXf0nt44xoig-Dd4a9dBFzlFglPLHunxGyr7GnhU/s320/c2_info.png" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNf1jei3GbY5FN0bel3SGTVRBklTBW9faLdVpNQLwXKafM4tJn19UGqEqZ3woE41kYBL3mFr4ZUBGFlGoFyAtKXC4eKfVEx4seRaT_ctw1pC4riVXRdU3NAydHeifMxLbbWgXI161T-zU/s1600/c3_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="399" data-original-width="528" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNf1jei3GbY5FN0bel3SGTVRBklTBW9faLdVpNQLwXKafM4tJn19UGqEqZ3woE41kYBL3mFr4ZUBGFlGoFyAtKXC4eKfVEx4seRaT_ctw1pC4riVXRdU3NAydHeifMxLbbWgXI161T-zU/s320/c3_info.png" width="320" /></a></div>
<br /></div>
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2) Luego se ingresa Estadísticas, Análisis Multivariado y Análisis de Conglomerados<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_PBhxJIPOY5VTmtA1YniRHF2lrCNAAN0GuoZOQkreS8E9mmS-vDrBEQ7YBzLA54Y1lU6GgSXJ_WrZZiv6ggqbroK7g1W8sFuLiat7ADkiZjQ3wY8G0DnH-iyVayz-PHjzwWI-CSq0RE8/s1600/c4_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="385" data-original-width="640" height="192" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_PBhxJIPOY5VTmtA1YniRHF2lrCNAAN0GuoZOQkreS8E9mmS-vDrBEQ7YBzLA54Y1lU6GgSXJ_WrZZiv6ggqbroK7g1W8sFuLiat7ADkiZjQ3wY8G0DnH-iyVayz-PHjzwWI-CSq0RE8/s320/c4_info.png" width="320" /></a></div>
<br />
Se ingresan las variables que se usarán en el proceso<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCM84TwR5YDm0kEKQOFaSasRZLuyfR1t5KyTHPspYVTr0UWPlFteZX0a03Z2YkjJ2M7RPljaSv0TBYrMgOpD04HjalFZTPAmaVTjn6lMeO2JDAHO4pndFa49xHhMXhsU2jqpevwbmgE-A/s1600/c8_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="515" data-original-width="501" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCM84TwR5YDm0kEKQOFaSasRZLuyfR1t5KyTHPspYVTr0UWPlFteZX0a03Z2YkjJ2M7RPljaSv0TBYrMgOpD04HjalFZTPAmaVTjn6lMeO2JDAHO4pndFa49xHhMXhsU2jqpevwbmgE-A/s320/c8_info.png" width="311" /></a></div>
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</div>
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</div>
Se ingresa información sobre el método de análisis, el tipo de distancia ysi se incluyen registros con datos faltantes<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVPqZZ8Rau3ALBXO7YlhzWE0oRPhoj6ken-TevkwGOoQqk5bVArOkHZqH_ZtoZqDJUzZR_6Xzc1TD96Gfzq7OUW1DtNckuYh_Iw52BZpPCQk5_e2AJmmMm78asKvIRTPSBIa7xDK7Z7wE/s1600/c9_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="508" data-original-width="614" height="264" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVPqZZ8Rau3ALBXO7YlhzWE0oRPhoj6ken-TevkwGOoQqk5bVArOkHZqH_ZtoZqDJUzZR_6Xzc1TD96Gfzq7OUW1DtNckuYh_Iw52BZpPCQk5_e2AJmmMm78asKvIRTPSBIa7xDK7Z7wE/s320/c9_info.png" width="320" /></a></div>
Se quita la marca donde dice Sólo registros completos<br />
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<b>Variante 1: método en agrupamiento promedio, y distancia euclidiana</b></div>
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Los resultados producidos son:</div>
Y al ingresar Aceptar, se produce lo sigiente:<br />
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<a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsbNr_3hOXxXsMbT8m20eyB5UwuuTWCFYhM6mIWzbcabusb-VJ1_Zg8Zre3I7E1IbSAp003TT5PI1PXDbzWA-WzgieOb1qAjR-KekUY4gm1e2rYjenvJNelw6NHg81mpaEuKyvCd7Mwq8/s1600/c10_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="401" data-original-width="363" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsbNr_3hOXxXsMbT8m20eyB5UwuuTWCFYhM6mIWzbcabusb-VJ1_Zg8Zre3I7E1IbSAp003TT5PI1PXDbzWA-WzgieOb1qAjR-KekUY4gm1e2rYjenvJNelw6NHg81mpaEuKyvCd7Mwq8/s320/c10_info.png" width="289" /></a></div>
<br /></div>
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<br /></div>
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<o:p></o:p></div>
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<b style="mso-bidi-font-weight: normal;">Variante 2: método de
agrupamiento promedio y distancia euclidiana al cuadrado<o:p></o:p></b></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghthjmVN7XjjxxgZoSihJnLQUTx_PoomSwhD0ZuXNcw_iuYwat_ldRKjnb5dlHTo_KjwiiIlJgFoeOEQOaEq3xPwOUVKVRBOCfGaV1TKfocVn665aCrPUWJBtEgJ9vhCoeEXrd9fnMH9A/s1600/c11_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="396" data-original-width="360" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghthjmVN7XjjxxgZoSihJnLQUTx_PoomSwhD0ZuXNcw_iuYwat_ldRKjnb5dlHTo_KjwiiIlJgFoeOEQOaEq3xPwOUVKVRBOCfGaV1TKfocVn665aCrPUWJBtEgJ9vhCoeEXrd9fnMH9A/s320/c11_info.png" width="290" /></a></div>
<br /></div>
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<b style="mso-bidi-font-weight: normal;">Variante 3: Método de
Ward y distancia euclidiana<o:p></o:p></b></div>
<div class="MsoNormal">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH7aza1St96D4pcU1i77zVe8A_NAqWkGeZn37sm9CmWAdhdrMYbjHgW97cwrBpRQtJ-_D5OteGqx0ULdnqKU7MVTHI2CfYiFhefyYtofAmw1UUbhsQ0HucG6jiuVsfBs3-wEvpzM2q1GA/s1600/c12_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="395" data-original-width="355" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH7aza1St96D4pcU1i77zVe8A_NAqWkGeZn37sm9CmWAdhdrMYbjHgW97cwrBpRQtJ-_D5OteGqx0ULdnqKU7MVTHI2CfYiFhefyYtofAmw1UUbhsQ0HucG6jiuVsfBs3-wEvpzM2q1GA/s320/c12_info.png" width="287" /></a></div>
<br /></div>
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Variante 4: Método de Ward y distancia euclidiana al
cuadrado<o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmwPImXBq9Cs7bmrlK1A-huxpu3mf4gZQlqjVjxaTVI9iStkVhvg_QAjTcmaLBWv0_l4TYL0Ge8tz-_W8Yor-YgjgF9OyD5Szq6FOSK5zORkszWVTxii65CD6S9YYmtTBQHVGY2vbJ6Gc/s1600/c13_info.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="395" data-original-width="358" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmwPImXBq9Cs7bmrlK1A-huxpu3mf4gZQlqjVjxaTVI9iStkVhvg_QAjTcmaLBWv0_l4TYL0Ge8tz-_W8Yor-YgjgF9OyD5Szq6FOSK5zORkszWVTxii65CD6S9YYmtTBQHVGY2vbJ6Gc/s320/c13_info.png" width="290" /></a></div>
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<br /></div>
Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-11304493774312213242018-04-09T11:39:00.000-06:002018-04-24T11:15:46.075-06:00ANÁLISIS DE CONGLOMERADOS PARA DATOS MULTIVARIADOS CON R:EL CASO DE LAS HAMBURGUESAS DE MCDONALDS®<br />
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span style="font-size: 16.0pt; line-height: 107%;">ANÁLISIS DE CONGLOMERADOSPARA DATOS
MULTIVARIADOS CON R: <o:p></o:p></span></b></div>
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<span style="font-size: 16.0pt; line-height: 107%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">EL CASO DE
LAS HAMBURGUESAS DE MCDONALDS®<o:p></o:p></span></div>
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<span style="font-size: 16.0pt; line-height: 107%;">Ing. Luis
Manfredo Reyes<o:p></o:p></span></div>
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<br /></div>
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Introducción:<o:p></o:p></div>
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En el análisis estadístico tradicional, se analizan una por
una las variables de interés, o a lo sumo se elaboran pruebas de hipótesis para
cada variable. Pero cuando se trata de más de dos variables, la situación se
hace muy difícil de manejar manualmente.-<o:p></o:p></div>
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De todos los métodos que se han desarrollado, hay uno que
permite agrupar los datos, mediante la medición de las distancias entre los
individuos y la creación de una gráfica (Dendograma)<o:p></o:p></div>
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<br /></div>
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La técnica se denomina Análisis de Conglomerados (Cluster). <o:p></o:p></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br /></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif;">El objetivo de éste artículo no es profundizar en los detalles del
análisis manual de la técnica, sino definir cómo se realiza en el paquete
estadístico R.<o:p></o:p></span></div>
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<br />
<a name='more'></a></div>
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Debido a que últimamente se ha popularizado el uso del
Lenguaje R en el mundo académico, no es de extrañar que exista la posibilidad
de realizar éste análisis con el mismo.<o:p></o:p></div>
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<br /></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #222222; font-family: "arial" , sans-serif;">El objetivo de éste artículo no es profundizar en los detalles del
análisis manual de la técnica, sino definir cómo se realiza en el paquete
estadístico R.<span style="font-size: 10pt;"><o:p></o:p></span></span></div>
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<br /></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">R es actualmente el programa estadístico más utilizado en el mundo
académico y de investigación, primero por su potencia de procesamiento, y en segundo
lugar por ser un software libre.<o:p></o:p></span></div>
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<br /></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">R es un software libre, que puede ser descargado del siguiente sitio:<o:p></o:p></span></div>
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<a href="http://www.r-project.org/"><span style="background: white; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">http://www.r-project.org/</span></a><span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Se asume que los lectores tienen conocimientos de cómo se utiliza el programa
R<o:p></o:p></span></div>
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Se ilustra el uso del mismo a partir de un ejemplo<o:p></o:p></div>
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<br /></div>
<div class="MsoNormal">
Es posible determinar gráficamente, cuales son los
individuos más parecidos o más distintos, mediante la aplicación de la técnica
de conglomerados (cluster)<o:p></o:p></div>
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<br /></div>
<div class="MsoNormal">
La siguiente tabla muestra el contenido nutricional de los
productos más conocidos de la marca de Hamburgesas McDonalds ® (fuente: <a href="https://www.zonadiet.com/tablas/mcdonalds.htm">https://www.zonadiet.com/tablas/mcdonalds.htm</a>)<o:p></o:p></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjPdIOhdSNAh-7CXbPc_k5RIpC08InjBhujVva-EHq06qMkBoS-80qewxVJ2I-Hk0IlH5Yc3wW4tm5PaCMN59lzTC9ytpzg7krSvtPQPYFzhpFPCkdf6Pcsq-BtSy5HBuBAeCiEbgkiEQ/s1600/cara2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="291" data-original-width="639" height="289" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjPdIOhdSNAh-7CXbPc_k5RIpC08InjBhujVva-EHq06qMkBoS-80qewxVJ2I-Hk0IlH5Yc3wW4tm5PaCMN59lzTC9ytpzg7krSvtPQPYFzhpFPCkdf6Pcsq-BtSy5HBuBAeCiEbgkiEQ/s640/cara2.png" width="640" /></a></div>
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<br /></div>
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<br /></div>
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NA significa información no disponible<o:p></o:p></div>
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<b style="mso-bidi-font-weight: normal;">McDonalds es marca
registrada de McDonalds International <o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
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<b style="mso-bidi-font-weight: normal;">EN ESTE EJEMPLO NO SE
DISCUTE LA CALIDAD NUTRICIONAL DE LOS PRODUCTOS, SINO EL GRADO DE PARECIDO O
DIFERENCIACION QUE PRESENTAN, DE ACUERDO A LOS DATOS DE LA TABLA<o:p></o:p></b></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><br /></b></div>
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<b style="mso-bidi-font-weight: normal;">SE APLICARÁ EN LOS EJERCICIOS EL MÉTODO JERÁRQUICO</b></div>
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<b style="mso-bidi-font-weight: normal;"><br /></b></div>
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Una práctica usual en el trabajo con R, es importar los
datos de un archivo de Excel®. Sin embargo, siendo pocos datos, también es
posible crear manualmente el archivo.<o:p></o:p></div>
<br />
<div class="MsoNormal">
1) Importar el archivo a R: (EL ARCHIVO EN ESTE EJEMPLO ESTÁ
EN FORMATO CSV)<o:p></o:p></div>
<div class="MsoNormal">
<span style="font-size: 14.0pt; line-height: 107%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-size: 14.0pt; line-height: 107%;"><b>Mcdonalds
<- read.table("E:/CHERNOFF/MACDONALDS.csv", header=TRUE,
sep=",", na.strings="NA", dec=".",
strip.white=TRUE)<o:p></o:p></b></span></div>
<div class="MsoNormal">
<span style="font-size: 14.0pt; line-height: 107%;"><b><br /></b></span></div>
<div class="MsoNormal">
4) Solicitar el análisis los datos de comparación están en
las columnas de la 2 a la 6 y la identificación de cada variable en la columna
1<o:p></o:p><br />
<br />
EN ÉSTA TÉCNICA, LOS INDIVIDUOS QUE TIENEN DATOS FALTANTES SON EXCLUÍDOS<br />
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Variante 1: método en
agrupamiento promedio, y distancia euclidiana<o:p></o:p></b></div>
<div class="MsoNormal">
HClust.1 <- hclust(dist(model.matrix(~-1+
Colesterol+Grasas+Hidratos+Kilocalorias+Porcion+Proteinas+Sodio, Mcdonalds)) ,
method= "average")<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
plot(HClust.1, main= "Cluster Dendrogram for Solution
HClust.1", xlab= <o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;"> </span>"Observation
Number in Data Set Mcdonalds", sub="Method=average;
Distance=euclidian")<o:p></o:p></div>
<div class="MsoNormal">
Los resultados producidos son:<o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEtCdxqcfdyKfHHOi0VsUsK9vGdA7wfIs2tbBzbOOtNanbKC5qXY4kVAIORJDVaVEHQK1oa-Nuno4RocW_f9TftNBjeprcAKyKMKUAm_3lPiOhxVquyVvtccKu5YFxiUuSv4Y0WY0lqIU/s1600/cluster+1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="697" data-original-width="683" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEtCdxqcfdyKfHHOi0VsUsK9vGdA7wfIs2tbBzbOOtNanbKC5qXY4kVAIORJDVaVEHQK1oa-Nuno4RocW_f9TftNBjeprcAKyKMKUAm_3lPiOhxVquyVvtccKu5YFxiUuSv4Y0WY0lqIU/s400/cluster+1.png" width="391" /></a></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><br /></b></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Interpretación: </b></div>
<div class="MsoNormal">
El individuo que más se diferencia de los demás es el Big Mac. El Cuarto de libra con queso y el McRoyal forman un grupo que después se une al big Mac.</div>
<div class="MsoNormal">
El cuarto de libra y el McPollo forman otro grupo, que luego se une a la quesoburguesa y posteriormente a la hamburguesa.</div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><br /></b></div>
<div class="MsoNormal">
Los dos grandes grupos que se forman son: Los productos "mas pesados" en contenidos, y los "menos pesados". </div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><br /></b></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Variante 2: método de
agrupamiento promedio y distancia euclidiana al cuadrado<o:p></o:p></b></div>
<div class="MsoNormal">
HClust.2 <- hclust(dist(model.matrix(~-1 + <o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;">
</span>Colesterol+Grasas+Hidratos+Kilocalorias+Porcion+Proteinas+Sodio,
Mcdonalds))^2 , method= "average")<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
plot(HClust.2, main= "Cluster Dendrogram for Solution
HClust.2", xlab= <o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;"> </span>"Observation
Number in Data Set Mcdonalds", sub="Method=average;
Distance=squared-euclidian")<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUUxmlzkjN4oFyMwnJ0_Oui713NKdbU-NDZum5wTUKhfeyuw7wZLg1IspviQzQ117AUMlYkhOVT09-Gjx9Q7SjUaLvjT1uAnWT1dNeFGjtLp8tUFP-8KaOq8ERFwCRQQYUbc1URe_OEaA/s1600/cluster+2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="709" data-original-width="681" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUUxmlzkjN4oFyMwnJ0_Oui713NKdbU-NDZum5wTUKhfeyuw7wZLg1IspviQzQ117AUMlYkhOVT09-Gjx9Q7SjUaLvjT1uAnWT1dNeFGjtLp8tUFP-8KaOq8ERFwCRQQYUbc1URe_OEaA/s400/cluster+2.png" width="383" /></a></div>
<div class="MsoNormal">
<o:p><br /></o:p></div>
<div class="MsoNormal">
<o:p>No hay variantes en los resultados, comparado con la variante 1</o:p></div>
<div class="MsoNormal">
<o:p><br /></o:p></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Variante 3: Método de
Ward y distancia euclidiana<o:p></o:p></b></div>
<div class="MsoNormal">
HClust.3 <- hclust(dist(model.matrix(~-1 + <o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;">
</span>Colesterol+Grasas+Hidratos+Kilocalorias+Porcion+Proteinas+Sodio,
Mcdonalds)) , method= "ward")<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
plot(HClust.3, main= "Cluster Dendrogram for Solution
HClust.3", xlab= <o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;"> </span>"Observation
Number in Data Set Mcdonalds", sub="Method=ward;
Distance=euclidian")<o:p></o:p></div>
<div class="MsoNormal">
<o:p><br /></o:p></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXf0w19EP0JKBwNDi6b-23f95-iHH7Ig3WYIvBPXea-q2GoFx9U-0s8pLRVD7GzN_eakHwJDPAgH6e_Dgz3_xFkvTsE3uwz3UWJWAgfq4Q6v_Kor1bmGZWTavjd9P49Xgcv4il1i7QHQo/s1600/cluster3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="717" data-original-width="681" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXf0w19EP0JKBwNDi6b-23f95-iHH7Ig3WYIvBPXea-q2GoFx9U-0s8pLRVD7GzN_eakHwJDPAgH6e_Dgz3_xFkvTsE3uwz3UWJWAgfq4Q6v_Kor1bmGZWTavjd9P49Xgcv4il1i7QHQo/s320/cluster3.png" width="303" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<o:p>No hay variantes en los resultados, comparado con la variante 1</o:p></div>
<div class="MsoNormal">
<o:p><br /></o:p></div>
<div class="MsoNormal">
Variante 4: Método de Ward y distancia euclidiana al
cuadrado<o:p></o:p></div>
<div class="MsoNormal">
HClust.4 <- hclust(dist(model.matrix(~-1 + <o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;">
</span>Colesterol+Grasas+Hidratos+Kilocalorias+Porcion+Proteinas+Sodio,
Mcdonalds))^2 , method= "ward")<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
plot(HClust.4, main= "Cluster Dendrogram for Solution
HClust.4", xlab= <o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;"> </span>"Observation
Number in Data Set Mcdonalds", sub="Method=ward;
Distance=squared-euclidian")<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwwhlIvyR0i0EamoQSe9B0ZxC7pRqU1KB8m6x8sdcWZtAVWjB29wxBFhVPQASd7VrBtH7p4K4cAgWbAmLBynfMqdfluxdSorp3YYYboa5k0yaZRrdhRXg2Hm-KQYcEVO9oVLkkqrrmPZI/s1600/cluster4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="707" data-original-width="687" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwwhlIvyR0i0EamoQSe9B0ZxC7pRqU1KB8m6x8sdcWZtAVWjB29wxBFhVPQASd7VrBtH7p4K4cAgWbAmLBynfMqdfluxdSorp3YYYboa5k0yaZRrdhRXg2Hm-KQYcEVO9oVLkkqrrmPZI/s320/cluster4.png" width="310" /></a></div>
<br />
La única variante en este caso es que las distancias a las que se agrupan son menores.<br />
La interpretación es la misma que en la variante 1<br />
<br />
Hay otras opciones que se pueden experimentar para observar los cambios que se producen en la gráfica.<br />
<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-67711396628042714942018-04-06T09:22:00.000-06:002018-04-06T09:22:31.852-06:00ANÁLISIS GRÁFICO DE DATOS MULTIVARIADOS CON R: CARITAS DE CHERNOFF<br />
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span style="font-size: 16.0pt; line-height: 107%;">ANÁLISIS GRÁFICO DE DATOS
MULTIVARIADOS CON R: <o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">CARITAS DE
CHERNOFF <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;"><br /></span></div>
<div class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">Ing. Luis
Manfredo Reyes<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Introducción:<o:p></o:p></div>
<div class="MsoNormal">
En el análisis estadístico tradicional, se grafican una por
una las variables de interés, o a lo sumo se elaboran gráficas de dispersión.
Pero cuando se trata de más de dos variables, la situación se hace muy difícil
de manejar.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
De todos los métodos que se han desarrollado, hay uno que
permite comparar grupos de datos, de una forma gráfica: Las caritas de Chernoff.<o:p></o:p></div>
<div class="MsoNormal">
</div>
<a name='more'></a><br />
<div class="MsoNormal">
Herman Chernoff,<span style="mso-spacerun: yes;"> </span>es
un matemático norteamericano, nacido en 1923. <o:p></o:p></div>
<div class="MsoNormal">
Las caritas de Chernoff son un método gráfico en el cual,
ciertas características cuantitativas de un grupo, se asocian con datos físicos
de la cara de una persona, con lo cual es posible realizar un dibujo que
representa dichas características, y realizar comparaciones.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Por ejemplo:<o:p></o:p></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTPICL5Q25rQP30hnoNqfS8U_Ckg-XaOkC7kmOqG5CgRwvLmdf-05MvUHeZ9FmEaqiZJRB-NXfrXk0SnipTO2LW5qLSaiBiAQ-UGlVAiI18JXFiwxDkJHbClBWFFJRIKaPga7MXN6YUqM/s1600/cara1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="300" data-original-width="589" height="162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTPICL5Q25rQP30hnoNqfS8U_Ckg-XaOkC7kmOqG5CgRwvLmdf-05MvUHeZ9FmEaqiZJRB-NXfrXk0SnipTO2LW5qLSaiBiAQ-UGlVAiI18JXFiwxDkJHbClBWFFJRIKaPga7MXN6YUqM/s320/cara1.png" width="320" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Fuente: https://flowingdata.com/2010/08/31/how-to-visualize-data-with-cartoonish-faces/<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Es posible determinar gráficamente, cuales son los
individuos más parecidos o más distintos, sin necesidad de realizar
comparaciones numéricas.<o:p></o:p></div>
<div class="MsoNormal">
En su diseño original, chernoff usó 18 variables asociadas a
características faciales.<o:p></o:p></div>
<div class="MsoNormal">
Debido a que últimamente se ha popularizado el uso del
Lenguaje R en el mundo académico, no es de extrañar que exista la posibilidad
de realizar éste análisis con el mismo.<o:p></o:p></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">El objetivo de éste artículo no es profundizar en los detalles del
análisis manual de la técnica, sino definir cómo se realiza en el paquete
estadístico R.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">R es actualmente el programa estadístico más utilizado en el mundo
académico y de investigación, primero por su potencia de procesamiento, y en
segundo lugar por ser un software libre.<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">R es un software libre, que puede ser descargado del siguiente sitio:<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<a href="http://www.r-project.org/"><span style="background: white; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">http://www.r-project.org/</span></a><span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Se asume que los lectores tienen conocimientos de cómo se utiliza el
programa R<o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br /></span></div>
<div class="MsoNormal">
Se ilustra el uso del mismo a partir de un ejemplo<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
La siguiente tabla muestra el contenido nutricional de los
productos más conocidos de la marca de Hamburgesas McDonalds ® (fuente: <a href="https://www.zonadiet.com/tablas/mcdonalds.htm">https://www.zonadiet.com/tablas/mcdonalds.htm</a>)<o:p></o:p></div>
<div class="MsoNormal">
Tabla 1: Contenido nutricional de productos McDonalds®<o:p></o:p></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; margin-left: .25pt; mso-padding-alt: 0cm 3.5pt 0cm 3.5pt; mso-yfti-tbllook: 1184; width: 601px;">
<tbody>
<tr style="height: 15.0pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td nowrap="" style="height: 15.0pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 117.0pt;" valign="bottom" width="156"><div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<br /></div>
</td>
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<b><span style="color: black; font-family: "arial" , "sans-serif"; font-size: 8.0pt;"><o:p><br /></o:p></span></b></div>
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<b><span style="color: black; font-family: "arial" , "sans-serif"; font-size: 8.0pt;"><o:p><br /></o:p></span></b></div>
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</td><td style="background: white; border-left: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 48.0pt;" width="64"></td><td style="background: white; border-left: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 55.0pt;" width="73"></td><td style="background: white; border-left: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 45.0pt;" width="60"><br /></td><td style="background: white; border-left: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 41.0pt;" width="55"></td><td style="background: white; border-left: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 49.0pt;" width="65"></td><td style="background: white; border-left: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 49.0pt;" width="65"></td><td style="background: white; border-left: none; border: solid windowtext 1.0pt; height: 15.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 3.5pt 0cm 3.5pt; width: 47.0pt;" width="63"><br /></td></tr>
</tbody></table>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><o:p></o:p></b></div>
<div class="MsoNormal">
<br />
El objetivo es aplicar la metodología de Chernoff, con el
objeto de determinar gráficamente el parecido o las diferencias entre los
productos.<o:p></o:p></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><br /></b></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">EN ESTE EJEMPLO NO SE
DISCUTE LA CALIDAD NUTRICIONAL DE LOS PRODUCTOS, SINO EL GRADO DE PARECIDO O
DIFERENCIACION QUE PRESENTAN, DE ACUERDO A LOS DATOS DE LA TABLA<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Una práctica usual en el trabajo con R, es importar los
datos de un archivo de Excel®. Sin embargo, siendo pocos datos, también es
posible crear manualmente el archivo.<o:p></o:p></div>
<div class="MsoNormal">
NOTA: Se requiere el uso del paquete “aplpack”, que es una
extensión de R, disponible en el sitio de R.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoListParagraph" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">11)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->En caso de no estar instalado el paquete, se
ejecuta el siguiente procedimiento:<o:p></o:p></div>
<div class="MsoNormal">
<span style="font-size: 22.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>install.packages(“aplpack”)<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
2)<span style="mso-spacerun: yes;"> </span>Luego se carga la
librería antes de definir el proceso:<o:p></o:p></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;"> </span><span style="font-size: 22.0pt; line-height: 107%;">library(aplpack)<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
3) Importar el archivo a R: (EL ARCHIVO EN ESTE EJEMPLO ESTÁ
EN FORMATO CSV)<o:p></o:p></div>
<div class="MsoNormal">
<span style="font-size: 14.0pt; line-height: 107%;">Mcdonalds
<- read.table("E:/CHERNOFF/MACDONALDS.csv", header=TRUE,
sep=",", na.strings="NA", dec=".",
strip.white=TRUE)<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
4) Solicitar el análisis los datos de comparación están en
las columnas de la 2 a la 6 y la identificación de cada variable en la columna
1<o:p></o:p></div>
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<span style="font-size: 18.0pt; line-height: 107%;">Caras<-faces(Mcdonalds[2:6])<o:p></o:p></span></div>
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<br /></div>
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Los resultados producidos son:<o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguK0n5bQ-95eTOL37b-teFEHMwN2zL0cGvhYZK_7lKkDcyyQJjuoriEpiEPHhT4mFMdfg2gcugESiU76ATFroUUjE2duahayLWeEWtQo3fX3WNh0cvV2HiliGY747NptkibLrCJ2WlxV8/s1600/cara3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="339" data-original-width="549" height="246" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguK0n5bQ-95eTOL37b-teFEHMwN2zL0cGvhYZK_7lKkDcyyQJjuoriEpiEPHhT4mFMdfg2gcugESiU76ATFroUUjE2duahayLWeEWtQo3fX3WNh0cvV2HiliGY747NptkibLrCJ2WlxV8/s400/cara3.png" width="400" /></a></div>
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<br /></div>
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Una tabla donde muestra qué características de la cara fueron
asignadas a las variables. <o:p></o:p></div>
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En los casos donde no existe dato (NA), el programa
reemplaza el dato faltante con la media de los demás datos<o:p></o:p></div>
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<br /></div>
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En la segunda salida, se muestran las caritas producidas
para cada producto:<o:p></o:p></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEsQ0tlmpP3EJXIJJbm96aOuMgL7g7G-86LbMyp0C1mMUu41H6S_YYzYR7YfvPyru2v0D0hoiuL4HcsGbQqIsXNAtHI1ncut2XZUPnCooWPybmOoZ6vMpyE3soCS3GagGw_keu4xOvQH0/s1600/cara4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="532" data-original-width="576" height="368" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEsQ0tlmpP3EJXIJJbm96aOuMgL7g7G-86LbMyp0C1mMUu41H6S_YYzYR7YfvPyru2v0D0hoiuL4HcsGbQqIsXNAtHI1ncut2XZUPnCooWPybmOoZ6vMpyE3soCS3GagGw_keu4xOvQH0/s400/cara4.png" width="400" /></a></div>
<br />
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INTERPRETACION:<o:p></o:p></div>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">1)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->El producto que es distinto a los demás es la
hamburguesa simple. (tiene los contenidos más bajos de nutrientes).<o:p></o:p></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">2)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->Los productos con mayor cantidad de calorías más
parecidos (altura de la cara) , son : el cuarto de libra con queso, el Big Mac
y la McNífica<o:p></o:p></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">3)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->En el contenido de grasas (ancho de la cara) los
más parecidos son el Big Mac y el Cuarto de Libra con Queso<o:p></o:p></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">4)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->En el contenido de proteína (altura de la boca)
el más bajo es la Hamburguesa simple y los mayores son cuarto de libra con
queso y Mc royal Bacon con queso<o:p></o:p></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">5)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->En el contenido de colesterol (ancho de la
boca), el que menos tiene es la hamburguesa simple y los que más el cuarto de
libra con queso y el Mc Royal.<o:p></o:p></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">6)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->En el contenido de sodio (sonrisa), el Mc Pollo
es el que tiene menos y los que más tienen son el cuarto de libra con queso y
el Mc royal Bacon con queso<o:p></o:p></div>
<div class="MsoListParagraphCxSpLast" style="mso-list: l1 level1 lfo2; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;">7)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->En base a todas las características, las tres
hamburguesas más parecidas son: Quarter pounder con queso, Big Mac y Mc Royal <o:p></o:p></div>
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La función faces no permite asignar las variables a las
características, pero en todo caso es muy fácil de interpretar.<o:p></o:p></div>
<div class="MsoNormal">
También ocurre que el programa repite la asignación de
variables a otras características (orejas, ojos, pelo). Pero eso tampoco cambia
la interpretación.<o:p></o:p></div>
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<br /></div>
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OTRAS OPCIONES:<o:p></o:p></div>
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CARITA ESTILO SANTA CLAUS<o:p></o:p></div>
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<span style="font-size: 18.0pt; line-height: 107%;">cara<-faces(Mcdonalds[,2:7],face.type=2)<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWO-gM2S4A24bqW3NCUQSdu7i3x6ws-7MbvG6yhnLuai3I8QAsL9w9xVY3YxZ0MTPS_lTf_yWz5MuLjQTNqAUg-wsKumKR0QgaQJHMIvyLwfGhH-oVMsjv6lWd3txAxXoELxypeSCbf4I/s1600/cara5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="558" data-original-width="543" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWO-gM2S4A24bqW3NCUQSdu7i3x6ws-7MbvG6yhnLuai3I8QAsL9w9xVY3YxZ0MTPS_lTf_yWz5MuLjQTNqAUg-wsKumKR0QgaQJHMIvyLwfGhH-oVMsjv6lWd3txAxXoELxypeSCbf4I/s400/cara5.png" width="388" /></a></div>
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<br /></div>
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<span style="color: black; font-family: "courier new"; font-size: 10.0pt;">CARITAS EN BLANCO Y NEGRO<o:p></o:p></span></div>
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<span style="color: black; font-family: "courier new"; font-size: 16.0pt;">cara<-faces(Mcdonalds[,2:7],face.type=0)<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzoNZ2JcQZMt1evqZUem8NMkFMcoWTO7cc9pEvRjYdB0018XgF-hiWYqfGq3ZFIpCAy1L0rravKhkK0va-kcQil4nXcCWYyWrqJrKOHCdZSBw5p5hxCiXs4_V_gIf9AaTHPsn-h8exjc0/s1600/cara6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="540" data-original-width="579" height="372" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzoNZ2JcQZMt1evqZUem8NMkFMcoWTO7cc9pEvRjYdB0018XgF-hiWYqfGq3ZFIpCAy1L0rravKhkK0va-kcQil4nXcCWYyWrqJrKOHCdZSBw5p5hxCiXs4_V_gIf9AaTHPsn-h8exjc0/s400/cara6.png" width="400" /></a></div>
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<span style="color: black; font-family: "courier new"; font-size: 10.0pt;">Bibliografía:<o:p></o:p></span></div>
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<span style="background: white; color: black; font-size: 13.5pt;">Chernoff, H. (1973): The
use of faces to represent statistiscal assoziation, JASA, 68, pp 361–368. <o:p></o:p></span></div>
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<br /></div>
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<span style="background: white; color: black; font-size: 13.5pt;">Las caritas son generadas
por un algoritmo definido en Ralston, A. and Rabinowitz, P. (1985): A first
course in numerical analysis, McGraw-Hill, pp 76ff. </span><a href="http://www.wiwi.uni-bielefeld.de/mitarbeiter/wolf/"><span style="background: white; color: purple; font-size: 13.5pt;">http://www.wiwi.uni-bielefeld.de/mitarbeiter/wolf/</span></a><span style="background: white; color: black; font-size: 13.5pt;"> : S/R - functions :
faces</span><span style="color: black; font-family: "courier new"; font-size: 10.0pt;"><o:p></o:p></span></div>
<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-44229522890337001732017-11-24T11:48:00.000-06:002017-11-24T15:26:55.279-06:00SOLUCIÓN DE SISTEMAS NO LINEALES DE ECUACIONES CON GEOGEBRA<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 0cm; margin-right: 0cm; margin-top: 9.0pt; mso-outline-level: 3;">
<a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.5pt;">SOLUCIÓN DE SISTEMAS NO LINEALES DE ECUACIONES CON GEOGEBRA<o:p></o:p></span><br />
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.5pt;"><br /></span></div>
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<a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
</span><span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;">Ing. Luis Manfredo
Reyes</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
</span><span style="color: #222222; font-family: "verdana" , "sans-serif"; font-size: 10.0pt;"><br />
<span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">He recibido una consulta donde me preguntan
¿cómo se pueden resolver sistemas no lineales de ecuaciones.</span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
</span><span style="color: #222222; font-family: "verdana" , "sans-serif"; font-size: 10.0pt;"><br />
<span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">Un sistema no lineal de ecuaciones es aquel
donde por lo menos una de sus ecuaciones no es de primer grado.</span></span></div>
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<span style="color: #222222; font-size: 10pt;"><span style="font-family: "verdana" , sans-serif;"></span></span></div>
<a name='more'></a><span style="font-family: "verdana" , sans-serif;"><br /></span>
<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;">Por ejemplo:</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhN_U0uHgk01RrrYZZBRJrsKh9pbdeyfCEbMMWaQPN0y_N_NOjnt5U06EnOcWAzKQGN40j_sShl2atBkPJPDyACs2Dnx5ogVc841ZaJLIoQDSnuboWBGil1ChZ2wWkrdjiDpCfGPUOHwbU/s1600/snl.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="163" data-original-width="324" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhN_U0uHgk01RrrYZZBRJrsKh9pbdeyfCEbMMWaQPN0y_N_NOjnt5U06EnOcWAzKQGN40j_sShl2atBkPJPDyACs2Dnx5ogVc841ZaJLIoQDSnuboWBGil1ChZ2wWkrdjiDpCfGPUOHwbU/s320/snl.png" width="320" /></a></div>
<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;"> </span><br />
<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;"><br /></span>
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<span style="color: #222222; font-family: "verdana" , "sans-serif"; font-size: 10.0pt;">En la
resolución manual de éste tipo de sistemas, se utiliza el método de sustitución
o el método gráfico.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "verdana" , "sans-serif"; font-size: 10.0pt;">Ahora se
procederá a resolver éste tipo de sistemas mediante Geogebra<o:p></o:p></span></div>
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<b><span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">GeoGebra</span></b><span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> es un software matemático. Fue creado con la idea de ser usado
para la educación en colegios y universidades. Fue creado por Markus
Hohenwarter, en el año 2001 en la </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><a href="https://es.wikipedia.org/wiki/Universidad_de_Salzburgo" title="Universidad de Salzburgo"><span style="color: #0b0080; font-size: 10.5pt;">Universidad de Salzburgo</span></a></span><span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, luego en la
Universidad de Atlantic, Florida, luego en la </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><a href="https://es.wikipedia.org/wiki/Universidad_Estatal_de_Florida" title="Universidad Estatal de Florida"><span style="color: #0b0080; font-size: 10.5pt;">Universidad Estatal de
Florida</span></a></span><span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> y actualmente , en la </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><a href="https://es.wikipedia.org/w/index.php?title=Universidad_de_Linz&action=edit&redlink=1" title="Universidad de Linz (aún no redactado)"><span style="color: #a55858; font-size: 10.5pt;">Universidad de Linz</span></a></span><span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, Austria.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">GeoGebra está escrito en </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n_Java" title="Lenguaje de programación Java"><span style="color: #0b0080; font-size: 10.5pt;">Java</span></a></span><span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> y por tanto
es posible usarlo en diferentes plataformas.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">El software tiene dos partes importantes: un procesador geométrico y un
procesador algebraico.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">GeoGebra permite el trazado de construcciones geométricas de todo tipo
así como la representación gráfica, el tratamiento algebraico y el cálculo de
funciones reales de variable real, sus derivadas, integrales, etc.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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</span><span style="color: #222222; font-family: "georgia" , "serif"; font-size: 18.0pt;">DISPONIBILIDAD:</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "georgia" , "serif"; font-size: 10.0pt;">El software puede ser descargado sin costo del sitio: </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><a href="http://www.geogebra.org/"><span style="color: #888888; font-family: "georgia" , serif; text-decoration-line: none;">www.geogebra.org</span></a></span><span style="color: #222222; font-family: "georgia" , "serif"; font-size: 10.0pt;"> </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "georgia" , "serif"; font-size: 10.0pt;">Existen versiones para: Windows, Linux, MacIntosh, Android e Ios.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "georgia" , "serif"; font-size: 10.0pt;">Existen algunas diferencias entre las versiones, pero la mayoría de
aspectos son iguales.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "georgia" , "serif"; font-size: 10.0pt;">La última versión disponible es la 5.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES-TRAD" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt; letter-spacing: -0.1pt;">En éste artículo se muestran aplicaciones del
Geogebra en el tema de la geometría analítica, partiendo de ejemplos clásicos
de cualquier curso.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Se asume que el lector tiene conocimientos del uso básico del programa<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "verdana" , "sans-serif";"><span style="font-size: large;">MÉTODO 1:
ANALÍTICO</span><span style="font-size: 10pt;"><o:p></o:p></span></span><br />
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<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;">En la consola de Geogebra, se abre la vista CAS<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;"> escribir:</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 18.0pt;">R</span><span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 18pt;">esuelve({x^2+y^2=25,x+y=7},{x,y})</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
</span><span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;">el conjunto de
ecuaciones debe quedar encerrado en llaves</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
</span><span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;">el conjunto de
incógnitas debe quedar encerrado en llaves</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><br />
</span><span style="color: #222222; font-family: "verdana" , sans-serif; font-size: 10pt;"><br />
y la solución es: </span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-size: large;">MÉTODO 2: SOLUCIÓN GRÁFICA:</span></span><span style="color: #222222; font-family: "arial" , sans-serif;"><br /><span style="font-size: 10pt;">
en el área de entrada de la vista algebraica ingresar lo
siguiente:</span><br /><span style="font-size: large;">
x^2+y^2 =25</span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"> enter Geogebra le asigna un
nombre, por ejemplo c y la define como cónica<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: large;">x+y=7 enter</span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">……. Geogebra le asigna un nombre, por ejemplo f<o:p></o:p></span></div>
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y la gráfica que produce es:<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Luego se encuentran las intersecciones:<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Interseca(c,f)<o:p></o:p></span></div>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-68608107770282051422017-07-29T15:47:00.002-06:002017-08-01T10:25:28.215-06:00DESCOMPOSICION EN FRACCIONES PARCIALES CON REDUCE<h3 class="post-title entry-title" itemprop="name" style="background-color: white; color: #222222; font-family: arial, tahoma, helvetica, freesans, sans-serif; font-size: 22px; font-stretch: normal; font-weight: normal; line-height: normal; margin: 0.75em 0px 0px; position: relative;">
DESCOMPOSICION EN FRACCIONES PARCIALES CON REDUCE</h3>
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Ing. Luis Manfredo Reyes</div>
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Introducción:</div>
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<span style="font-size: 13.2px;">En la técnica de integración por descomposición en fracciones parciales, es necesario transformar la expresión original en una equivalente que sea la suma de varias fracciones, para poder aplicar algún teorema de integración.</span></div>
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POR SUPUESTO REDUCE PRESENTAR LA RESPUESTA FINAL, pero usualmente en un exámen se exige el procedimiento. El programa no presenta la solución paso a paso, para no malacostumbrar al estudiante a tener todo ya hecho.</div>
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<b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">REDUCE</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> es un software de </span><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_computacional" title="Álgebra computacional"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">álgebra computacional</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> (</span><a href="https://es.wikipedia.org/wiki/Sistema_algebraico_computacional" title="Sistema algebraico computacional"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">CAS</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">) desarrollado especialmente para usos en </span><a href="https://es.wikipedia.org/wiki/F%C3%ADsica" title="Física"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">física</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">. Comenzó a ser desarrollado desde 1960 por </span><a href="https://es.wikipedia.org/w/index.php?title=Anthony_Hearn&action=edit&redlink=1" title="Anthony Hearn (aún no redactado)"><span style="color: #a55858; font-family: "arial" , sans-serif; font-size: 10.5pt;">Anthony Hearn</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, desde entonces, muchos colaboradores de todo el mundo han contribuido a su desarrollo.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> Actualmente está disponible en las plataformas de </span><a href="https://es.wikipedia.org/wiki/Unix" title="Unix"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Unix</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, </span><a href="https://es.wikipedia.org/wiki/Linux" title="Linux"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Linux</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, </span><a href="https://es.wikipedia.org/wiki/Windows" title="Windows"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Windows</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, o </span><a href="https://es.wikipedia.org/wiki/Mac_OS" title="Mac OS"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Mac OS</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">,,Android e IOS <o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">el software puede ser descargado sin costo del sitio: </span><a href="http://reduce-algebra.sourceforge.net/"><span style="color: #888888; font-family: "arial" , sans-serif; font-size: 10.5pt;">http://reduce-algebra.sourceforge.net/</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">La versión para android está disponible en el Play Market<o:p></o:p></span></div>
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<a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span lang="ES-TRAD" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; letter-spacing: -0.1pt;">En éste artículo se muestran aplicaciones del Reduce en el tema de la descomposición de una expresión en fracciones parciales, partiendo de ejemplos clásicos de cualquier curso de álgebra</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt;"><o:p></o:p></span></div>
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<span lang="ES-TRAD" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; letter-spacing: -0.1pt;">Se asume que el usuario tiene instalado el programa y conoce el uso básico del mismo</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt;">El programa puede operar en modo consola, o en modo windows<o:p></o:p></span></div>
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CASO 1: FACTORES LINEALES NO REPETIDOS</div>
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Descomponga en fracciones parciales:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzNw-DfssVnSZ6pFahLtFanicVNGdjwIebTPysssDS1SaNAL7VoOAKpwoTiCYhkyZbskoH7pgkMso4ouhSRkh2ir-f1zpJJAhwM27BFIj2Qm1vWdkwb-k_9nHhCMVa_e0wUEvI2Clhglk/s1600/fp1.png" imageanchor="1" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzNw-DfssVnSZ6pFahLtFanicVNGdjwIebTPysssDS1SaNAL7VoOAKpwoTiCYhkyZbskoH7pgkMso4ouhSRkh2ir-f1zpJJAhwM27BFIj2Qm1vWdkwb-k_9nHhCMVa_e0wUEvI2Clhglk/s1600/fp1.png" style="border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.098) 1px 1px 5px; padding: 5px; position: relative;" /></a></div>
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En el área de Entrada, se escribe:</div>
<span style="font-size: large;">pf((4*x-1)/(x^2+x-2),x);</span><span style="font-size: 13.2px;"> y oprimir Enter. La respuesta es la siguiente:</span></div>
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<span style="font-size: 13.2px;">CASO 2: FACTORES LINEALES REPETIDOS</span></div>
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Descomponga en fracciones parciales:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXBECi_3xIne8ba5RKsp_YZSCWQBHCYOWWs94Ns7jwzXd1D_32-wWBUxBQVyYVotjYEXVYtS7EZwZ4VMIYz4MJQ0DJZ8bnV_l5mAvpyv5hsxAKEpdQVW_EwO_0AjQVXLkd58XB_J5Ynbg/s1600/fp3.png" imageanchor="1" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="80" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXBECi_3xIne8ba5RKsp_YZSCWQBHCYOWWs94Ns7jwzXd1D_32-wWBUxBQVyYVotjYEXVYtS7EZwZ4VMIYz4MJQ0DJZ8bnV_l5mAvpyv5hsxAKEpdQVW_EwO_0AjQVXLkd58XB_J5Ynbg/s1600/fp3.png" style="border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.098) 1px 1px 5px; padding: 5px; position: relative;" width="320" /></a></div>
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<span style="font-size: 13.2px;">Escribir lo siguiente:</span></div>
<span style="font-size: large;">pf((x^2+2*x+4)/(x^3+3*x^2+3*x+1),x);</span><span style="font-size: 13.2px;"> y oprimir Enter</span><br />
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<span style="font-size: 13.2px;">Se presenta la respuesta sin factorizar los denominadores</span></div>
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<span style="font-size: 13.2px;">De otra manera, usar el comando off exp antes</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEM07vKpLHL7ORFh1n_rMqRC4D93UOkEU3-6o9omB5Ia2ddnjt27PptmrvtTk8Rs4CjWa8Vq-ZnMNwgD8mEAk1apTgLYdLyMFdMIFOpvkjTJeTfFtK0iDANQ_Tk6Jpom9a3sqTpuKT-hQ/s1600/fp_reduce2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="239" data-original-width="475" height="201" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEM07vKpLHL7ORFh1n_rMqRC4D93UOkEU3-6o9omB5Ia2ddnjt27PptmrvtTk8Rs4CjWa8Vq-ZnMNwgD8mEAk1apTgLYdLyMFdMIFOpvkjTJeTfFtK0iDANQ_Tk6Jpom9a3sqTpuKT-hQ/s400/fp_reduce2.png" width="400" /></a></div>
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CASO 3: FACTORES CUADRÁTICOS NO REPETIDOS</div>
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Descomponga en fracciones parciales:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijy3Rpe8Twsul_jSZ7SzJHcj58Rc-yxHMFVYHKON6_hpPJwLAaB_5TFujzlsT70Q2Fj9Lr1-jdpm0_prdmA3zA6vhKORPb1vQnAq82V0H5BgJ-dadqean1I52oaHchwKpe1qUWAt3Qd64/s1600/fp6.png" imageanchor="1" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="193" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijy3Rpe8Twsul_jSZ7SzJHcj58Rc-yxHMFVYHKON6_hpPJwLAaB_5TFujzlsT70Q2Fj9Lr1-jdpm0_prdmA3zA6vhKORPb1vQnAq82V0H5BgJ-dadqean1I52oaHchwKpe1qUWAt3Qd64/s1600/fp6.png" style="border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.098) 1px 1px 5px; padding: 5px; position: relative;" width="320" /> </a></div>
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Ingresar lo siguiente:</div>
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<span style="font-size: large;">pf((4*x^2-8*x+1)/(x^3-x+6));</span><span style="font-size: 13.2px;"> y oprimir enter, la</span><span style="font-size: 13.2px;"> respuesta es:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9S4KG4j6xaMxEgTOlYMFBZxkPJpr18aqJAKh5mjeOf0tgrkOESjWJemxfGtVJPPzZpdF5tGzbIzVwoAZnBARgIxwQt7T0LXdvLI8gaz-CgMGzNjqJn4UGpKhY_lE0ejkOQ4tI38Q3pFg/s1600/fp_reduce3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="112" data-original-width="446" height="100" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9S4KG4j6xaMxEgTOlYMFBZxkPJpr18aqJAKh5mjeOf0tgrkOESjWJemxfGtVJPPzZpdF5tGzbIzVwoAZnBARgIxwQt7T0LXdvLI8gaz-CgMGzNjqJn4UGpKhY_lE0ejkOQ4tI38Q3pFg/s400/fp_reduce3.png" width="400" /></a></div>
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CASO 4: FACTORES CUADRÁTICOS REPETIDOS</div>
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Descomponga en factores parciales:</div>
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<a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><img height="311" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQXp5lmYINzvZ3MisCBRtdg-gFqGGj1h0fCx991cNzkFYLnnX3b1fRUte7aooZDnf_o0zy8xGNoFYzG-bAz7hzxAs0R0qO5L2TOIBQEQtBotOqBrdsxAoiz6Gg1KqNsFwVMgXgnBDOQxc/s640/fp5.png" width="640" /></div>
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<span style="font-size: 13.2px; text-align: justify;">Ingresar:</span><span style="font-size: large; text-align: justify;"> pf((1-x+2x^2-x^3)/(x*(x^2+1)^2),x);</span></div>
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y la respuesta es:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihwRW3NbSfPQNOLOoLrpIDDU_zQCjV3qxZDs53f6e0tQWIO3CS-WQYknqd0jLwi34aFCxp_EpiDAVPCyk2HU52TwzroibFKoIGSGapWLw_9IOgRfMKLIZIVDvXt8pWHtvYgS4zMY3EN00/s1600/fp_reduce4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="104" data-original-width="448" height="92" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihwRW3NbSfPQNOLOoLrpIDDU_zQCjV3qxZDs53f6e0tQWIO3CS-WQYknqd0jLwi34aFCxp_EpiDAVPCyk2HU52TwzroibFKoIGSGapWLw_9IOgRfMKLIZIVDvXt8pWHtvYgS4zMY3EN00/s400/fp_reduce4.png" width="400" /></a></div>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-88754471379437126912017-07-29T15:29:00.000-06:002017-07-29T15:29:11.002-06:00ÁLGEBRA LINEAL CON REDUCE<div class="MsoNormal" style="background-color: white; margin: 0cm 0cm 0pt;">
<span lang="ES" style="font-size: 21.33px;"><span style="color: #222222; font-family: "arial" , sans-serif;"><b>ÁLGEBRA LINEAL CON REDUCE</b></span></span></div>
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<span lang="ES" style="font-size: 21.33px;"><span style="color: #222222; font-family: "arial" , sans-serif;"><b>Ing. Luis Manfredo Reyes</b></span></span></div>
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<i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.26px;"><br /></span></i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.26px;">El Algebra lineal es una rama de la matemática en la cual se opera con entidades llamadas Matrices y Vectores.</span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.26px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.26px;">En éste documento se discuten las principales operaciones quese pueden realizar usando el programa Mathematica</span><br />
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<span lang="ES">Siendo el paquete computacional REDUCE uno de los más importantes en el campo de la computación aplicada a la matemática, no es de extrañar que contenga utilidades para la resolución de aplicaciones de álgebra lineal.</span></div>
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<span lang="ES">En este documento se analizan los casos más importantes de operaciones con vectores y matrices, aplicando el paquete. Sin embargo, se debe tomar en cuenta que, éste software no es un sustituto del estudio ni de la resolución manual. Es solamente una herramienta de apoyo.</span></div>
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<b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">REDUCE</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> es un software de </span><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_computacional" title="Álgebra computacional"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">álgebra computacional</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> (</span><a href="https://es.wikipedia.org/wiki/Sistema_algebraico_computacional" title="Sistema algebraico computacional"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">CAS</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">) desarrollado especialmente para usos en </span><a href="https://es.wikipedia.org/wiki/F%C3%ADsica" title="Física"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">física</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">. Comenzó a ser desarrollado desde 1960 por </span><a href="https://es.wikipedia.org/w/index.php?title=Anthony_Hearn&action=edit&redlink=1" title="Anthony Hearn (aún no redactado)"><span style="color: #a55858; font-family: "arial" , sans-serif; font-size: 10.5pt;">Anthony Hearn</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, desde entonces, muchos colaboradores de todo el mundo han contribuido a su desarrollo.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> Actualmente está disponible en las plataformas de </span><a href="https://es.wikipedia.org/wiki/Unix" title="Unix"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Unix</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, </span><a href="https://es.wikipedia.org/wiki/Linux" title="Linux"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Linux</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, </span><a href="https://es.wikipedia.org/wiki/Windows" title="Windows"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Windows</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, o </span><a href="https://es.wikipedia.org/wiki/Mac_OS" title="Mac OS"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Mac OS</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">,,Android e IOS <o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">el software puede ser descargado sin costo del sitio: </span><a href="http://reduce-algebra.sourceforge.net/"><span style="color: #888888; font-family: "arial" , sans-serif; font-size: 10.5pt;">http://reduce-algebra.sourceforge.net/</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">La versión para android está disponible en el Play Market<o:p></o:p></span></div>
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<a href="https://www.blogger.com/blogger.g?blogID=3997797568400374468" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span lang="ES-TRAD" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; letter-spacing: -0.1pt;">En éste artículo se muestran aplicaciones del Reduce en el tema del álgebra, partiendo de ejemplos clásicos de cualquier curso de álgebra</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt;"><o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt; letter-spacing: -0.1pt;">Se asume que el usuario tiene instalado el programa y conoce el uso básico del mismo</span></div>
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<b style="font-family: arial, tahoma, helvetica, freesans, sans-serif; font-size: 13.2px;"><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 16pt;"> OPERACIONES CON MATRICES Y VECTORES</span></b></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;"><b> Definición de una matriz:</b></span></div>
<span style="font-family: "arial" , sans-serif;">Por ejemplo, se desea crear la matriz a:</span><br />
<span style="font-family: "arial" , sans-serif;"><br /></span>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs2OHnu1caNUuwOuY5SUrfggfFpgpVBXxMHQUwvziOkQOoYke91iPgwhtpoQD_dFIzcH73TFMooKNP_dVoZk5f4sIXjIl0J-1kCQwWKNmfa48dq1UAV044WCVLZ1sYG-UtgVQjDOP3r88/s1600/m28.jpg" imageanchor="1" style="color: #888888; margin-left: 1em; margin-right: 1em;"><img border="0" i="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs2OHnu1caNUuwOuY5SUrfggfFpgpVBXxMHQUwvziOkQOoYke91iPgwhtpoQD_dFIzcH73TFMooKNP_dVoZk5f4sIXjIl0J-1kCQwWKNmfa48dq1UAV044WCVLZ1sYG-UtgVQjDOP3r88/s1600/m28.jpg" style="border: 1px solid rgb(238, 238, 238); box-shadow: 1px 1px 5px rgba(0,0,0,0.098); padding: 5px; position: relative;" true="" /></a></div>
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<span style="font-family: "arial" , sans-serif;">Para especificar el tamaño de usa matriz se utiliza: </span><br />
<span lang="ES" style="color: black; font-family: "arial" , sans-serif; font-size: 13.2px;">m</span><b style="font-family: arial, tahoma, helvetica, freesans, sans-serif; font-size: 13.2px;"><span lang="ES" style="color: red; font-family: "arial" , sans-serif;">atrix a(2,3); enter</span></b></div>
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<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"></span><br /></div>
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<span style="font-family: "arial" , sans-serif;">para ingresar los datos y almacenarlos en un objeto llamado "a", se escribe</span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;"><span lang="ES" style="color: black; font-family: "arial" , sans-serif; font-size: 13.2px;">a:= mat((</span><b style="font-family: arial, tahoma, helvetica, freesans, sans-serif; font-size: 13.2px;"><span lang="ES" style="color: red; font-family: "arial" , sans-serif;">1,2,3),(4,5,6)); enter</span></b></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKn9WHtnQ9735bCvGJrOydXeIuUVr8JObZE_R7BIyGRn2YbPd-PWsX3HqttVMEM-3uT5AxwAfpQSKMlQ8lHNCEpIqi41JYDVjfPAJasjK4qt-JK8qXgjf-kVIW9DVbpYBSdKWqes7UMco/s1600/red_alglin_1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="143" data-original-width="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKn9WHtnQ9735bCvGJrOydXeIuUVr8JObZE_R7BIyGRn2YbPd-PWsX3HqttVMEM-3uT5AxwAfpQSKMlQ8lHNCEpIqi41JYDVjfPAJasjK4qt-JK8qXgjf-kVIW9DVbpYBSdKWqes7UMco/s1600/red_alglin_1.png" /></a></div>
<span lang="ES" style="font-family: "arial" , sans-serif;"><br /></span><span lang="ES" style="font-family: "arial" , sans-serif;"><b>Inversa de una matriz:</b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;">Solamente se aplica a matrices cuadradas</span><br />
se puede crear el objeto y luego usarlo :<br />
<br />
matrix b(4,4); enter<br />
<b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">b:=mat((1,2,3,4),(2,3,4,0),(3,4,0,0),(4,0,0,0));</span></b><br />
<b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">b^-1 ; enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqF0oyHFmgkbOQggox_u9NYdsntu11Nn26Nqa3_hxSQQr8B1EZTRqzS0XcLllMIL-uHJ0JGETCv0h00VvjNH2aZElh9ZEpJoiJ4Bhwbda9-QAmAMaS509V9_Yh96IO1sChfqGrdAVe1_Y/s1600/red_al2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="493" data-original-width="499" height="395" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqF0oyHFmgkbOQggox_u9NYdsntu11Nn26Nqa3_hxSQQr8B1EZTRqzS0XcLllMIL-uHJ0JGETCv0h00VvjNH2aZElh9ZEpJoiJ4Bhwbda9-QAmAMaS509V9_Yh96IO1sChfqGrdAVe1_Y/s400/red_al2.png" width="400" /></a></div>
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o se puede definir directamente:</div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"> mat((</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">1,2,3,4),(2,3,4,0),(3,4,0,0),(4,0,0,0))^-1 ; enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuiKS_7c-pubZyZNVzi_RRC94sKiJs3mMQ80mtDHxqrqraOp4JBaWzNY8hWKONkg8T_JfZY8s0OVutaFQzUOWtc1H6Kobjr32uk694ZVmxKEJhlDFaC8hxnw13jIG7EWQTuvzkr8wQ774/s1600/red_al3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="281" data-original-width="463" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuiKS_7c-pubZyZNVzi_RRC94sKiJs3mMQ80mtDHxqrqraOp4JBaWzNY8hWKONkg8T_JfZY8s0OVutaFQzUOWtc1H6Kobjr32uk694ZVmxKEJhlDFaC8hxnw13jIG7EWQTuvzkr8wQ774/s400/red_al3.png" width="400" /></a></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif; position: relative; top: 63pt;"></span><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span><br /></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;"><b>Determinante de una matriz:</b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;">Solamente aplica a matices cuadradas </span></div>
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matrix m(3,3); enter</div>
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<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"> m:=mat((</span><b><span style="color: red; font-family: "arial" , sans-serif;">a,0,0),(b,c,0),(d,e,f)); enter</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif; font-size: 13.2px;"> det(m)</span><b style="font-size: 13.2px;"><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">; enter</span></b></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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a c f<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif; position: relative; top: 3.5pt;"><shape id="_x0000_i1073" o:bullet="t" style="height: 14.25pt; width: 27.75pt;" type="#_x0000_t75"><imagedata o:title="" src="file:///C:\Users\VAIO\AppData\Local\Temp\msohtmlclip1\01\clip_image095.wmz"></imagedata></shape></span></div>
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<span style="color: black; font-family: "arial" , sans-serif;"><b>Transpuesta de una matriz:</b></span><br />
<span style="color: black; font-family: "arial" , sans-serif;">Se aplica a cualquier matriz aunque no sea cuadrada</span><br />
<span style="color: black; font-family: "arial" , sans-serif;">usando la misma matriz m del ejemplo anterior</span><br />
<span style="color: black; font-family: "arial" , sans-serif;"><br /></span></div>
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<span style="color: black; font-family: "arial" , sans-serif;">tp(m); enter</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDMg7MnfRJpwsy4nwlxtj1W4tjkPnq_m013RM08fCmw3fjuHlzKIVbNQcnZrOu6PNiMVs8TMEARCrJIEJ3Zd6BDQ0YK-e-Ns4ah8fE4n7LDCbdeq6UD2HqBX-TyNeUvn7VDL5RGWiie2Y/s1600/red_al4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="126" data-original-width="424" height="95" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDMg7MnfRJpwsy4nwlxtj1W4tjkPnq_m013RM08fCmw3fjuHlzKIVbNQcnZrOu6PNiMVs8TMEARCrJIEJ3Zd6BDQ0YK-e-Ns4ah8fE4n7LDCbdeq6UD2HqBX-TyNeUvn7VDL5RGWiie2Y/s320/red_al4.png" width="320" /></a></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;">Operaciones con vectores:</span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;"></span><br />
<pre style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-bottom-color: initial; border-bottom-style: initial; border-image: initial; border-left-color: rgb(229, 229, 229); border-left-style: solid; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 0px 1px; box-sizing: inherit; color: #555555; font-family: monospace, sans-serif; font-size: 13px; outline: 0px; overflow: auto; padding: 15px; vertical-align: baseline; white-space: pre-wrap; word-wrap: break-word;">1: load_package listvecops;
>>
>> 2: A := {a1,a2,a3,a4};
>>
>> a := {a1,a2,a3,a4}
>>
>> 3: B := {b1,b2,b3,b4};
>>
>> b := {b1,b2,b3,b4}
>>
>> 4: c*A ;
>>
>> {a1*c,
>>
>> a2*c,
>>
>> a3*c,
>>
>> a4*c}
>>
>> 5: A + B;
>>
>> {a1 + b1,
>>
>> a2 + b2,
>>
>> a3 + b3,
>>
>> a4 + b4}
>>
>> 6: A*B;
>>
>> {a1*b1,
>>
>> a2*b2,
>>
>> a3*b3,
>>
>> a4*b4}
>>
>> 7: c1*A + c2*B;
>>
>> {a1*c1 + b1*c2,
>>
>> a2*c1 + b2*c2,
>>
>> a3*c1 + b3*c2,
>>
>> a4*c1 + b4*c2}
>></pre>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-33704319641421149362017-07-15T15:18:00.002-06:002017-07-17T10:25:24.679-06:00ALGEBRA CON REDUCE<div class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 6.0pt;">
<b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 18.0pt;">ALGEBRA CON REDUCE<o:p></o:p></span></b></div>
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<b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">Ing. Luis Manfredo Reyes<o:p></o:p></span></b></div>
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<b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">REDUCE</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> es un software de </span><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_computacional" title="Álgebra computacional"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">álgebra computacional</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> (</span><a href="https://es.wikipedia.org/wiki/Sistema_algebraico_computacional" title="Sistema algebraico computacional"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">CAS</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">) desarrollado especialmente para usos en </span><a href="https://es.wikipedia.org/wiki/F%C3%ADsica" title="Física"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">física</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">. Comenzó a ser desarrollado desde
1960 por </span><a href="https://es.wikipedia.org/w/index.php?title=Anthony_Hearn&action=edit&redlink=1" title="Anthony Hearn (aún no redactado)"><span style="color: #a55858; font-family: "arial" , sans-serif; font-size: 10.5pt;">Anthony Hearn</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, desde entonces, muchos colaboradores de todo el mundo han contribuido
a su desarrollo.<o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> Actualmente está disponible en las plataformas de </span><a href="https://es.wikipedia.org/wiki/Unix" title="Unix"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Unix</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, </span><a href="https://es.wikipedia.org/wiki/Linux" title="Linux"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Linux</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, </span><a href="https://es.wikipedia.org/wiki/Windows" title="Windows"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Windows</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, o </span><a href="https://es.wikipedia.org/wiki/Mac_OS" title="Mac OS"><span style="color: #0b0080; font-family: "arial" , sans-serif; font-size: 10.5pt;">Mac OS</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">,,Android e IOS <o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">el software puede ser descargado sin costo del sitio: </span><a href="http://reduce-algebra.sourceforge.net/"><span style="color: #888888; font-family: "arial" , sans-serif; font-size: 10.5pt;">http://reduce-algebra.sourceforge.net/</span></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">La versión para android está disponible en el Play Market<o:p></o:p></span></div>
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<span lang="ES-TRAD" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt; letter-spacing: -0.1pt;">En éste artículo se muestran aplicaciones del Reduce
en el tema del álgebra, partiendo
de ejemplos clásicos de cualquier curso de álgebra</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES-TRAD" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt; letter-spacing: -0.1pt;">Se asume que el usuario tiene instalado el programa
y conoce el uso básico del mismo</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">El programa puede operar en modo
consola, o en modo windows<o:p></o:p></span></div>
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<b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 14.0pt;"> FACTORIZACION DE
EXPRESIONES: (modo real)</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">:</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 20.0pt;">6x<sup>2</sup>+18x-2</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 20.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Para modo consola: En la zona
de entrada escribir factorize[6*x^2+18*x-2]; y oprimir enter<o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">En modo Windows: seleccionar
la pestaña algebra, seleccionar factorizar, y en la ventana que aparece
ingresar el polinomio</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">el numero 1 en cada factor indica que el factor se repite una vez</span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 20.0pt;">X<sup>7</sup>-1</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 20.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 20.0pt;">I</span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">ngresar:
factorize[x^7-1]; y enter. En modo Windows: seleccionar la pestaña algebra,
seleccionar factorizar, y en la ventana que aparece ingresar el polinomio</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 20.0pt;">x<sup>3</sup>+y<sup>3</sup></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 20.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">ingresar: factorize[x^3+y^3];
y oprimir enter. En modo Windows: seleccionar la pestaña algebra, seleccionar
factorizar, y en la ventana que aparece ingresar el polinomio</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , sans-serif;"><span style="font-size: large;">x<sup>3</sup>+5</span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Es irreducible en los enteros<o:p></o:p></span></div>
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<b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">DESARROLLO DE PRODUCTOS</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">Se debe escribir la operación y finalizarla con punto y coma</span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"> </span><b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 18.0pt;">(4x-1)<sup>3</sup></span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">(x-2)<sup>4</sup></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">(3x-2) (2x-4)</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"> </span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 16pt;">(Sen x-Tan x)</span><sup style="color: #222222; font-family: Arial, sans-serif;">4</sup></div>
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<b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">SIMPLIFICACIÓN DE EXPRESIONES</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">Raíces de polinomios</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: large;">x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1</span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">No hay raíces
reales</span></div>
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<b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">SOLUCION DE ECUACIONES DE UNA
VARIABLE</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">El comando para resolver
ecuaciones solamente está disponible en vista CAS</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Resolver </span><b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">x<sup>2</sup>+5x-6=0</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span style="font-family: monospace; white-space: nowrap;">Ln(sen (x+3))^5=8</span><br />
<span style="font-family: monospace; white-space: nowrap;"><span style="font-size: large;"> solve(log(sin(x+3))^5 = 8,x); </span></span><br />
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<b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 16.0pt;">SOLUCION DE SISTEMAS DE ECUACIONES</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<b><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">GRÁFICAS DE FUNCIONES</span></b><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Reduce puede graficar funciones de una y dos
variables </span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;">Ejemplo:</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<span lang="ES-TRAD" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10.0pt; letter-spacing: -0.15pt;">Dada
la función: f(x)= 2x<sup>3</sup>-6x<sup>2</sup>-8x-1</span><br />
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Una de tres variables: <span style="font-size: large;">z=x²-y²</span></div>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-37879309636039626892017-07-11T11:39:00.000-06:002017-07-11T21:22:18.178-06:00SOFTWARE MATEMÁTICO REDUCE: OTRA OPCIÓN PARA EL ESTUDIO DE LA MATEMÁTICA<b>SOFTWARE MATEMÁTICO REDUCE: OTRA OPCIÓN PARA EL ESTUDIO DE LA MATEMÁTICA</b><br />
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Ing. Luis Manfredo Reyes<br />
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Introducción:<br />
Partiendo de la famosa frase de Charles Babbagge: "ojalá estos cálculos me los hiciera una máquina de vapor", a lo largo de la historia siempre han existido deseos de los matemáticos de contar con máquinas capaces de resolver operaciones aritméticas y algebráicas para auxiliarse en sus labores.<br />
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En el caso de los programas matemáticos, no solamente ha existido el deseo de auxilio aritmético, sino también simbólico.<br />
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<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">Un </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">sistema algebraico computacional</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> o </span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">sistema de álgebra computacional</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> (</span><b style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">CAS</b><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">, del inglés </span><i style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">computer algebra system</i><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">) es un programa de computarora o de calculadora avanzada que facilita el cálculo simbólico.</span><br />
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<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> La principal diferencia entre una máquina CAS y una tradicional es la capacidad de la primera para trabajar con </span><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n" style="background: none rgb(255, 255, 255); color: #0b0080; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Ecuación">ecuaciones</a><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> y fórmulas simbólicamente, en lugar de numéricamente. </span><br />
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<span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">Por ejemplo, </span><i style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;">a + b</i><span style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px;"> es interpretada siempre como "la suma de dos variables", y no como "la suma de dos números" (con valores asignados).</span><br />
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Los primeros trabajos en sistemas CAS fueron desarrollados por el físico y Premio Nobel <a href="https://es.wikipedia.org/wiki/Martinus_J._G._Veltman" style="background: none; color: #0b0080; text-decoration-line: none;" title="Martinus J. G. Veltman">Martin Veltman</a>, quien en 1963 desarrolló un programa para matemática simbólica, llamado <a class="new" href="https://es.wikipedia.org/w/index.php?title=Schoonship&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Schoonship (aún no redactado)">Schoonship</a>, especializado en Física de Altas Energías. </div>
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Los primeros sistemas populares fueron <a href="https://es.wikipedia.org/wiki/Reduce" style="background: none; color: #0b0080; text-decoration-line: none;" title="Reduce">Reduce</a> y <a class="new" href="https://es.wikipedia.org/w/index.php?title=Macsyma&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Macsyma (aún no redactado)">Macsyma</a>. Hoy en día, una versión de software libre de Macsyma llamada <a href="https://es.wikipedia.org/wiki/Maxima" style="background: none; color: #0b0080; text-decoration-line: none;" title="Maxima">Maxima</a> es mantenida activamente. Posteriormente, en los <a href="https://es.wikipedia.org/wiki/A%C3%B1os_1980" style="background: none; color: #0b0080; text-decoration-line: none;" title="Años 1980">años 1980</a> adquirió popularidad <a href="https://es.wikipedia.org/wiki/Derive" style="background: none; color: #0b0080; text-decoration-line: none;" title="Derive">Derive</a>.</div>
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<b>REDUCE</b> es un software de <a class="mw-redirect" href="https://es.wikipedia.org/wiki/%C3%81lgebra_computacional" style="background: none; color: #0b0080; text-decoration-line: none;" title="Álgebra computacional">álgebra computacional</a> (<a href="https://es.wikipedia.org/wiki/Sistema_algebraico_computacional" style="background: none; color: #0b0080; text-decoration-line: none;" title="Sistema algebraico computacional">CAS</a>) desarrollado especialmente para usos en <a href="https://es.wikipedia.org/wiki/F%C3%ADsica" style="background: none; color: #0b0080; text-decoration-line: none;" title="Física">física</a>. Comenzó a ser desarrollado desde 1960 por <a class="new" href="https://es.wikipedia.org/w/index.php?title=Anthony_Hearn&action=edit&redlink=1" style="background: none; color: #a55858; text-decoration-line: none;" title="Anthony Hearn (aún no redactado)">Anthony Hearn</a>, desde entonces, muchos colaboradores de todo el mundo han contribuido a su desarrollo.</div>
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Actualmente está disponible en las plataformas de <a href="https://es.wikipedia.org/wiki/Unix" style="background: none; color: #0b0080; text-decoration-line: none;" title="Unix">Unix</a>, <a class="mw-redirect" href="https://es.wikipedia.org/wiki/Linux" style="background: none; color: #0b0080; text-decoration-line: none;" title="Linux">Linux</a>, <a class="mw-redirect" href="https://es.wikipedia.org/wiki/Windows" style="background: none; color: #0b0080; text-decoration-line: none;" title="Windows">Windows</a>, o <a href="https://es.wikipedia.org/wiki/Mac_OS" style="background: none; color: #0b0080; text-decoration-line: none;" title="Mac OS">Mac OS</a>,,Android e IOS </div>
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el software puede ser descargado sin costo del sitio: <a href="http://reduce-algebra.sourceforge.net/">http://reduce-algebra.sourceforge.net/</a></div>
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La versión para android está disponible en el Play Market</div>
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Para usuarios avanzados, hay una versión tipo consola, donde se ingresan comandos directamente</div>
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Para usuarios acostumbrados al estilo de menús de Windows, hay una versión con ése estilo, que genera el código necesario</div>
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Las pestañas File y Edit tienen los típicos comandos de windows para manejo de archivos.</div>
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En la pestaña Algebra se tienen los comandos para: factorización, simplificación, solución de ecuaciones y sistemas de ecuaciones<br />
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En la pestaña Calculus se tienen comandos para cálculo de límites, derivadas, integrales.<br />
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La pestaña Matrix tiene los comandos necesarios para el trabajo con matrices.<br />
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En la pestaña Graphs se pueden trabajar gráficas en 2 y 3 dimensiones, usando el paquete libre </div>
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GNUPLOT.</div>
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Tambien hay una gran cantidad de paquetes auxiliares que han sido desarrollados po usuarios en todo e mundo<br />
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En otros documentos se discutirá el uso específico de Reduce en cada uno de los temas específicos.</div>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-69430567107907362212017-07-10T11:43:00.000-06:002017-07-10T11:45:19.429-06:00FUNCIONES COMPUTACIONALES DE APROXIMACIÓN A LAS DISTRIBUCIONES DE PROBABILIDAD: NORMAL, t, CHI CUADRADO Y F<div class="separator" style="clear: both; text-align: center;">
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<br />Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-7890589437286644132017-06-19T16:20:00.002-06:002017-06-23T16:24:46.435-06:00CALCULO DE VALORES Y VECTORES PROPIOS CON GEOGEBRA<b><span style="font-size: large;">CALCULO DE VALORES Y VECTORES PROPIOS CON GEOGEBRA</span></b><br />
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<span style="font-size: 26pt; line-height: 39.8667px;">ING. LUIS MANFREDO REYES</span></div>
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<span style="font-size: 12pt; line-height: 18.4px;">1.<span style="font-size: 7pt; line-height: normal;"> </span></span><span style="font-size: 12pt; line-height: 18.4px;">INTRODUCCION: Los valores y vectores propios, son propiedades importantes de las matrices cuadradas de tamaño nxn y tienen aplicaciones en el cálculo y la estadística. En éste documento se estudiará el uso de GEOGEBRA para el cálculo de valores y vectores propios, <o:p></o:p></span></div>
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<span style="font-size: 12pt; line-height: 18.4px;">2.<span style="font-size: 7pt; line-height: normal;"> </span></span><br />
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<span style="font-size: 12pt; line-height: 18.4px;">Para detalles sobre el calculo manual de valores y vectores propios, y sus aplicaciones se puede consultar:</span><br />
<span style="line-height: 18.4px;"><b>http://reyesestadistica.blogspot.com/2013/04/calculo-de-valores-y-vectores-propios-y.html</b></span><br />
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<b><span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"><o:p></o:p></span></b></div>
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<b><span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">GeoGebra</span></b><span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> es un software matemático. Fue creado con la idea de ser usado para la educación en colegios y universidades. Fue creado por Markus Hohenwarter, en el año 2001 en la </span><span lang="ES"><a href="https://es.wikipedia.org/wiki/Universidad_de_Salzburgo" title="Universidad de Salzburgo"><span style="color: #0b0080; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">Universidad de Salzburgo</span></a></span><span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, luego en la Universidad de Atlantic, Florida, luego en la </span><span lang="ES"><a href="https://es.wikipedia.org/wiki/Universidad_Estatal_de_Florida" title="Universidad Estatal de Florida"><span style="color: #0b0080; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">Universidad Estatal de Florida</span></a></span><span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> y actualmente , en la </span><span lang="ES"><a href="https://es.wikipedia.org/w/index.php?title=Universidad_de_Linz&action=edit&redlink=1" title="Universidad de Linz (aún no redactado)"><span style="color: #a55858; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">Universidad de Linz</span></a></span><span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">, Austria.<o:p></o:p></span></div>
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<span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">GeoGebra está escrito en </span><span lang="ES"><a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n_Java" title="Lenguaje de programación Java"><span style="color: #0b0080; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">Java</span></a></span><span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;"> y por tanto es posible usarlo en diferentes plataformas.<o:p></o:p></span></div>
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<span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">El software tiene dos partes importantes: un procesador geométrico y un procesador algebraico.<o:p></o:p></span></div>
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<span lang="ES" style="color: #252525; font-family: "arial" , "sans-serif"; font-size: 10.5pt;">GeoGebra permite el trazado de construcciones geométricas de todo tipo así como la representación gráfica, el tratamiento algebraico y el cálculo de funciones reales de variable real, sus derivadas, integrales, etc.<o:p></o:p></span></div>
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<span lang="ES" style="font-family: "georgia" , serif; font-size: 18pt;">DISPONIBILIDAD:<o:p></o:p></span></div>
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<span lang="ES" style="font-family: "georgia" , serif;">El software puede ser descargado sin costo del sitio: </span><span lang="ES"><a href="http://www.geogebra.org/"><span style="font-family: "georgia" , "serif";">www.geogebra.org</span></a></span><span lang="ES" style="font-family: "georgia" , serif;"> <o:p></o:p></span></div>
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<span lang="ES" style="font-family: "georgia" , serif;">Existen versiones para: Windows, Linux, MacIntosh, Android e Ios.<o:p></o:p></span></div>
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<span lang="ES" style="font-family: "georgia" , serif;">Existen algunas diferencias entre las versiones, pero la mayoría de aspectos son iguales.<o:p></o:p></span></div>
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<span lang="ES" style="font-family: "georgia" , serif;">La última versión disponible es la 5.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif;"><span style="letter-spacing: -0.133333px;">Geogebra no posee comandos directos para calcular valores propios (eigenvalues) o vectores propios (eigenvectors), pero el procedimiento se puede realizar utilizando las hojas de trabajo desarrolladas por los usuarios</span></span><br />
<span style="font-family: "arial" , sans-serif;"><span style="letter-spacing: -0.133333px;"><br /></span></span>
<span style="font-family: "arial" , sans-serif;"><span style="letter-spacing: -0.133333px;">PARA MATRICES 2X2: https://www.geogebra.org/m/N78q77VK</span></span><br />
<span style="font-family: "arial" , sans-serif;"><span style="letter-spacing: -0.133333px;">PARA MATRICES 3X3: https://www.geogebra.org/m/ZJtywDTd</span></span><br />
<span style="font-family: "arial" , sans-serif;"><span style="letter-spacing: -0.133333px;">Autor Lee Robinson</span></span></div>
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<b><span style="font-size: 12pt; line-height: 18.4px;"><span style="font-size: 7pt; font-weight: normal; line-height: normal;"> </span></span></b><b><span style="font-size: 12pt; line-height: 18.4px;">Ejemplo<o:p></o:p></span></b></div>
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<span style="font-size: 12pt; line-height: 18.4px;">Calcular los valores y vectores propios de la matriz:<o:p></o:p></span><br />
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Cargar la hoja de tabajo: https://www.geogebra.org/m/rMXd7p3V<br />
Ingresar los valores en la parte inferior, al lado del boton RESET y luego dar click en este boton:<br />
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El deslizador del lado derecho debe moverse hacia abajo para mostrar las operaciones realizadas:<br />
Los valores propios son -1 y 2<br />
Los vectores propios son S1 y S2 y sus multiplos<br />
En la parte grafica se pueden ver los vectores proyectados en el plano.<br />
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ejemplo con una matriz 3x3<br />
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Cargar la hoja de trabajo: https://www.geogebra.org/m/ZJtywDTd<br />
En la parte inferior ingresar la matriz y oprimir el boton RESET<br />
El deslizador de la izquierda moverlo al fondo para mostrar las operaciones realizadas:<br />
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los valores propios son -1, 3, 4<br />
y los vectores propios son s1,s2,s3 y sus multiplos<br />
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<b>PROCEDIMIENTO MANUAL:</b></div>
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Se puede realizar el calculo en forma manual, aplicando una serie de operaciones matriciales y algebraicas. EL PROCEDIMIENTO SE REALIZA EN LA VISTA CAS, la cual no esta disponible en tablets ni telefonos.</div>
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Ejemplo: (obtenido en Geogebra Tube)</div>
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1. Ingresar la matriz en la vista CAS: q={{3,3},{9,5}}<br />
2. Ingresar la matriz identidad 2x2: p={{1,0},{0,1}}<br />
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3. Realizar la operacion: q-t*p<br />
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4. Calcular la determinante del resultado: Determinante[$1}<br />
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5. Igualar a cero y resolver: Soluciones[$2=0,t]<br />
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6. Seleccionar el primer elemento de la respuesta: Elemento[$3,1]<br />
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7. Formar el vector x,y que contendra los valores propios: Vector[x,y]<br />
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8. Operar q*el vector x,y: q*$5<br />
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9. Operar $4*$5 y resolver el sistema<br />
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Los valores propios son: 1.27 y 4.73</div>
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Realizar las sustituciones por los valores propios: -<br />
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Vectores propios (3,1.73) y (3,1.73)<br />
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-37822715947695283832017-06-13T12:01:00.001-06:002017-06-13T19:06:38.493-06:00ÁLGEBRA LINEAL CON MATHEMATICA<div class="MsoNormal" style="background-color: white; margin: 0cm 0cm 0pt;">
<span lang="ES" style="font-size: 21.3333px;"><span style="color: #222222; font-family: "arial" , sans-serif;"><b>ÁLGEBRA LINEAL CON MATHEMATICA</b></span></span></div>
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<span lang="ES" style="font-size: 21.3333px;"><span style="color: #222222; font-family: "arial" , sans-serif;"><b>Ing. Luis Manfredo Reyes</b></span></span></div>
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<i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span></i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">El Algebra lineal es una rama de la matemática en la cual se opera con entidades llamadas Matrices y Vectores.</span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En éste documento se discuten las principales operaciones quese pueden realizar usando el programa Mathematica</span><br />
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<span lang="ES">Siendo el paquete computacional Mathematica ® (un producto de Wolfram Research) uno de los más importantes en el campo de la computación aplicada a la matemática, no es de extrañar que contenga utilidades para la resolución de aplicaciones de álgebra lineal.</span></div>
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<span lang="ES">En este documento se analizan los casos más importantes de operaciones con vectores y matrices, aplicando el paquete. Sin embargo, se debe tomar en cuenta que, éste software no es un sustituto del estudio ni de la resolución manual. Es solamente una herramienta de apoyo.</span></div>
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<span lang="ES"><br /></span><span lang="ES">MATHEMATICA ES UN SOFTWARE COMERCIAL, POR LO QUE TIENE UN COSTO (Y ELEVADO POR CIERTO). No se recomienda el uso "pirata" del mismo.</span></div>
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<b><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 16pt;"> OPERACIONES CON MATRICES Y VECTORES</span></b></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;"><b> Definición de una matriz:</b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;"><b>forma directa:</b></span></div>
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<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">M</span><b><span lang="ES" style="color: red; font-family: "arial" , sans-serif;">atrixForm[{1,2,3},{4,5,6}}]; shift+enter</span></b><span lang="ES" style="color: black; font-family: "arial" , sans-serif;"></span><br />
<b><span lang="ES" style="color: red; font-family: "arial" , sans-serif;"><br /></span></b>
<b><span lang="ES" style="color: red; font-family: "arial" , sans-serif;">el comando MatrixForm[] se usa para presentar el resultado en forma de matriz, de lo contrario lo hace en forma de lista de listas</span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 13.2px;">FORMA INDIRECTA: se asigna a una variable</span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;"><span lang="ES" style="color: black; font-family: "arial" , sans-serif; font-size: 13.2px;"> M={{</span><b style="font-family: arial, tahoma, helvetica, freesans, sans-serif; font-size: 13.2px;"><span lang="ES" style="color: red; font-family: "arial" , sans-serif;">1,2,3},{4,5,6}} shift+enter</span></b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;"><br /></span><span lang="ES" style="font-family: "arial" , sans-serif;"><b>Inversa de una matriz:</b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;">Solamente se aplica a matrices cuadradas</span><br />
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">MatrixForm[Inverse[{{</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">a,c},{b,d}}] shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"> MatrixForm[Inverse[{{</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">1,2,3,4},{2,3,4,0},{3,4,0,0},{4,0,0,0}}]] shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;"><b>Determinante de una matriz:</b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;">Solamente aplica a matices cuadradas </span></div>
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<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"> m={{</span><b><span style="color: red; font-family: "arial" , sans-serif;">a,0,0},{b,c,0},{d,e,f}} shift+enter</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span><br />
<b><span style="color: red; font-family: "arial" , sans-serif;">MatrixForm[m] shift+enter</span></b></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> Det[m]</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">; shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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a c f<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif; position: relative; top: 3.5pt;"><shape id="_x0000_i1073" o:bullet="t" style="height: 14.25pt; width: 27.75pt;" type="#_x0000_t75"><imagedata o:title="" src="file:///C:\Users\VAIO\AppData\Local\Temp\msohtmlclip1\01\clip_image095.wmz"></imagedata></shape></span></div>
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<span style="color: black; font-family: "arial" , sans-serif;"><b>Transpuesta de una matriz:</b></span><br />
<span style="color: black; font-family: "arial" , sans-serif;">Se aplica a cualquier matriz aunque no sea cuadrada</span></div>
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<span style="color: black; font-family: "arial" , sans-serif;">Transpose[m] shift+enter</span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;">Producto punto de vectores:</span></div>
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<span style="color: black; font-family: "arial" , sans-serif;">> a={</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">x,y,1,2} shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif; position: relative; top: 30.5pt;"></span><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> b={</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">3,4,5,6} shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"><span style="font-size: 13.2px;">> </span><span style="font-size: large;">a.b</span></span><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif; font-size: 13.2px;"><b> shift+enter</b></span><span style="font-size: 13.2px;"><span lang="EN-US" style="color: #222222; font-family: "arial" , sans-serif;"></span></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif;">Producto Cruz:</span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif;">Es más cómodo asignar el vector a una variable</span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"> a=</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;"> {1,2,3} shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">b={</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">2,3,4} shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">Cross[a,b] shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Adición (suma) y sustracción (resta) de matrices:<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Para que éstas operaciónes sean válidas, las matrices deben tener las mismas dimensiones<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">La operación se puede realizar directamente, o bien almacenar las matrices en objetos<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 18pt;">a={{2,-1,-2},{0,4,1},{-3,3,5}} </span><span style="font-family: "arial" , sans-serif; font-size: 18pt;">shift+enter</span><br />
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<span style="font-family: "arial" , "sans-serif"; font-size: 18pt;">b={{-3,0,1},{4,3,3},{0,0,4}} shift+enter</span><br />
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<span style="font-family: "arial" , "sans-serif"; font-size: 18pt;">A+B shift+enter<o:p></o:p></span><br />
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<span style="font-family: "arial" , sans-serif;"><span style="font-size: 24px;">A-B shift+enter</span></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">PRODUCTO DE MATRICES:<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Es necesario cumplir con la condición de conformabilidad: el número de columnas de la primera matriz debe ser igual al número de filas de la segunda matriz.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Ejemplo: con las mismas matrices a y b efectúe: axb<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;"> <span style="font-size: large;">a</span><span style="font-size: 24px;">.b; shift+enter</span></span></div>
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Eigenvalores (Valores propios) y Eigenvectores<br />
Para la misma matriz b del ejemplo anterior:<br />
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-66524407474303284242017-06-07T18:07:00.000-06:002017-06-14T15:34:15.470-06:00ÁLGEBRA LINEAL CON MAXIMA<div class="MsoNormal" style="background-color: white; margin: 0cm 0cm 0pt;">
<span lang="ES" style="font-size: 21.3333px;"><span style="color: #222222; font-family: "arial" , sans-serif;"><b>ÁLGEBRA LINEAL CON MAXIMA</b></span></span></div>
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<span lang="ES" style="font-size: 21.3333px;"><span style="color: #222222; font-family: "arial" , sans-serif;"><b>Ing. Luis Manfredo Reyes</b></span></span></div>
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<i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span></i>
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">El Algebra lineal es una rama de la matemática en la cual se opera con entidades llamadas Matrices y Vectores.</span><br />
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En éste documento se discuten las principales operaciones quese pueden realizar usando el programa Maxima.</span><br />
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<i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Maxima es un sistema de cálculo simbólico, que fue desarrollado inicialmente <span class="apple-converted-space"> </span>en el lenguaje de programación <span class="apple-converted-space"> </span>Lisp.</span></i><span class="apple-converted-space"><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"> </span></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">Maxima es un software que se derivó del sistema original Macsyma, desarrollado en el MIT (Massachusetts Institute of Technology) entre los años 1968 y 1982 como parte de un proyecto de investigación en computación simbólica</span><span class="apple-converted-space" style="font-family: "arial" , sans-serif; font-size: 10pt;"> </span><span style="font-family: "arial" , sans-serif; font-size: 10pt;"> llamado “MAC”.</span><span class="apple-converted-space" style="font-family: "arial" , sans-serif; font-size: 10pt;"> </span><br />
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En un gesto digno de agradecimiento, el MIT otorgó <span class="apple-converted-space"> </span>una copia del código fuente original del software <span class="apple-converted-space"> </span>al DOE (Department of Energy) en 1982, en una versión conocida como DOE-Macsyma.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Una de estas copias fue custodiada por el Profesor William F. Schelter de la Universidad de Texas desde el año 1982 hasta su fallecimiento en 2001.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En 1998 gracias a las gestiones y perseverancia de Schelter, se <span class="apple-converted-space"> </span>logró obtener el permiso del Departamento de Energía para distribuir el programa bajo la llamada licencia GNU-GPL, iniciando en el año 2000 el proyecto Maxima en SourceForge con el fin de mantener y seguir desarrollando DOE-Macsyma, ahora con el nombre de Maxima.<span class="apple-converted-space"> </span></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">El Software puede descargarse, instalarse, utilizarse en forma gratuita, siempre que no se use con fines comerciales, ni se cobre por su uso. El sitio es:<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><a href="http://sourceforge.net/projects/maxima/files/latest/download?source=files"><span lang="ES" style="color: #888888; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">http://sourceforge.net/projects/maxima/files/latest/download?source=files</span></a><o:p></o:p></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">Maxima posee un<span class="apple-converted-space"> </span> amplio conjunto de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática"><span lang="ES" style="color: #888888; text-decoration-line: none;">funciones</span></a><span class="apple-converted-space"><span lang="ES" style="color: #222222; mso-ansi-language: ES;"> </span></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">para hacer manipulación simbólica de </span><a href="http://es.wikipedia.org/wiki/Polinomio" title="Polinomio"><span lang="ES" style="color: #888888; text-decoration-line: none;">polinomios</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas)"><span lang="ES" style="color: #888888; text-decoration-line: none;">matrices</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, funciones racionales,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración"><span lang="ES" style="color: #888888; text-decoration-line: none;">integración</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Derivada" title="Derivada"><span lang="ES" style="color: #888888; text-decoration-line: none;">derivación</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, manejo de gráficos en 2D y 3D, manejo de números de punto flotante y grandes, expansión en series de potencias y de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Fourier" title="Fourier"><span lang="ES" style="color: #888888; text-decoration-line: none;">Fourier</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, entre otras funcionalidades.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><o:p></o:p></span></div>
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<span style="color: #222222;"><br /></span><span style="color: #222222;">EXISTE UNA VERSIÓN DE MAXIMA PARA ANDROID E IOS, tanto para tabletas como para teléfonos inteligentes.<o:p></o:p></span></div>
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<b><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 16pt;"> OPERACIONES CON MATRICES Y VECTORES</span></b></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;"><b> Definición de una matriz:</b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;"><b>forma directa:</b></span></div>
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<span lang="ES" style="color: black; font-family: "arial" , sans-serif;">> m</span><b><span lang="ES" style="color: red; font-family: "arial" , sans-serif;">atrix([[1,2,3],[4,5,6]]); shift+enter</span></b><span lang="ES" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;">Otra forma:</span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">< < 1 | 2 | 3 >, < 4 | 5 | 6 > >; shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;">FORMA INDIRECTA: se asigna a una variable</span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;"><span lang="ES" style="color: black; font-family: "arial" , sans-serif; font-size: 13.2px;"> M: m</span><b style="font-family: arial, tahoma, helvetica, freesans, sans-serif; font-size: 13.2px;"><span lang="ES" style="color: red; font-family: "arial" , sans-serif;">atrix([[1,2,3],[4,5,6]]); shift+enter</span></b></span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;"><br /></span>
<span lang="ES" style="font-family: "arial" , sans-serif;">Inversa de una matriz:</span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;">Solamente se aplica a matrices cuadradas</span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;"><br /></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> invert(matrix([</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">a,c],[b,d])); shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> invert(matrix([</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">1,2,3,4],[2,3,4,0],[3,4,0,0],[4,0,0,0])); shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;">Determinante de una matriz:</span><br />
<span lang="ES" style="font-family: "arial" , sans-serif;">Solamente aplica a matices cuadradas </span></div>
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<span lang="ES" style="color: black; font-family: "arial" , sans-serif;"> </span><span style="color: black; font-family: "arial" , sans-serif;">M: ma</span><b><span style="color: red; font-family: "arial" , sans-serif;">trix([a,0,0],[[b,c,0],[d,e,f]); shift+enter</span></b><span style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> d</span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">eterminant(M); shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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acf<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif; position: relative; top: 3.5pt;"><shape id="_x0000_i1073" o:bullet="t" style="height: 14.25pt; width: 27.75pt;" type="#_x0000_t75"><imagedata o:title="" src="file:///C:\Users\VAIO\AppData\Local\Temp\msohtmlclip1\01\clip_image095.wmz"></imagedata></shape></span></div>
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<span style="color: black; font-family: "arial" , sans-serif;">Transpuesta de una matriz:</span><br />
<span style="color: black; font-family: "arial" , sans-serif;">Se aplica a cualquier matriz aunque no sea cuadrada</span></div>
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<span style="color: black; font-family: "arial" , sans-serif;">transpose(M); shift+enter</span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif;">Producto punto de vectores:</span></div>
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<span style="color: black; font-family: "arial" , sans-serif;">> </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">V1 : matrix([x,y,1,2]);</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif; position: relative; top: 30.5pt;"></span><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">V2 :matriz([3,4,5,6]);</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> V1 . V2 </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">;</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif;">Producto Cruz:</span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif;">Es más cómodo asignar la matriz a una variable</span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">V1 : matrix([1,2,3]; shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">V2 : matrix([2,3,4]); shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;">> </span><b><span lang="EN-US" style="color: red; font-family: "arial" , sans-serif;">CrossProduct(V1, V2); shift+enter</span></b><span lang="EN-US" style="color: black; font-family: "arial" , sans-serif;"></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Adición (suma) y sustracción (resta) de matrices:<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Para que éstas operaciónes sean válidas, las matrices deben tener las mismas dimensiones<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">La operación se puede realizar directamente, o bien almacenar las matrices en objetos<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 18pt;">A: matrix([[2,-1,-2],[0,4,1],[-3,3,5]]); </span><span style="font-family: "arial" , sans-serif; font-size: 18pt;">shift+enter</span><br />
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<span style="font-family: "arial" , "sans-serif"; font-size: 18pt;">B: matrix([[-3,0,1],[4,3,3],[0,0,4]]); shift+enter</span><br />
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<span style="font-family: "arial" , "sans-serif"; font-size: 18pt;">A+B; shift+enter<o:p></o:p></span><br />
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<span style="font-family: "arial" , sans-serif;"><span style="font-size: 24px;">A-B; shift+enter</span></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;"><o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">PRODUCTO DE MATRICES:<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Es necesario cumplir con la condición de conformabilidad: el número de columnas de la primera matriz debe ser igual al número de filas de la segunda matriz.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;">Ejemplo: con las mismas matrices A y B efectúe: AxB<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; mso-ansi-language: ES-GT;"> <span style="font-size: 24px;">A*B; shift+enter</span></span></div>
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EIGENVALORES (VALORES PROPIOS) Y EIGENVECTORES<br />
Usando la misma matriz B del ejemplo anterior:<br />
<b> B: matrix([[-3,0,1],[4,3,3],[0,0,4]]); </b> shift+enter<br />
eigenvalues(B); shift+enter<br />
eigenvectors(B); shift+enter<br />
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[-4,-3,3]<br />
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[[1,25,7],[-3,2,0],[0,1,0]]<br />
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-51968679396791632212017-06-05T17:54:00.001-06:002017-06-06T11:11:15.046-06:00VISUALIZACIÓN Y CÁLCULO DE VOLÚMENES DE SÓLIDOS DE REVOLUCIÓN CON MAXIMA<div class="MsoNormal">
<span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><b>VISUALIZACIÓN Y CÁLCULO DE VOLÚMENES DE SÓLIDOS DE REVOLUCIÓN CON MAXIMA</b></span><br />
<span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Ing. Luis Manfredo Reyes<o:p></o:p></span><br />
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<span lang="ES" style="font-family: "times" , serif;">Ing. Luis Manfredo Reyes Chàvez, </span></div>
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<span lang="ES" style="font-family: "times" , serif;">Ares Físico-Matemática</span></div>
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<span lang="ES" style="font-family: "times" , serif;">F</span><span style="font-family: "times" , serif;">acultad de Ciencias Químicas y Farmacia USAC</span></div>
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<span lang="ES" style="font-family: "times" , serif;">luismanfredo2000@gmail.com</span></div>
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<span lang="ES" style="font-family: "times" , serif;"></span><b><span lang="EN-US" style="font-family: "times" , serif;"></span></b></div>
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<b><span lang="EN-US" style="font-family: "times" , serif;">INTRODUCCION</span></b><span lang="EN-US" style="font-family: "times" , serif;">:</span></div>
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<span lang="ES" style="font-family: "times" , serif;">Uno de los temas màs difíciles en la enseñanza del càlculo integral es el del càlculo de volúmenes de sòlidos de revoluciòn, debido a la dificultad de visualizar los sòlidos, que requiere en muchos casos bastante imaginación.</span><br />
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<span style="font-family: "arial" , sans-serif; font-size: 13.3333px;">En éste documento se discute el uso de Maxima para realizar la labor. </span><br />
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<span style="font-family: "arial" , sans-serif; font-size: 13.3333px;">Los ejempos han sido tomados, traducidos y adaptados del documento: "wxmaxima for calculus II" del autor Zachary Hannan del Solano Community College</span></div>
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<i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Maxima es un sistema de cálculo simbólico, que fue desarrollado inicialmente <span class="apple-converted-space"> </span>en el lenguaje de programación <span class="apple-converted-space"> </span>Lisp.</span></i><span class="apple-converted-space"><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"> </span></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 10pt;">Maxima es un software que se derivó del sistema original Macsyma, desarrollado en el MIT (Massachusetts Institute of Technology) entre los años 1968 y 1982 como parte de un proyecto de investigación en computación simbólica</span><span class="apple-converted-space" style="font-family: "arial" , sans-serif; font-size: 10pt;"> </span><span style="font-family: "arial" , sans-serif; font-size: 10pt;"> llamado “MAC”.</span><span class="apple-converted-space" style="font-family: "arial" , sans-serif; font-size: 10pt;"> </span><br />
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En un gesto digno de agradecimiento, el MIT otorgó <span class="apple-converted-space"> </span>una copia del código fuente original del software <span class="apple-converted-space"> </span>al DOE (Department of Energy) en 1982, en una versión conocida como DOE-Macsyma.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Una de estas copias fue custodiada por el Profesor William F. Schelter de la Universidad de Texas desde el año 1982 hasta su fallecimiento en 2001.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span></div>
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<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En 1998 gracias a las gestiones y perseverancia de Schelter, se <span class="apple-converted-space"> </span>logró obtener el permiso del Departamento de Energía para distribuir el programa bajo la llamada licencia GNU-GPL, iniciando en el año 2000 el proyecto Maxima en SourceForge con el fin de mantener y seguir desarrollando DOE-Macsyma, ahora con el nombre de Maxima.<span class="apple-converted-space"> </span></span></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">El Software puede descargarse, instalarse, utilizarse en forma gratuita, siempre que no se use con fines comerciales, ni se cobre por su uso. El sitio es:<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><a href="http://sourceforge.net/projects/maxima/files/latest/download?source=files"><span lang="ES" style="color: #888888; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">http://sourceforge.net/projects/maxima/files/latest/download?source=files</span></a><o:p></o:p></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">Maxima posee un<span class="apple-converted-space"> </span> amplio conjunto de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática"><span lang="ES" style="color: #888888; text-decoration-line: none;">funciones</span></a><span class="apple-converted-space"><span lang="ES" style="color: #222222; mso-ansi-language: ES;"> </span></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">para hacer manipulación simbólica de </span><a href="http://es.wikipedia.org/wiki/Polinomio" title="Polinomio"><span lang="ES" style="color: #888888; text-decoration-line: none;">polinomios</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas)"><span lang="ES" style="color: #888888; text-decoration-line: none;">matrices</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, funciones racionales,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración"><span lang="ES" style="color: #888888; text-decoration-line: none;">integración</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Derivada" title="Derivada"><span lang="ES" style="color: #888888; text-decoration-line: none;">derivación</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, manejo de gráficos en 2D y 3D, manejo de números de punto flotante y grandes, expansión en series de potencias y de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Fourier" title="Fourier"><span lang="ES" style="color: #888888; text-decoration-line: none;">Fourier</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, entre otras funcionalidades.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><o:p></o:p></span></div>
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<span style="color: #222222;"><br /></span><span style="color: #222222;">EXISTE UNA VERSIÓN DE MAXIMA PARA ANDROID E IOS, tanto para tabletas como para teléfonos inteligentes.<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "times" , serif;">PROCEDIMIENTO PARA VISUALIZAR SÓLIDOS DE REVOLUCIÓN:</span></b></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "times" , serif;">EJEMPLOS:</span></b></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "times" , serif;">Ejemplo 1:</span></b></div>
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<span lang="ES" style="font-family: "times" , serif;">Calcule el volumen generado con la rotación de y=x<sup>2 </sup>entre x=0<sup>, </sup>x=2 alrededor del eje x</span></div>
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Ingresar la siguiente orden:</div>
<span style="font-family: "times new roman";">load(draw)$</span><br />
<span style="font-family: "times new roman";">draw3d(</span><br />
<span style="font-family: "times new roman";">axis_3d=false,</span><br />
<span style="font-family: "times new roman";">dimensions=[600,600],</span><br />
<span style="font-family: "times new roman";">view=[85,10],</span><br />
<span style="font-family: "times new roman";">xrange=[0,3],</span><br />
<span style="font-family: "times new roman";">yrange=[-4,4],</span><br />
<span style="font-family: "times new roman";">zrange=[-4,4],</span><br />
<span style="font-family: "times new roman";">xtics=false,</span><br />
<span style="font-family: "times new roman";">ytics=false,</span><br />
<span style="font-family: "times new roman";">ztics=false,</span><br />
<span style="font-family: "times new roman";">color=black,</span><br />
<span style="font-family: "times new roman";">nticks=600,</span><br />
<span style="font-family: "times new roman";">line_width=2,</span><br />
<span style="font-family: "times new roman";">parametric(t,0,0,t,0,3),</span><br />
<span style="font-family: "times new roman";">parametric(0,t,0,t,-4,4),</span><br />
<span style="font-family: "times new roman";">parametric(0,0,t,t,-4,4),</span><br />
<span style="font-family: "times new roman";">color=dark_grey,</span><br />
<span style="font-family: "times new roman";">parametric_surface(r,r^2*cos(t),r^2*sin(t),r,0,2,t,0,2*%pi),</span><br />
<span style="font-family: "times new roman";">color=black,</span><br />
<span style="font-family: "times new roman";">parametric_surface(r,r^2*cos(t),r^2*sin(t),r,1.3,1.4,t,0,2*%pi),</span><br />
<span style="font-family: "times new roman";">color=dark_red,</span><br />
<span style="font-family: "times new roman";">parametric_surface(1.4,u^2*cos(t),u^2*sin(t),u,0,1.4,t,0,2*%pi)</span><br />
<span style="font-family: "times new roman";">); shift+enter</span><br />
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<span style="font-family: "times new roman";"><br /></span></div>
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<span style="font-family: "times new roman";"><br /></span></div>
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<span style="font-family: "times new roman";">El resultado obtenido es el siguiente:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitLe6tKNQlFsLndx0esXcfMrsL2W0IdpZDT79PVAuZ-5S-Q2yeSeGaHJCytzrswFvQQQdhcS8JNVCWjFHmgqnTxQhpT0ZFBlLbTX6aC64BdcnMTi4qULQ5eVKpof_N0pDPkzZMCDApMGg/s1600/volmaxima1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="683" data-original-width="605" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitLe6tKNQlFsLndx0esXcfMrsL2W0IdpZDT79PVAuZ-5S-Q2yeSeGaHJCytzrswFvQQQdhcS8JNVCWjFHmgqnTxQhpT0ZFBlLbTX6aC64BdcnMTi4qULQ5eVKpof_N0pDPkzZMCDApMGg/s400/volmaxima1.png" width="353" /></a></div>
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<span style="font-family: "times new roman";"><br /></span></div>
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<span style="font-family: "times new roman";">Para definir los discos, se utilizan funciones paramétricas para el círculo</span></div>
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<span style="font-family: "times new roman";">El disco de análisis se presenta en color negro</span></div>
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<span style="font-family: "times new roman";"><br /></span></div>
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<span style="font-family: "times new roman";">Para calcular el volumen se ingresa:</span></div>
</div>
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<div style="font-family: "times new roman";">
<b><span style="font-size: large;">integrate(%pi*(x^2)^2,x,0,2); shift+enter</span></b></div>
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<b><span style="font-size: large;"><br /></span></b>
</div>
<div style="font-family: "times new roman";">
<b><span style="font-size: large;">(32 %pi)/5</span></b></div>
<div style="font-family: "times new roman";">
<span style="font-size: large;">si se desea resultado decimal se usa el comando float:</span></div>
<div style="font-family: "times new roman";">
<b><span style="font-size: large;">float(integrate(%pi*(x^2)^2,x,0,2)); shift+enter</span></b></div>
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<br /></div>
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<div style="font-family: "times new roman";">
EJEMPLO 2:</div>
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<span lang="ES" style="font-family: "times" , serif;">Calcule el volumen generado con la rotación de y=x<sup>2 </sup>entre x=0<sup>, </sup>x=2 alrededor del eje y</span></div>
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<div style="font-family: "times new roman";">
Ingresar la siguiente orden:<br />
<div>
load(draw) $</div>
</div>
</div>
<span style="font-family: "times new roman";">draw3d(</span><br />
<span style="font-family: "times new roman";">axis_3d=false,</span><br />
<span style="font-family: "times new roman";">dimensions=[600,600],</span><br />
<span style="font-family: "times new roman";">view=[85,10],</span><br />
<span style="font-family: "times new roman";">xrange=[-3,3],</span><br />
<span style="font-family: "times new roman";">yrange=[-3,3],</span><br />
<span style="font-family: "times new roman";">zrange=[0,9],</span><br />
<span style="font-family: "times new roman";">xtics=false,</span><br />
<span style="font-family: "times new roman";">ytics=false,</span><br />
<span style="font-family: "times new roman";">ztics=false,</span><br />
<span style="font-family: "times new roman";">color=black,</span><br />
<span style="font-family: "times new roman";">nticks=600,</span><br />
<span style="font-family: "times new roman";">line_width=2,</span><br />
<span style="font-family: "times new roman";">parametric(t,0,0,t,-3,3),</span><br />
<span style="font-family: "times new roman";">parametric(0,t,0,t,-3,3),</span><br />
<span style="font-family: "times new roman";">parametric(0,0,t,t,0,6),</span><br />
<span style="font-family: "times new roman";">color=dark_grey,</span><br />
<span style="font-family: "times new roman";">parametric_surface(r*cos(t),r*sin(t),r^2,r,0,2,t,0,2*%pi),</span><br />
<span style="font-family: "times new roman";">color=black,</span><br />
<span style="font-family: "times new roman";">parametric_surface(r*cos(t),r*sin(t),r^2,r,1.3,1.4,t,0,2*%pi),</span><br />
<span style="font-family: "times new roman";">color=dark_red,</span><br />
<span style="font-family: "times new roman";">parametric_surface(r*cos(t),r*sin(t),1.4^2,r,0,1.4,t,0,2*%pi)</span><br />
<span style="font-family: "times new roman";">);</span><br />
<span style="font-family: "times new roman";">Y se obtiene e siguiente resultado:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAvW9vaQ4gSp1is-m8FQhZ8u0omqHuJW_yNNk2a_2Q6EvjNrD6PEkvIAJwaeSkDxo-GQ6ksBlhWlkbZCCRGufyVV6PnpsqCX-E6W77JMON5e5oF1tf6Dwk8NUd2zxnfxv540katJIASJA/s1600/volmax2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="605" data-original-width="597" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAvW9vaQ4gSp1is-m8FQhZ8u0omqHuJW_yNNk2a_2Q6EvjNrD6PEkvIAJwaeSkDxo-GQ6ksBlhWlkbZCCRGufyVV6PnpsqCX-E6W77JMON5e5oF1tf6Dwk8NUd2zxnfxv540katJIASJA/s400/volmax2.png" width="393" /></a></div>
<span style="font-family: "times new roman";"><br /></span>
<span style="font-family: "times new roman";">Para calcular el volumen se ingresa la orden:</span><br />
<span style="font-family: "times new roman";">integrate(%pi*(sqrt(y)^2,y,0,2) </span><br />
<span style="font-family: "times new roman";">y a respuesta es: </span><br />
<div style="font-family: "times new roman";">
8*%pi</div>
<div style="font-family: "times new roman";">
<br /></div>
<div style="font-family: "times new roman";">
EJEMPLO 3: Calcule el volumen del sólido obtenido cuando la región formada por y=0.3x, y=sen x rota alrededor del eje x.<br />
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Inicialmente, se grafican las funciones en 2 dimensiones para observar la intersección y se calcula la intersección.<br />
<br />
load(draw)$<br />
f(x):=0.3*x$<br />
g(x):=sin(x)$ shitf+enter<br />
find_root(f(x)-g(x),x,1.4); shit enter<br />
2.3564411<br />
<br />
draw2d( xaxis=true, yaxis=true,xtics=false,ytics=false,<br />
dimensions=[600,600], xrange=[-%pi,%pi], yrange=[-1.2,1.2],<br />
fill_color=grey, filled_func=true, filled_func=f(x),<br />
explicit(g(x),x,0,2.356), filled_func=false,<br />
line_width=2,color=black,key="f(x)",<br />
explicit(f(x),x,-%pi,%pi), color=red,<br />
key="g(x)", expicit(g(x),x,-%pi,%pi) );<br />
<br />
Resultado:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixJLbIjC601tLlEetVwmuM1n53kIGjmoyWy9N-s-9iI-QT50XK84vXmyZ-onweK5E6xqwT_UhDp7tn1KkNjiyK3fIvmeKZe3GZZcLW8_upbRduPg-WtaXpl5Km84uPpxxW878EdgEQRcg/s1600/volmax3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="340" data-original-width="391" height="278" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixJLbIjC601tLlEetVwmuM1n53kIGjmoyWy9N-s-9iI-QT50XK84vXmyZ-onweK5E6xqwT_UhDp7tn1KkNjiyK3fIvmeKZe3GZZcLW8_upbRduPg-WtaXpl5Km84uPpxxW878EdgEQRcg/s320/volmax3.png" width="320" /></a></div>
<br />
Ahora se procede a la rotación:<br />
draw3d( xxis_3d=false, yaxis=true,xtics=false,ytics=false,ztics=false,<br />
dimensions=[600,600],view=[85,10], xrange=[-%pi,%pi], yrange=[-1.2,1.2], zrange=[-1.2,1.2],<br />
color=black, nticks=200, line_width=2,<br />
parametric(t,0,0,t,-%pi,%pi),<br />
parametric(0,t,0,t,-1.2,1.2),<br />
parametric(0,0,t,t,-1.2,1.2),<br />
color=grey, transparent=true,<br />
parametric_surface(r,g(r)*sin(t), g(r)*cos(t),r,-2.356,2.356,t,0,2*%pi<br />
transparent=false, color=dark_grey,<br />
parametric_surface(r,f(r)*sin(t), f(r)*cos(t),r,-2.356,2.356,t,0,2*%pi<br />
clor=black,<br />
parametric_surface(1.6,u*sin(t), u*cos(t),u,f(1.6), g(1.6),t,0,2*%pi<br />
); shift+enter<br />
<br />
y e resultado obtenido es:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUSaDkzM2rUnCttnsjK59WmTgfP_BkXQY0gMHPu1vM2WNctoEt_0FEn0U126HmNAd4C3AbLNKga7Sk62T8AxgqtPSqyRUtzJRTqEIPCV2O-ve4LaLrzxBoj4JdqLnPrkh9lVIqwj1Bl3A/s1600/volmax4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="364" data-original-width="391" height="297" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUSaDkzM2rUnCttnsjK59WmTgfP_BkXQY0gMHPu1vM2WNctoEt_0FEn0U126HmNAd4C3AbLNKga7Sk62T8AxgqtPSqyRUtzJRTqEIPCV2O-ve4LaLrzxBoj4JdqLnPrkh9lVIqwj1Bl3A/s320/volmax4.png" width="320" /></a></div>
<br />
<br />
Para calcuar el volumen, se ingresa la integral:<br />
float(2*integrate(%pi*(sin(x)^2-(0.3*x)^2,x,0,2.356)); shift+enter<br />
<br />
y el volumen deseado es: 6.50733<br />
<br />
<br /></div>
</div>
</div>
</div>
</div>
Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-66569919311690001412017-05-31T16:33:00.001-06:002017-05-31T18:42:03.860-06:00WXMAXIMA: INTERFASE ESTILO WINDOWS PARA MAXIMAWXMAXIMA: INTERFASE ESTILO WINDOWS PARA MAXIMA<br />
Ing. Luis Manfredo Reyes<br />
<br />
Introducción:<br />
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<i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Maxima es un sistema de cálculo simbólico, que fue desarrollado inicialmente <span class="apple-converted-space"> </span>en el lenguaje de programación <span class="apple-converted-space"> </span>Lisp.</span></i><span class="apple-converted-space"><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"> </span></span><br />
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Maxima es un software que se derivó del sistema original Macsyma, desarrollado en el MIT (Massachusetts Institute of Technology) entre los años 1968 y 1982 como parte de un proyecto de investigación en computación simbólica<span class="apple-converted-space"> </span> llamado “MAC”.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En un gesto digno de agradecimiento, el MIT otorgó <span class="apple-converted-space"> </span>una copia del código fuente original del software <span class="apple-converted-space"> </span>al DOE (Department of Energy) en 1982, en una versión conocida como DOE-Macsyma.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Una de estas copias fue custodiada por el Profesor William F. Schelter de la Universidad de Texas desde el año 1982 hasta su fallecimiento en 2001.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En 1998 gracias a las gestiones y perseverancia de Schelter, se <span class="apple-converted-space"> </span>logró obtener el permiso del Departamento de Energía para distribuir el programa bajo la llamada licencia GNU-GPL, iniciando en el año 2000 el proyecto Maxima en SourceForge con el fin de mantener y seguir desarrollando DOE-Macsyma, ahora con el nombre de Maxima.<span class="apple-converted-space"> </span></span><br />
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"></span><br />
<a name='more'></a><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">El Software puede descargarse, instalarse, utilizarse en forma gratuita, siempre que no se use con fines comerciales, ni se cobre por su uso. El sitio es:<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><a href="http://sourceforge.net/projects/maxima/files/latest/download?source=files"><span lang="ES" style="color: #888888; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">http://sourceforge.net/projects/maxima/files/latest/download?source=files</span></a><o:p></o:p></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">Maxima posee un<span class="apple-converted-space"> </span> amplio conjunto de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática"><span lang="ES" style="color: #888888; text-decoration-line: none;">funciones</span></a><span class="apple-converted-space"><span lang="ES" style="color: #222222; mso-ansi-language: ES;"> </span></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">para hacer manipulación simbólica de </span><a href="http://es.wikipedia.org/wiki/Polinomio" title="Polinomio"><span lang="ES" style="color: #888888; text-decoration-line: none;">polinomios</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas)"><span lang="ES" style="color: #888888; text-decoration-line: none;">matrices</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, funciones racionales,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración"><span lang="ES" style="color: #888888; text-decoration-line: none;">integración</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Derivada" title="Derivada"><span lang="ES" style="color: #888888; text-decoration-line: none;">derivación</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, manejo de gráficos en 2D y 3D, manejo de números de punto flotante y grandes, expansión en series de potencias y de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Fourier" title="Fourier"><span lang="ES" style="color: #888888; text-decoration-line: none;">Fourier</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, entre otras funcionalidades.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><o:p></o:p></span></div>
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<span style="color: #222222;"><br /></span><span style="color: #222222;">EXISTE UNA VERSIÓN DE MAXIMA PARA ANDROID E IOS, tanto para tabletas como para teléfonos inteligentes.<o:p></o:p></span><br />
<div>
<span style="color: #222222;"><br /></span>
<span style="color: #222222;">Una queja frecuente de los usuarios de Maxima, es que se deben ingresar comandos manualmente, algunos con una estructura poco "elegante".</span><br />
<span style="color: #222222;"><br /></span>
<span style="color: #222222;">Por ejemplo, para resolver la integral de xcos x entre '1 y pi, se ingresa:</span><br />
<span style="color: #222222;">Integrate[x*Cos[x],{x,-1,3.1416}]</span><br />
<span style="color: #222222;"><br /></span>
<span style="color: #222222;">El equipo de trabajo de Sourceforge, ha desarrollado una interfase estilo Windows, para facilitar el uso del sistema y atraer mas usuarios.</span><br />
<span style="color: #222222;">Se llama Wxmaxima y es parte del instalador de Maxima que se descarga para Windows, Linux y Mac.</span><br />
<span style="color: #222222;"><br /></span>
<span style="color: #222222;">Ambiente de trabajo: la ventana de Wxmaxima es la siguiente:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFlDrFee1P632K3iaIfNMymhbbdcQKSf3HtTDun7RGDWSkTbZLyqcT3RpLKlO54jjlRt4FUa5QTzLRpKCqp4QdktBnx13Lr2Be1Zxdy6nYSDZIVi_-36DfmOxrFoha3ClCZVhWqQR_ha8/s1600/wx1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="653" data-original-width="957" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFlDrFee1P632K3iaIfNMymhbbdcQKSf3HtTDun7RGDWSkTbZLyqcT3RpLKlO54jjlRt4FUa5QTzLRpKCqp4QdktBnx13Lr2Be1Zxdy6nYSDZIVi_-36DfmOxrFoha3ClCZVhWqQR_ha8/s400/wx1.png" width="400" /></a></div>
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<span style="color: #222222;"><br /></span></div>
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<span style="color: #222222;">Las pestaña de Archivos tiene las típicas funciones de Windows, Mac y Linux</span></div>
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<span style="color: #222222;"><br /></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7uZ3Y0WKOGo-2vVZB-LRimyTl4Wk1kxd9acK0jL1fRFT_3AlIX0aZvSsc_E9ZtJF7CVAnjxj7bHH7ELe67Ah90nrAMXCwpyjuWnn9UnurzQ5FsrNLXESuu_NkelOT3d3ym24f1E-WPiI/s1600/wx2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="300" data-original-width="389" height="246" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7uZ3Y0WKOGo-2vVZB-LRimyTl4Wk1kxd9acK0jL1fRFT_3AlIX0aZvSsc_E9ZtJF7CVAnjxj7bHH7ELe67Ah90nrAMXCwpyjuWnn9UnurzQ5FsrNLXESuu_NkelOT3d3ym24f1E-WPiI/s320/wx2.png" width="320" /></a></div>
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<span style="color: #222222;">Tambien la pestaña Editar tiene las típicas opciones de cualquier programa para Windows, Linux y Mac:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3p1xBaMh1UjwQNaQ0sxzCUeGHJnRoc87UpWOwnfAC9WYoxXXERMG1rY00Qgcy1sEbSIT_z-vSXT2NkOsbPZjMbGa7WVatd3ZAsAax2GTGvIi-AUrxyB1TGXCsLqIeHKVznG6i_IQA5p8/s1600/wx3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="428" data-original-width="389" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3p1xBaMh1UjwQNaQ0sxzCUeGHJnRoc87UpWOwnfAC9WYoxXXERMG1rY00Qgcy1sEbSIT_z-vSXT2NkOsbPZjMbGa7WVatd3ZAsAax2GTGvIi-AUrxyB1TGXCsLqIeHKVznG6i_IQA5p8/s320/wx3.png" width="290" /></a></div>
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<span style="color: #222222;"><br /></span></div>
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<span style="color: #222222;">La pestaña de Ecuaciones contiene muchas opciones para resolver diversos tipos de ecuaciones:</span></div>
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</div>
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlBpKk-X0TVgfxnAv26d2YITYL6acuU9veYsBtvzfOuwC8Sdt4Q91BoEn_xK4xvUTBUadEkgpTcbumXO9PyegQcEwGfjcx6ldnA4naJS1sOmou8eO7N5-aZ1DZojvShsby0pzteBDcql4/s1600/wx4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="378" data-original-width="389" height="310" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlBpKk-X0TVgfxnAv26d2YITYL6acuU9veYsBtvzfOuwC8Sdt4Q91BoEn_xK4xvUTBUadEkgpTcbumXO9PyegQcEwGfjcx6ldnA4naJS1sOmou8eO7N5-aZ1DZojvShsby0pzteBDcql4/s320/wx4.png" width="320" /></a></div>
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</div>
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<span style="color: #222222;">En cada una de las opciones, se usa el formato algebraico propio de Maxima.</span></div>
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<span style="color: #222222;">Por ejemplo, para resolver la ecuación: </span>4x<sup>2</sup>+2x-8=0</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjt38zcTVmXsqTYmOA1ZBE9HxJ-naW7PAL0IlNWBxT4zldI8pUmr1QSOTav0vN8wEd1j7NbTFg2rQv4O_iWMmD2VUUIi19vjb3ObkB0yKTfiWjWItq3YXnvzUcCYHK5-X87EtOKpwNkMW0/s1600/wx5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="151" data-original-width="341" height="141" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjt38zcTVmXsqTYmOA1ZBE9HxJ-naW7PAL0IlNWBxT4zldI8pUmr1QSOTav0vN8wEd1j7NbTFg2rQv4O_iWMmD2VUUIi19vjb3ObkB0yKTfiWjWItq3YXnvzUcCYHK5-X87EtOKpwNkMW0/s320/wx5.png" width="320" /></a></div>
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La interfase genera el codigo de Maxima y presenta la respuesta:</div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJefxeQkJrOWeMDklErpUDUGRS5YsTRXIaGXiv8KesKxndzfPLiLcGYFT-qsp-IyfVcXkYbL9wHypZLfbUPZXyl-nFfM1V4KuAxxmtA2CJtJPZQz7JYkwxOjc_H68BueYKmMlI9lfbClU/s1600/wx6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="159" data-original-width="382" height="133" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJefxeQkJrOWeMDklErpUDUGRS5YsTRXIaGXiv8KesKxndzfPLiLcGYFT-qsp-IyfVcXkYbL9wHypZLfbUPZXyl-nFfM1V4KuAxxmtA2CJtJPZQz7JYkwxOjc_H68BueYKmMlI9lfbClU/s320/wx6.png" width="320" /></a></div>
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La siguiente pestana permite realizar diversas operaciones con matrices:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcsikV10C4tEy-9X59nTmXJRsKL67JtkR2hWTczytg-BMROpS8DHrswwhvYWZYEOtDkXyWMEC0Os7W951q_x8Ai3zKY1k_QkhK7ojJ898pbIPom4kVYD1Q-zFU6YLkrf4X6QlOxxSefgQ/s1600/wx7.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="345" data-original-width="341" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcsikV10C4tEy-9X59nTmXJRsKL67JtkR2hWTczytg-BMROpS8DHrswwhvYWZYEOtDkXyWMEC0Os7W951q_x8Ai3zKY1k_QkhK7ojJ898pbIPom4kVYD1Q-zFU6YLkrf4X6QlOxxSefgQ/s320/wx7.png" width="316" /></a></div>
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En la siguiente seccion estan las opciones de calculo importantes:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgterpK4db5GBODh27wE_rjqfUq4TYonQnQemsJ7eV-5AKDgPqxrX2GwfoKKuMC6QEz9CJEcR4bx4oWxoFPy4EBQJnnWAifCjulzz_Kn7Y6O7SKnkK5Prh2u6GtJjOqKkN4NMjhgABS7T0/s1600/wx9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="407" data-original-width="341" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgterpK4db5GBODh27wE_rjqfUq4TYonQnQemsJ7eV-5AKDgPqxrX2GwfoKKuMC6QEz9CJEcR4bx4oWxoFPy4EBQJnnWAifCjulzz_Kn7Y6O7SKnkK5Prh2u6GtJjOqKkN4NMjhgABS7T0/s320/wx9.png" width="268" /></a></div>
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En la seccion Simplificar se presentan las opciones de manipulacion algebraica mas comunes</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0NOxdQi6819BVVl-8jq1tACb_JmIQ9gMR94be93woYQRHohd1N0slFYq22zsinBZwvMGWdmr9UVO4z2IOAoOD8SJShVittraKonrcgroRJqzvkH-h3Cv90nIzk0QjrXEMBOPHD9yeyns/s1600/wx10.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="376" data-original-width="341" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0NOxdQi6819BVVl-8jq1tACb_JmIQ9gMR94be93woYQRHohd1N0slFYq22zsinBZwvMGWdmr9UVO4z2IOAoOD8SJShVittraKonrcgroRJqzvkH-h3Cv90nIzk0QjrXEMBOPHD9yeyns/s320/wx10.png" width="290" /></a></div>
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En la seccion Graficos, se presentan los comandos para graficas en 2 y 3 dimensiones:</div>
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Tambien existen opciones para la manipulacion numerica de datos:</div>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-9236241955215580732017-05-31T11:16:00.000-06:002017-05-31T11:16:03.142-06:00VISUALIZACIÓN Y CÁLCULO DE VOLÚMENES DE SÓLIDOS DE REVOLUCIÓN CON MATHEMATICA<div class="MsoNormal">
<span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">VISUALIZACIÓN Y CÁLCULO DE VOLÚMENES DE SÓLIDOS DE REVOLUCIÓN CON MATHEMATICA</span><br />
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<span style="background: white; color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Ing. Luis Manfredo Reyes<o:p></o:p></span><br />
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<span lang="ES" style="font-family: "times" , serif;">Ing. Luis Manfredo Reyes Chàvez, </span></div>
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<span lang="ES" style="font-family: "times" , serif;">Ares Físico-Matemática</span></div>
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<span lang="ES" style="font-family: "times" , serif;">F</span><span style="font-family: "times" , serif;">acultad de Ciencias Químicas y Farmacia USAC</span></div>
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<span lang="ES" style="font-family: "times" , serif;">luismanfredo2000@gmail.com</span></div>
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<span lang="ES" style="font-family: "times" , serif;"><br /></span><a href="https://www.blogger.com/null" name="more"></a></div>
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<span lang="ES" style="font-family: "times" , serif;"></span><b><span lang="EN-US" style="font-family: "times" , serif;"></span></b></div>
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<b><span lang="EN-US" style="font-family: "times" , serif;">INTRODUCCION</span></b><span lang="EN-US" style="font-family: "times" , serif;">:</span></div>
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<span lang="ES" style="font-family: "times" , serif;">Uno de los temas màs difíciles en la enseñanza del càlculo integral es el del càlculo de volúmenes de sòlidos de revoluciòn, debido a la dificultad de visualizar los sòlidos, que requiere en muchos casos bastante imaginación.</span></div>
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<span lang="ES">Siendo el paquete computacional Mathematica ® (un producto de Wolfram Research) uno de los más importantes en el campo de la computación aplicada a la matemática, no es de extrañar que contenga utilidades para la resolución de aplicaciones de cálculo integral.</span></div>
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<span lang="ES">En este documento se analizan los casos más importantes de cálculo de volúmenes de sólidos de revolución. . aplicando el paquete. Sin embargo, se debe tomar en cuenta que, éste software no es un sustituto del estudio ni de la resolución manual. Es solamente una herramienta de apoyo.</span></div>
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<span lang="ES"><br /></span><span lang="ES">MATHEMATICA ES UN SOFTWARE COMERCIAL, POR LO QUE TIENE UN COSTO (Y ELEVADO POR CIERTO). No se recomienda el uso "pirata" del mismo.</span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "times" , serif;">PROCEDIMIENTO PARA VISUALIZAR SÓLIDOS DE REVOLUCIÓN:</span></b></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "times" , serif;">EJEMPLOS:</span></b></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "times" , serif;">Ejemplo 1:</span></b></div>
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<span lang="ES" style="font-family: "times" , serif;">Calcule el volumen generado con la rotación de y=x<sup>2 </sup>entre x=0<sup>, </sup>x=2 alrededor del eje x</span></div>
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Ingresar la siguiente orden:</div>
<b><span style="font-size: large;">RevolutionPlot3D[x^2, {x, 0, 2}, RevolutionAxis -> {1, 0, 0}]</span></b></div>
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<shape id="_x0000_s1030" style="height: 196.5pt; left: 0px; margin-left: 273.9pt; margin-top: 2.6pt; position: absolute; width: 174pt; z-index: 251658752;" type="#_x0000_t75"><imagedata o:title="" src="file:///C:\DOCUME~1\Usurio\CONFIG~1\Temp\msohtmlclip1\01\clip_image031.png"></imagedata><wrap type="square"></wrap></shape><span lang="ES" style="font-family: "times" , serif;"></span></div>
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shift+enter<br />
la orden tiene tres partes: la función, los límites de rotación y el eje sobre el que rota. Cuando es sobre el eje x, se ingresa {1,0,0}<br />
La gráfica producida se puede rotar en cualquier dirección.<br />
El resultado es éste:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGp4sf4j8-dWDvsjHCcFydBch1sTzKLU3cvFawTo8Z_KN4CGiRavvTTrxHx2wQgSfIV59HDkQq90kUQlYot7LYYc9y_mL7yN1hTrR9FIyL2KSWVSFkDkN5zIXBO-zLy4cn8h7pswGk-VQ/s1600/solidmathe1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="485" data-original-width="538" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGp4sf4j8-dWDvsjHCcFydBch1sTzKLU3cvFawTo8Z_KN4CGiRavvTTrxHx2wQgSfIV59HDkQq90kUQlYot7LYYc9y_mL7yN1hTrR9FIyL2KSWVSFkDkN5zIXBO-zLy4cn8h7pswGk-VQ/s400/solidmathe1.png" width="400" /></a></div>
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Para calcular el volumen se ingresa:<br />
<b><span style="font-size: large;">Integrate[%pi*(x^2)^2,{x,0,2}] shift+enter</span></b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkdlMHA_LW0Pz2hkd4QqGaW1-f7LcUeoOsxhyqp3jKkrhAIPqj4FSEja8U31gi5OI1PZ6oP2tcwLBNSSlQHNinUgFLG2TPTlsj-kqt4TX_VQu0Zc8iix9bSAYyDlUVPkn4yWfYn0LCbwY/s1600/solidmathe2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="134" data-original-width="362" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkdlMHA_LW0Pz2hkd4QqGaW1-f7LcUeoOsxhyqp3jKkrhAIPqj4FSEja8U31gi5OI1PZ6oP2tcwLBNSSlQHNinUgFLG2TPTlsj-kqt4TX_VQu0Zc8iix9bSAYyDlUVPkn4yWfYn0LCbwY/s640/solidmathe2.png" width="640" /></a></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="ES" style="font-family: "times" , serif;">Ejemplo 2:</span></b></div>
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<span lang="ES" style="font-family: "times" , serif;">La región formada por: f(x)=sen x y el eje x entre x=-1 y x=</span><span lang="ES" style="font-family: "arial" , sans-serif;">π</span><span lang="ES" style="font-family: "times" , serif;"> rota alrededor de y=-1. Calcule el volumen del sólido resultante</span></div>
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Ingresar: </div>
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<span style="font-family: times new roman;"><b>RevolutionPlot3D[Sin[x] -1, {x, -1, 3.1416}, RevolutionAxis -> {-1, 0, 0}] shift+enter</b></span><br />
<span style="font-family: times new roman;">A la función original se debe restar la distancia al eje de rotación (1 en éste caso), y el eje de rotación se especifica {-1,0,0}</span><br />
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<shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"> El gráfico producido es el siguiente:</shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe1zKRwzRx9Iqu2hZQUuyhQ6baue7fzuDyUgUjJRQUEgeGz25uE3r_4xQ9_q84Kqzl5eFU9x6xAX2yDby99-cmup_SZYKaVODL3mD3Ob5b3Hi80YXvYvDoz6OIzG9b8RKS79MBtDdKegk/s1600/solidmathe3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="441" data-original-width="637" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe1zKRwzRx9Iqu2hZQUuyhQ6baue7fzuDyUgUjJRQUEgeGz25uE3r_4xQ9_q84Kqzl5eFU9x6xAX2yDby99-cmup_SZYKaVODL3mD3Ob5b3Hi80YXvYvDoz6OIzG9b8RKS79MBtDdKegk/s400/solidmathe3.png" width="400" /></a></div>
<shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"> </shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"> </shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"> </shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75">La integral para calcular el volumen es: <b>Integrate[%pi*(Sin[x]^2-1),{x,-1,%pi}], </b>y el resultado obtenido es:<b> 1.6498 </b></shape><shape id="_x0000_s1031" style="height: 201pt; left: 0px; margin-left: 0px; margin-top: -0.3pt; position: absolute; width: 300pt; z-index: 251659776;" type="#_x0000_t75"><br /></shape></div>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0tag:blogger.com,1999:blog-3997797568400374468.post-71247414095136961632017-05-29T15:51:00.000-06:002017-05-30T11:41:54.261-06:00VISUALIZACIÓN Y CÁLCULO DE ÁREAS ENTRE GRÁFICAS DE FUNCIONES USANDO MAXIMA<div class="MsoNormal" style="margin-left: .5in;">
<b><span lang="ES" style="font-size: 20pt; line-height: 30.6667px;">VISUALIZACIÓN Y CÁLCULO DE ÁREAS ENTRE GRÁFICAS DE FUNCIONES USANDO MAXIMA<o:p></o:p></span></b><br />
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<span lang="ES">Ing. Luis Manfredo Reyes<o:p></o:p></span></div>
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<span lang="ES">Otra de las aplicaciones de la integral que se enseña en los cursos de cálculo , es la de calcular el área entre la gráfica de funciones (incorrectamente llamada “área entre curvas”). </span></div>
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<span lang="ES">Para ello es necesario realizar las gráficas de las funciones, determinar los límites de integración y luego “armar” la integral que representa el área.<o:p></o:p></span></div>
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<span lang="ES">Aunque existen muchas opciones para realizar gráficas, por su facilidad de uso y su condición de software libre, aquí se discute el uso de Maxima<o:p></o:p></span><br />
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<i><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Maxima es un sistema de cálculo simbólico, que fue desarrollado inicialmente <span class="apple-converted-space"> </span>en el lenguaje de programación <span class="apple-converted-space"> </span>Lisp.</span></i><span class="apple-converted-space"><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"> </span></span><br />
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Maxima es un software que se derivó del sistema original Macsyma, desarrollado en el MIT (Massachusetts Institute of Technology) entre los años 1968 y 1982 como parte de un proyecto de investigación en computación simbólica<span class="apple-converted-space"> </span> llamado “MAC”.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En un gesto digno de agradecimiento, el MIT otorgó <span class="apple-converted-space"> </span>una copia del código fuente original del software <span class="apple-converted-space"> </span>al DOE (Department of Energy) en 1982, en una versión conocida como DOE-Macsyma.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">Una de estas copias fue custodiada por el Profesor William F. Schelter de la Universidad de Texas desde el año 1982 hasta su fallecimiento en 2001.<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><br />
<span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">En 1998 gracias a las gestiones y perseverancia de Schelter, se <span class="apple-converted-space"> </span>logró obtener el permiso del Departamento de Energía para distribuir el programa bajo la llamada licencia GNU-GPL, iniciando en el año 2000 el proyecto Maxima en SourceForge con el fin de mantener y seguir desarrollando DOE-Macsyma, ahora con el nombre de Maxima.<span class="apple-converted-space"> </span></span><br />
<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">El Software puede descargarse, instalarse, utilizarse en forma gratuita, siempre que no se use con fines comerciales, ni se cobre por su uso. El sitio es:<span class="apple-converted-space"> </span></span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><a href="http://sourceforge.net/projects/maxima/files/latest/download?source=files"><span lang="ES" style="color: #888888; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;">http://sourceforge.net/projects/maxima/files/latest/download?source=files</span></a><o:p></o:p></div>
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<span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><br /></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">Maxima posee un<span class="apple-converted-space"> </span> amplio conjunto de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática"><span lang="ES" style="color: #888888; text-decoration-line: none;">funciones</span></a><span class="apple-converted-space"><span lang="ES" style="color: #222222; mso-ansi-language: ES;"> </span></span><span lang="ES" style="color: #222222; mso-ansi-language: ES;">para hacer manipulación simbólica de </span><a href="http://es.wikipedia.org/wiki/Polinomio" title="Polinomio"><span lang="ES" style="color: #888888; text-decoration-line: none;">polinomios</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas)"><span lang="ES" style="color: #888888; text-decoration-line: none;">matrices</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, funciones racionales,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración"><span lang="ES" style="color: #888888; text-decoration-line: none;">integración</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">,<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Derivada" title="Derivada"><span lang="ES" style="color: #888888; text-decoration-line: none;">derivación</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, manejo de gráficos en 2D y 3D, manejo de números de punto flotante y grandes, expansión en series de potencias y de<span class="apple-converted-space"> </span></span><a href="http://es.wikipedia.org/wiki/Fourier" title="Fourier"><span lang="ES" style="color: #888888; text-decoration-line: none;">Fourier</span></a><span lang="ES" style="color: #222222; mso-ansi-language: ES;">, entre otras funcionalidades.</span><span style="color: #222222; font-family: "arial" , "sans-serif"; font-size: 10pt; line-height: 14.2667px;"><o:p></o:p></span></div>
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<span style="color: #222222;"><br /></span><span style="color: #222222;">EXISTE UNA VERSIÓN DE MAXIMA PARA ANDROID E IOS, tanto para tabletas como para teléfonos inteligentes.<o:p></o:p></span><br />
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<b>GRAFICAS DE FUNCIONES Y CÁLCULO DEL ÁREA ENTRE GRÁFICAS<o:p></o:p></b><br />
<b>METODO 1: PROCESO MANUAL</b></div>
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<span lang="ES-MX">Ejemplo; calcule el área limitada por las gráficas de y= x+3, y=</span><span style="text-align: start;">x</span><sup style="text-align: start;">2</sup><br />
<sup style="text-align: start;"><br /></sup><sup style="text-align: start;"><br /></sup>Ingresar las ecuaciones en la zona de entrada. . Si es necesario, se debe acomodar la ecuación para ponerla en la forma y=f(x)</div>
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<span lang="ES-MX" style="font-size: large;"><b><br /></b></span>
<span lang="ES-MX"><b><span style="font-size: large;">plot2d([x+3,x^2],[x,-3,3]); shift enter</span></b></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3xhvVNHB9yQv41iwhgVRDP_ijLi5cx37Niauqi1eWpkdW68IcQ7qpPcV9o2n0uaC45cwK6UHSq9sn9kXx5eS8Tl5iFVkERnlzaGAX2KULROSAZyWy5gQYvqbMLV0BtCuCHG7e3nTOR5g/s1600/entremax1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="472" data-original-width="660" height="228" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3xhvVNHB9yQv41iwhgVRDP_ijLi5cx37Niauqi1eWpkdW68IcQ7qpPcV9o2n0uaC45cwK6UHSq9sn9kXx5eS8Tl5iFVkERnlzaGAX2KULROSAZyWy5gQYvqbMLV0BtCuCHG7e3nTOR5g/s320/entremax1.png" width="320" /></a></div>
<span lang="ES-MX">Es necesario encontrar los interceptos, con el siguiente comando:</span><br />
<span lang="ES-MX"><b><span style="font-size: large;">solve(x+3=x^2,x); shift enter</span></b></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvlCXAa0ANypmM4NErewmEcl9bHzHV1akOneXAi4_DPM4AB3GkCBtDjQ2ATK8Ux5YL_ieltUhw_F4yqn4oTeVxuGkyBdXTD_7hGIbXHdCp7samRr602l0UeWBk1o5mcA7bZhQDF0rlNws/s1600/entremax2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="139" data-original-width="535" height="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvlCXAa0ANypmM4NErewmEcl9bHzHV1akOneXAi4_DPM4AB3GkCBtDjQ2ATK8Ux5YL_ieltUhw_F4yqn4oTeVxuGkyBdXTD_7hGIbXHdCp7samRr602l0UeWBk1o5mcA7bZhQDF0rlNws/s640/entremax2.png" width="640" /></a></div>
Para calcular el área, se ingresa la orden: <b>integrate((x+3)-x^2,-1.302,2.302);</b> shift enter<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-sLUjhyluPZ5cxFIiTaSRzqebvGRHu-KtZFtzRqgO7QKKeCqwOo58hLCKg9rGcvewN_4XECw_tMXCmk1iRvtpOzKcV6VFGxiUZiQpC0Xh905Si7Ii6CNQrBbCDitlYcE0iZXaBW1Dm_M/s1600/entremax3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="217" data-original-width="581" height="236" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-sLUjhyluPZ5cxFIiTaSRzqebvGRHu-KtZFtzRqgO7QKKeCqwOo58hLCKg9rGcvewN_4XECw_tMXCmk1iRvtpOzKcV6VFGxiUZiQpC0Xh905Si7Ii6CNQrBbCDitlYcE0iZXaBW1Dm_M/s640/entremax3.png" width="640" /></a></div>
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<span lang="ES-MX"><b>MÉTODO 2: USANDO EL PAQUETE DRAW PARA SOMBREAR EL ÁREA</b></span><br />
<span lang="ES-MX">Existe una enorme cantidad de paquetes adicionales creados por usuarios para ser incorporados al sistema, uno de ellos es <b>draw</b></span><br />
ejemplo: Calcule el área limitada por las gráficas de:<b> y= cos x, y=2+sen x</b> entre -pi/2 y pi/2<br />
ejemplo tomado del sitio: https://people.richland.edu/james/spring15/m122/projects/draw.html<br />
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<span lang="ES-MX">se carga el paquete con el comando load y se ingresa:</span><br />
<span lang="ES-MX">load(draw)$ shift enter</span><br />
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<pre class="maxima" style="border-radius: 8px; border: thin solid rgb(102, 51, 153); margin: 0.5em 4em; padding: 1em; text-align: start;"><code style="font-size: 16.25px; line-height: 1.5em;">draw2d(
fill_color = red,
filled_func = 2+sin(x),
explicit(cos(x),x,-%pi/2,%pi/2),
filled_func = false,
line_width = 2,
color = black,
explicit(cos(x),x,-%pi,%pi),
color = blue,
explicit(2+sin(x),x,-%pi,%pi)
)$</code></pre>
<span lang="ES-MX">Al oprimir Shift enter, se obtiene lo siguiente:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRNrW-YQlwLdrWUi36rmQIGpEOqkI2Ol8b7VcLqE7dC2P0CKAKmN-0uY-KkqkZLyJRNjP1klT8tk3stJvSz3r5O_cLweRvRGFYGyqqGGhYpHYvRJ84zq-X0ODz8fneo1x2orrA0IzFm5s/s1600/entremax4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="255" data-original-width="312" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRNrW-YQlwLdrWUi36rmQIGpEOqkI2Ol8b7VcLqE7dC2P0CKAKmN-0uY-KkqkZLyJRNjP1klT8tk3stJvSz3r5O_cLweRvRGFYGyqqGGhYpHYvRJ84zq-X0ODz8fneo1x2orrA0IzFm5s/s1600/entremax4.png" /></a></div>
<span lang="ES-MX">Para calcular el área, se ingresa; integrate(2+sin(x)-cos(x),x,</span><span style="font-size: 16.25px; text-align: start;">-%pi/2,%pi/2);</span><br />
<span style="font-size: 16.25px; text-align: start;">o si se desea resultado decimal, se antepone float</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZjWdGj2auLWhvE3wMnCftHE2yuL8XceTuyh_als0RV9WD86OM17kIjrW4ZOeczbeMr1x5-chZO-f7Q5_kJ9xJE2Z5MBMqfY5toIUGct8XPUzz7FK1eo_zYlZN4NPf4Xpcmb5FRkTH8PM/s1600/entremax5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="86" data-original-width="455" height="120" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZjWdGj2auLWhvE3wMnCftHE2yuL8XceTuyh_als0RV9WD86OM17kIjrW4ZOeczbeMr1x5-chZO-f7Q5_kJ9xJE2Z5MBMqfY5toIUGct8XPUzz7FK1eo_zYlZN4NPf4Xpcmb5FRkTH8PM/s640/entremax5.png" width="640" /></a></div>
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Ing. Luis Reyeshttp://www.blogger.com/profile/02219201261132395664noreply@blogger.com0