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Cell[CellGroupData[{

Cell["Alvaro S. N\[UAcute]\[NTilde]ez.", "Author"],

Cell[CellGroupData[{

Cell[TextData[{
 "Introducci\[OAcute]n a ",
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 "."
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Cell[TextData[{
 "En esta secci\[OAcute]n comenzaremos el estudio de las capacidades de ",
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 " mediante una discusi\[OAcute]n de los elementos m\[AAcute]s \
b\[AAcute]sicos del programa. Desprovisto de las herramientas de programaci\
\[OAcute]n ",
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Cell["Aritmetica b\[AAcute]sica", "Subsection"],

Cell["\<\
El primer c\[AAcute]lculo que podemos hacer es la suma, 2+2=4. Escribimos la \
operaci\[OAcute]n (2+2) directamente en la hoja de c\[AAcute]lculo y la \
seguimos con las teclas (\[ShiftKey]\[EnterKey]) simultan\[EAcute]amente. El \
resultado es:\
\>", "Text"],

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Cell["\<\
De este ejemplo podemos rescatar la siguiente notaci\[OAcute]n a seguir en \
estas notas, los inputs ser\[AAcute]n indicados mediante negrillas y los \
outputs mediante letras simples. \
\>", "Text"],

Cell["\<\
El resto de las operaciones aritm\[EAcute]ticas tiene una notaci\[OAcute]n \
auto-explicativa: \
\>", "Text"],

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Cell[TextData[{
 "de este \[UAcute]ltimo ejemplo se rescata que las operaciones son evaluadas \
de acuerdo al orden: *, /, +, -. Este orden es el orden esta heredado de C++ \
lenguaje madre de ",
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 " y del cual se heredan muchas otras convenciones sint\[AAcute]cticas."
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Cell["Otras operaciones simples son, por ejemplo:", "Text"],

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 " y como muestra el ejemplo anterior su manipulaci\[OAcute]n es tan simple \
como la de variables reales."
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Cell["\<\
Debemos destacar la diferencia entre estos dos inputs y outputs:\
\>", "Text"],

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Cell[TextData[{
 "Los n\[UAcute]meros enteros son tratados en ",
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 " de manera simb\[OAcute]lica, todas las operaciones se hacen en ellos de \
manera exacta. Al escribir 2. o 3. estamos indicando a ",
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\[OpenCurlyDoubleQuote]reales\[CloseCurlyDoubleQuote]. Al evaluar Sqrt[2.] ",
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 " automaticamente invoca los algoritmos num\[EAcute]ricos internos para \
evaluar aproximadamente la ra\[IAcute]z de 2."
}], "Text"],

Cell["\<\
Si nos encontramos en la necesidad de convertir un n\[UAcute]mero simb\
\[OAcute]lico en un n\[UAcute]mero aproximado, podemos recurrir a la funci\
\[OAcute]n N[o]:\
\>", "Text"],

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Cell[TextData[{
 "Adem\[AAcute]s ",
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Cell["cuyos valore num\[EAcute]ricos pueden obtenerse usando N: ", "Text"],

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Cell["\<\
Ahora, N tiene una opci\[OAcute]n que indica el n\[UAcute]mero de decimales \
deseados:\
\>", "Text"],

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Cell["Manipulaciones algebraicas y trigonom\[EAcute]tricas elementales", \
"Subsection"],

Cell[TextData[{
 "Sin duda, que la caracteristica distintiva de ",
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 ", con respecto a una calculadora simple (y con respecto a la mayoria de los \
lenguajes de programaci\[OAcute]n) es la capacidad para manejar variables \
simbolicas. Como se logra esto es algo que comenzaremos a comprender en las \
secciones siguientes, mientras debemos conformarnos con observar esta \
capacidad:"
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Cell["\<\
Al tipear una expresi\[OAcute]n algebraica nada ocurre, salvo por un \
reordenamiento de la expresi\[OAcute]n. El comando Expand, desarrolla el \
producto algebraico\
\>", "Text"],

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Cell["\<\
El simbolo de porcentaje hace referencia al \[UAcute]ltimo valor adquirido \
por el Out. En este caso el resultado anterior es elevado a la quinta \
potencia y desarrollado\
\>", "Text"],

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Cell["Las relaciones trigonom\[EAcute]tricas b\[AAcute]sicas estan incluidas: \
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Cell["Calculo elemental y avanzado", "Subsection"],

Cell["\<\
Las reglas de c\[AAcute]lculo estan incorporadas. Las integrales se \
especifican mediante el comando Integrate. En el caso de integrales \
indefinidas la sintaxis es:\
\>", "Text"],

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Cell["En el caso de integrales definidas", "Text"],

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Cell[TextData[{
 "Este \[UAcute]ltimo ejemplo ilustra  ",
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 " maneja una serie de funciones trascendentales. La funciones \
\[CapitalGamma] (Gamma[z]), ",
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Cell["\<\
Desde luego las operaciones no estan confinadas a la evaluaci\[OAcute]n de \
operaciones simples. Por ejemplo podemos determinar los primeros \
t\[EAcute]rminos de una expansi\[OAcute]n en serie:\
\>", "Text"],

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Cell["\<\
Este comando puede incluso usarse para la determinaci\[OAcute]n de una serie \
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Stirling se determinan facilmente.\
\>", "Text"],

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Cell["\<\
La sintaxis de los diversos operadores, (el lugar de los distintos parentisis \
y par\[AAcute]metros) puede parecer demasiado complejo como para recordar. \
Con el tiempo y mucha practica, ser\[AAcute] f\[AAcute]cil recordar la \
estrucutra los operadores m\[AAcute]s usados. Una vez que se dominen los \
aspectos m\[AAcute]s avanzados de programaci\[OAcute]n se comenzada a ver una \
racionalidad bastante coherente en lo que este momento debe verse como una \
convenci\[OAcute]n arbitraria e incomoda. Mientras uno puede usar la ayuda \
que viene incluida con el FrontEnd. Para pedir ayuda sobre un comando espec\
\[IAcute]fica se puede usar: ?Comando.\
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Cell["\<\
Adem\[AAcute]s el FrontEnd tiene incluido el programa HelpBrowser que permite \
recorrer un listado de funciones y comandos con detalles sobre uso y su \
sintaxis, acompa\[NTilde]ado de un gran n\[UAcute]mero de ejemplos. \
\>", "Text"]
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\[AAcute]s simple de las cuales es sin duda el comando Plot."
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Cell[TextData[{
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\[AAcute]fico. El valor de Out no es el gr\[AAcute]fico mismo, sino una \
estructura de tipo -Graphics-, a este tipo de expresiones uno puede aplicar \
otras funciones de ",
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Cell["\<\
Para funciones de dos variables los comandos Plot3D, DensityPlot y \
ContourPlot comparten sintaxis y pueden ser utilizados a conveniencia.\
\>", "Text"],

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  Axes->True,
  BoxRatios->{1, 1, 0.4},
  Method->{"RotationControl" -> "Globe"},
  PlotRange->
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    3.141592653589793}, {-3.141592653589793, 
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  PlotRangePadding->{
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    Scaled[0.02]}]], "Output",
 CellChangeTimes->{3.479818985012792*^9},
 CellLabel->"Out[47]="]
}, Open  ]],

Cell["\<\
Analogamente, el valor del Out en este caso es -SurfaceGraphics-. Esta \
diferencia se debe a que, aunque en su representaci\[OAcute]n gr\[AAcute]fica \
un gr\[AAcute]fico en 3D es una imagen igual que uno de 2D, internamente hay \
grandes diferencias en la forma en que se almacenan los datos que estan \
representados en la imagen. En principio es posible acceder directamente a \
dichos datos y estudiar (y optativamente modificar) las especificaciones.\
\>", "Text"],

Cell[CellGroupData[{

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 "La funci\[OAcute]n DensityPlot entrega una estructura del tipo  \
-DensityGraphics-. Adem\[AAcute]s del acceso directo, a las extructuras gr\
\[AAcute]ficas, es posible modificar la apariencia de un gr\[AAcute]fico \
indirectamente mediante las opciones de la funci\[OAcute]n que lo construye. \
En ",
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mediante el uso de la funci\[OAcute]n Options."
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opciones en la ayuda on-line, y en la serie de documentos adjuntos a estas ",
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Cell["\<\
Finalmente un gr\[AAcute]fico de lineas de nivel se hace con el comando \
ContourPlot:\
\>", "Text"],

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La soluci\[OAcute]n algebraica de sistemas de ecuaciones simples se puede \
llevar a cabo mediante el comando Solve[ecuaci\[OAcute]n, incognita]:\
\>", "Text"],

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trabajar automaticamente con listas. En el intertanto ser\[AAcute] suficiente \
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Cell["\<\
Las regla para resolver ecuaciones cuadr\[AAcute]ticas tambi\[EAcute]n surge \
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Solve puede en principio \[OpenCurlyDoubleQuote]resolver\
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de la ecuaci\[OAcute]n. Si uno quiere evaluar num\[EAcute]ricamente uno puede \
intentar con el comando N. Por otro lado uno puede evaluar directamente la \
soluci\[OAcute]n num\[EAcute]rica ya sea con NSolve, o con Solve cambiando \
los param\[EAcute]tros de num\[EAcute]ros exactos a n\[UAcute]meros \
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deben entender  especificamente a ecuaciones algebraicas o ecuaciones que se \
pueden convertir en tales (i.e. ecuaciones que ",
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Cell["\<\
La advertencia Solve::ifun: hace referencia al hecho que la ecuaci\[OAcute]n \
tiene en realidad una gran cantidad de soluciones y sugiere usar la funci\
\[OAcute]n Reduce:\
\>", "Text"],

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Cell["\<\
Para ecuaciones no algebraicas (trascendentales) ni Solve ni NSolve son de \
ayuda:\
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Cell["\<\
En dichos casos la funci\[OAcute]n que debemos usar es FindRoot, que solo \
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Vemos que ahora debemos incluir un par\[AAcute]metro adicional, un valor de \
inicio para buscar la soluci\[OAcute]n. Es este caso el valor de inicio no es \
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Esto se debe a que la soluci\[OAcute]n es unica, como indica el siguiente gr\
\[AAcute]fico:\
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En otros casos el valor de la soluci\[OAcute]n es sensible al valor inicial:\
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Cell["M\[AAcute]s adelante veremos detalles de como funciona FindRoot.", \
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Cell["\<\
Como vemos las listas mantienen su orden y por lo tanto corresponden a \
arreglos de datos fijos (arrays en C++). No debemos confundirlas con \
conjuntos. Una lista tiene estructura.\
\>", "Text"],

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Cell["\<\
Para crear una lista tenemos diversos comandos. Lo m\[AAcute]s simple es \
escribir directamente los elementos entre { }:\
\>", "Text"],

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Cell["\<\
Si queremos una lista de elementos consecutivos podemos usar Range:\
\>", "Text"],

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Cell["\<\
Por otro lado quiz\[AAcute]s queramos una lista donde los elementos sean \
cierta funci\[OAcute]n de los indices:\
\>", "Text"],

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Cell["\<\
Podemos obtener el primer elemento de una lista cualquiera, (el \
\[UAcute]ltimo, los n-primeros, los n-\[UAcute]ltimos, etc) mediante:\
\>", "Text"],

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Cell["Adem\[AAcute]s podemos eliminar ciertos elementos de la lista:", "Text"],

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Cell["Esto elimina el segundo elemento de la lista:", "Text"],

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Cell["el tercero de atr\[AAcute]s para adelante", "Text"],

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Cell["Este comando elimina el tercer y quinto elemento:", "Text"],

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Cell["Sin duda el m\[AAcute]s utilizado es Part:", "Text"],

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Cell[TextData[{
 "Ejemplo: Un rectificador de Voltaje ",
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  ButtonData:>"Maeder99",
  ButtonNote->"Maeder99"],
 "."
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Cell["\<\
La relaci\[OAcute]n entre voltaje y corriente en un diodo puede ser \
aproximada razonablemente bien mediante la forma funcional:\
\>", "Text"],

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Cell[TextData[{
 "donde ",
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 " es la corriente residual, ",
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Cell[TextData[{
 "En primer lugar  invocamos algunos Standard Packages (c\[OAcute]digo de ",
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 " adicional disponible) con la distribuci\[OAcute]n usual. Estos \
corresponden a material que no es requerido comunmente por la gran masa de \
usuarios. En este curso vamos a usar varios de ellos."
}], "Text"],

Cell["\<\
El primero es el de unidades. Contiene un gran listado de unidades y factores \
de conversi\[OAcute]n. Junto con la funci\[OAcute]n Convert[variable, \
newunits].\
\>", "Text"],

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Cell["\<\
Un listado de constante f\[IAcute]sicas esta disponible en el package de \
PhysicalUnits. \
\>", "Text"],

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Cell["\<\
Por \[UAcute]ltimo para adquirir la capacidad de hacer gr\[AAcute]ficos \
logaritmicos y otras variantes complicadas, de un modo directo llamamos al \
Package Graphics.\
\>", "Text"],

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Cell["Verificamos las constantes f\[IAcute]sicas que nos interesan:", "Text"],

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Cell["\<\
Tomamos el factor que convertira la variables de la exponencial en \
n\[UAcute]meros adimensionales:\
\>", "Text"],

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Cell["Ingresamos la relaci\[OAcute]n del diodo:", "Text"],

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Cell["con una corriente caracter\[IAcute]stica (este valor es \
t\[IAcute]pico):", "Text"],

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Cell["\<\
Ahora determinamos una gr\[AAcute]fica de la corriente versus la \
ca\[IAcute]da de voltaje a trav\[EAcute]s del diodo:\
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Al conectar el diodo en serie con una resistencia obtenemos una ecuaci\
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Una manera de revelar esta estructura interna es la funci\[OAcute]n FullForm:\
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Para determinar solamente el encabezado de la expresion podemos usar la funci\
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Los ladrillos b\[AAcute]sicos para construir expresiones son los \
\[AAcute]tomos (que son expresiones por definici\[OAcute]n). Los \
\[AAcute]tomos son los n\[UAcute]meros enteros, los reales, los simbolos y \
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\t(a) Simbolos son todas las palabras que comienzan con letras, ya sea \
incluidas por el usuario o las que forman parte de los c\[OAcute]digos de los \
programas internos.\
\>", "Text"],

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\t(b) Strings secuencias de caracteres ASCII entre comillas. Son usadas para \
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Aunque los \[AAcute]tomos no son expresiones definidas recursivamente (son \
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de expresiones) de todos modos tienen, por definici\[OAcute]n, un encabezado:\
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Para seguir que tipo de conversiones se hacen durante la ejecuci\[OAcute]n de \
cualquier comando es posible usar el comando Trace:"
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Objetos mat\[EAcute]maticos m\[AAcute]s complejos se representan mediante \
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Cell["Estructura de las expresiones", "Subsection"],

Cell["\<\
Dado que una lista es una expresi\[OAcute]n con encabezado List, no deber\
\[IAcute]a sorprender que uno puede referirse a partes de una \
expresi\[OAcute]n usando los mismos comandos asociados con listas.\
\>", "Text"],

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Cell["\<\
Esto se debe a que las formas internas de ll y suma solo difieren en el \
encabezado:\
\>", "Text"],

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El resumen los comandos asociados a listas no solo funcionan en listas sino \
que en cualquier expresi\[OAcute]n.\
\>", "Text"],

Cell["\<\
Hay una serie de comandos para acceder a partes de expresiones. El \
m\[AAcute]s importante es el comando Part.\
\>", "Text"],

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Cell["Uno puede comparar expresiones:", "Text"],

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Cell["\<\
Mientras no definamos el significado de Encabezado el programa dejara las \
comparaciones (y todas las expresiones que dependan de su valor) sin evaluar.\
\
\>", "Text"],

Cell["El comando flatten reduce los encabezados anidados:", "Text"],

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Cell["Por ejemplo una matriz de 2x2:", "Text"],

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Cell["Estilos de Programaci\[OAcute]n", "Section"],

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Cell["Programaci\[OAcute]n basada en reglas de sustituci\[OAcute]n", \
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Cell[TextData[{
 "El principio b\[AAcute]sico de que todo es una expresi\[OAcute]n es usado \
como la principal herramienta para programar en ",
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reglas de transformaci\[OAcute]n."
}], "Text"],

Cell["\<\
EL concepto de comparaci\[OAcute]n de patrones es clave para esta tarea. Un \
patron es una expresi\[OAcute]n que tiene la forma:\
\>", "Text"],

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Cell["\<\
Como vemos su forma abreviada nombre:clase. Las clases de patrones son:\
\>", "Text"],

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Cell["De este modo tenemos:", "Text"],

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Cell[TextData[{
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Cell["Este patron representa cualquier secuencia de enteros.", "Text"],

Cell["\<\
Podemos determinar si cierta expresi\[OAcute]n satisface el patron usando \
MatchQ[]:\
\>", "Text"],

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Cell["\<\
Ahora podemos entender mejor la forma para definir una funci\[OAcute]n:\
\>", "Text"],

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tal que cuando evaluada en cualquier expresi\[OAcute]n de ",
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Integrate. Esto ilustra como es posible que ",
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Cell["\<\
Podemos introducir reglas de transformaci\[OAcute]n creadas por nosotros. Su \
notaci\[OAcute]n es: \
\>", "Text"],

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Cell["\<\
Para implementar una transformaci\[OAcute]n usando estas reglas podemos usar \
Replace (/.)\
\>", "Text"],

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Cell["\<\
El verdadero poder de las reglas de transformaci\[OAcute]n se hace manifiesto \
al tratar con patrones.\
\>", "Text"],

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Cell["\<\
Este comando transforma el primer primo de la lista en su cubo:\
\>", "Text"],

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Cell["Mientras que este lo hace con todos los primos de la lista:", "Text"],

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Cell["Programaci\[OAcute]n procedural", "Subsection"],

Cell[CellGroupData[{

Cell["Modulos y variables locales", "Subsubsection"],

Cell[TextData[{
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variables en ",
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 " son globales por omisi\[OAcute]n, es decir que los valores que toman son \
accequibles y modificables desde cualquier punto de un programa. Esto trae \
grandes ventajas de comodidad para programar, pero tambi\[EAcute]n algunos \
riesgos asociados a la posibilidad de confundir ciertos valores o cambiar \
algunos sin plena conciencia. Este riesgo se incrementa si, como ocurre en \
problemas complejos, se hace necesario compartir codigos entre diversos \
programadores. En leguajes como C, C++ o FORTRAN las variables tienen valores \
locales asociadas a cada bloque de programaci\[OAcute]n. ",
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Cell["\<\
Consideremos el siguiente ejemplo, donde calculamos el cuadrado del seno del \
argumento en dos pasos, primero destinamos el valor de la variable y para \
evaluar el seno del argumento y luego elevamos dicha variable al cuadrado. \
Este tipo de paso intermedio es com\[UAcute]n al evaluar funciones muy \
complejas.\
\>", "Text"],

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Cell["Esta definici\[OAcute]n cumple el objetivo deseado:", "Text"],

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Cell["\<\
Sin embargo tenemos el problema que el valor de y esta modificado:\
\>", "Text"],

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Cell["el valor cambia cada vez que llamamos a la funci\[OAcute]n:", "Text"],

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Cell["\<\
Esto es claramente inaceptable y debemos usar Module para corregir este \
comportamiento:\
\>", "Text"],

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Cell["\<\
En la sintaxis de Module las variables locales se especifican mediante {}. \
Nuevamente la funci\[OAcute]n entrega el valor deseado:\
\>", "Text"],

Cell[BoxData[
 RowBox[{"F", "[", "a", "]"}]], "Input"],

Cell["\<\
Pero esta vez el valor de la variable auxiliar no se confunde en el contexto \
global:\
\>", "Text"],

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell["Recursiones, Iteraciones y Loops", "Subsubsection"],

Cell[TextData[{
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " permite la implementaci\[OAcute]n de funciones recursivas, i.e. cuyos \
valores dependen de la funci\[OAcute]n misma en otros argumentos. El ejemplo \
can\[OAcute]nico de este tipo de funciones es la funci\[OAcute]n factorial, \
n!=1*2*3*...*n, que obedece la relaci;\[OAcute]n recursiva:"
}], "Text"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{
    RowBox[{"fac", "[", "n", "]"}], "=", 
    RowBox[{"n", "*", 
     RowBox[{"fac", "[", 
      RowBox[{"n", "-", "1"}], "]"}]}]}], ","}], 
  TraditionalForm]], "NumberedEquation"],

Cell["\<\
que puede implementarse de manera directa en de la siguiente forma:\
\>", "Text"],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"Fac", "[", "1", "]"}], "=", "1"}], ";"}], "\n", 
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Cell[BoxData[
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Cell[BoxData[
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  RowBox[{
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Cell["\<\
Como vemos, la funci\[OAcute]n utitliza la relaci\[OAcute]n m\[AAcute]s \
general (Fac(n)) hasta que encuentra un caso particular (Fac(1)), es facil \
ver que de no haber especificado el valor Fac[1] la funci\[OAcute]n se \
hubiese llamado a si misma infinitas veces.\
\>", "Text"],

Cell["\<\
La serie de Fibbonacci nos entrega otro ejemplo de relaci\[OAcute]n recursiva:\
\>", "Text"],

Cell[BoxData[{
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  RowBox[{
   RowBox[{"Fib", "[", "1", "]"}], "=", "1"}], ";"}], "\n", 
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Cell[BoxData[
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Cell[TextData[{
 "Un ultimo ejemplo, menos trivial, es el problema de determinar el maximo \
comun divisor entre dos numeros.",
 StyleBox["\[InvisibleSpace]", "RefSep"],
 ButtonBox["[Maeder (1999)]",
  BaseStyle->"Citation",
  ButtonData:>"Maeder99",
  ButtonNote->"Maeder99"],
 " Usaremos la siguiente propiedad del m\[AAcute]ximo com\[UAcute]n divisor:"
}], "Text",
 CellTags->":bib:Maeder99"],

Cell[BoxData[
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    "0"}]}]}]], "NumberedEquation"],

Cell["Esta relaci\[OAcute]n genera inmediatamente la propiedad:", "Text"],

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Cell[TextData[{
 "que se puede implementar directamente en ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ":"
}], "Text"],

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 RowBox[{
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   RowBox[{"mcd", "[", 
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
 "Los iteradores son fundamentales para la programaci\[OAcute]n. ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " tiene una serie de comandos que permiten hacer iteraciones, cual de ellos \
usar depende de cada problema, la sintaxis de todos es bastante similar de \
modo que solo discutiremos los m\[AAcute]s usados."
}], "Text"],

Cell["\<\
El principal comando para formar loops es Do[operaci\[OAcute]n, iterador]:\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Do", "[", 
  RowBox[{
   RowBox[{"Print", "[", 
    RowBox[{"i", "^", "2"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"i", ",", "1", ",", "6"}], "}"}]}], "]"}]], "Input"],

Cell["\<\
La falta de Out[1] indica que el iterador Do en si mismo no entrega valores \
al ciclo principal, solo realiza operaciones como asignaciones y otros.\
\>", "Text"],

Cell[BoxData[
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    RowBox[{"{", 
     RowBox[{"n", ",", "1", ",", "3", ",", "0.5`"}], "}"}]}], "]"}], 
  ";"}]], "Input"],

Cell["\<\
Otro iterador que es muy \[UAcute]til es el comando For cuya sintaxis esta \
basada en la sintaxis del  iterador for de C: \
For[comienzo,decisi\[OAcute]n,paso,comandos]\
\>", "Text"],

Cell[BoxData[
 RowBox[{"For", "[", 
  RowBox[{
   RowBox[{"i", "=", "1"}], ",", 
   RowBox[{"i", "<", "4"}], ",", 
   RowBox[{"i", "++"}], ",", 
   RowBox[{"Print", "[", "i", "]"}]}], "]"}]], "Input",
 CellTags->"For"],

Cell["\<\
Notemos que en cada item de For podemos hacer diversas operaciones. En este \
ejemplo cambiamos \
\>", "Text"],

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}, Open  ]],

Cell[CellGroupData[{

Cell["Programaci\[OAcute]n funcional", "Subsection"],

Cell["\<\
Un estilo de programaci\[OAcute]n funcional es aquel que se basa en la \
implementaci\[OAcute]n de funciones de alto orden. Es decir un funciones que \
dependen de funciones cuyos parametros y opciones son, a su vez funciones. \
Algunos ejemplos del la secci\[OAcute]n satisfacen esta condici\[OAcute]n, \
por ejemplo, Integrate toma como argumento una funci\[OAcute]n func y entrega \
como resultado otra funci\[OAcute]n la primitiva de func.\
\>", "Text"],

Cell[TextData[{
 "Comenzaremos nuestro estudio de como ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " facilita la implementaci\[OAcute]n de este estilo con el problema de la \
aplicaci\[OAcute]n de una funci\[OAcute]n."
}], "Text"],

Cell[CellGroupData[{

Cell["Aplicaci\[OAcute]n de una funci\[OAcute]n", "Subsubsection"],

Cell["\<\
En primer lugar tenemos la forma m\[AAcute]s primitiva de definir y evaluar \
una funci\[OAcute]n. En este ejemplo la funci\[OAcute]n f es definida de tal \
manera que entrega una lista consistente en los digitos de la parte entera de \
su argumento.\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", 
  RowBox[{"IntegerDigits", "[", 
   RowBox[{"Floor", "[", "x", "]"}], "]"}]}]], "Input"],

Cell["La evaluaci\[OAcute]n sigue directamente el formato usual:", "Text"],

Cell[BoxData[
 RowBox[{"f", "[", "2.5", "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"f", "[", "32.4", "]"}]], "Input"],

Cell[BoxData[
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  RowBox[{"\[Pi]", "^", "4"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"f", "[", 
  RowBox[{"\[ExponentialE]", "*", 
   RowBox[{"\[Pi]", "^", "5"}]}], "]"}]], "Input"],

Cell["\<\
Sin embargo es posible usar otras formas, a veces m\[AAcute]s convenientes, \
para pasar un argumento a la funci\[OAcute]n. Por ejemplo podemos aplicar la \
funci\[OAcute]n tras la evaluaci\[OAcute]n del argumento. \
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"10", "+", "\[Pi]"}], "//", "f"}]], "Input"],

Cell["\<\
tambi\[EAcute]n podemos usar la notaci\[OAcute]n @ (que ser\[AAcute] \
explicada a continauci\[OAcute]n):\
\>", "Text"],

Cell[BoxData[
 RowBox[{"f", "@", 
  SuperscriptBox["\[ExponentialE]", "\[Pi]"]}]], "Input"],

Cell["\<\
Si tenemos una lista de argumentos a los cuales aplicar la funci\[OAcute]n es \
conveniente usar la funci\[OAcute]n Map:\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Map", "[", 
  RowBox[{"f", ",", 
   RowBox[{"{", 
    RowBox[{"\[Pi]", ",", "\[ExponentialE]", ",", 
     SuperscriptBox["\[Pi]", "\[ExponentialE]"], ",", 
     SuperscriptBox["\[ExponentialE]", "\[Pi]"], ",", 
     SuperscriptBox["\[ExponentialE]", 
      SuperscriptBox["\[Pi]", "\[ExponentialE]"]]}], "}"}]}], "]"}]], "Input"],

Cell["\<\
El resultado es el mismo que aplicar la funci\[OAcute]n a cada uno de los \
elementos de la lista por separado.\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"{", 
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    RowBox[{"f", "[", "\[Pi]", "]"}], ",", 
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      SuperscriptBox["\[Pi]", "\[ExponentialE]"]], "]"}]}], "}"}], 
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Cell["\<\
Por supuesto, Map puede actuar sobre cualquier expresi\[OAcute]n, aunque su \
encabezado no sea List. Por ejemplo en este producto el resultado \
\>", "Text",
 CellTags->":note:caveat on Map"],

Cell[BoxData[
 RowBox[{"Map", "[", 
  RowBox[{"Gamma", ",", 
   RowBox[{"a", "*", "b", "*", "c", "*", "d"}]}], "]"}]], "Input"],

Cell[TextData[{
 "Este resultado se entiende al expresar el producto como la \
expresi\[OAcute]n de ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ":"
}], "Text"],

Cell[BoxData[
 RowBox[{
  TagBox[
   RowBox[{"Map", "[", 
    RowBox[{"Gamma", ",", 
     RowBox[{"Times", "[", 
      RowBox[{"a", ",", "b", ",", "c", ",", "d"}], "]"}]}], "]"}],
   FullForm], "//", "FullForm"}]], "Input"],

Cell["\<\
Una abreviaci\[OAcute]n muy conveniente del comando Map es /@ (slash-at):\
\>", "Text",
 CellTags->":note:abreviaciones"],

Cell[BoxData[
 RowBox[{"f", "/@", 
  RowBox[{"{", 
   RowBox[{"\[Pi]", ",", "\[ExponentialE]", ",", 
    SuperscriptBox["\[Pi]", "\[ExponentialE]"], ",", 
    SuperscriptBox["\[ExponentialE]", "\[Pi]"], ",", 
    SuperscriptBox["\[ExponentialE]", 
     SuperscriptBox["\[Pi]", "\[ExponentialE]"]]}], "}"}]}]], "Input"],

Cell["\<\
Nos ahorramos escribir el comando y adem\[AAcute]s unos parentesis, pero lo m\
\[AAcute]s importante es que una vez internalizada, esta abreviaci\[OAcute]n \
permite una sintaxis mucho m\[AAcute]s fluida.  \
\>", "Text"],

Cell["\<\
Un \[UAcute]ltimo ejemplo ilustra la forma en que usaremos Map reiteradamente \
para hacer c\[AAcute]lculo num\[EAcute]rico. \
\>", "Text"],

Cell[BoxData[
 RowBox[{"ListPlot", "[", 
  RowBox[{"Map", "[", 
   RowBox[{"Tanh", ",", 
    RowBox[{"Range", "[", 
     RowBox[{
      RowBox[{"-", "10"}], ",", "10", ",", "0.01"}], "]"}]}], "]"}], 
  "]"}]], "Input"],

Cell["\<\
El problema que vemos con este gr\[AAcute]fico es que el eje horizontal esta \
completamente fuera de escala, en lugar de representar la variable en \
terminos de su valor num\[EAcute]rico, lo hace en terminos de su posici\
\[OAcute]n en la discretizaci\[OAcute]n arbitraria del intervalo (-10,10). \
Por ejemplo si escogemos una nueva unidad para la divisi\[OAcute]n, el eje \
horizontal cambia completamente.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"ListPlot", "[", 
  RowBox[{"Tanh", "/@", 
   RowBox[{"Range", "[", 
    RowBox[{
     RowBox[{"-", "10"}], ",", "10", ",", "0.008"}], "]"}]}], "]"}]], "Input"],

Cell["\<\
Para arreglar esta molestia podemos definir una funci\[OAcute]n que en lugar \
de entregar la Tanh nos entregue las coordenadas del punto que queremos \
graficar, algo como:\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", 
  RowBox[{"{", 
   RowBox[{"x", ",", 
    RowBox[{"Tanh", "[", "x", "]"}]}], "}"}]}]], "Input"],

Cell["y usar esta funci\[OAcute]n auxiliar en el mapeo:", "Text"],

Cell[BoxData[
 RowBox[{"ListPlot", "[", 
  RowBox[{"f", "/@", 
   RowBox[{"Range", "[", 
    RowBox[{
     RowBox[{"-", "10"}], ",", "10", ",", "0.008"}], "]"}]}], "]"}]], "Input"],

Cell["\<\
sin embargo al programar funcionalmente es muy incomodo tener que escribir \
una declaraci\[OAcute]n por cada funci\[OAcute]n que se necesita.\
\>", "Text"],

Cell["\<\
Veremos a continuaci\[OAcute]n como remediar este defecto. Antes estudiaremos \
otra forma de pasar una funci\[OAcute]n a diversos argumentos. Esta vez \
usamos el comando Apply. El resultado de Apply[g,expr] es expr con su \
encabezado cambiado por g.\
\>", "Text"],

Cell["Recordemos cual es el encabezado de una lista:", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"Range", "[", 
   RowBox[{"1", ",", "10"}], "]"}], "//", "Head"}]], "Input"],

Cell["Esto es debido a que en su representaci\[OAcute]n interna:", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"Range", "[", 
   RowBox[{"1", ",", "10"}], "]"}], "//", "FullForm"}]], "Input"],

Cell["Ahora si aplicamos Apply con el comando Plus:", "Text"],

Cell[BoxData[
 RowBox[{"Apply", "[", 
  RowBox[{"Plus", ",", 
   RowBox[{"Range", "[", 
    RowBox[{"1", ",", "10"}], "]"}]}], "]"}]], "Input"],

Cell["\<\
Este resultado se puede entender al estudiar el accionar de Apply con Trace:\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Apply", "[", 
    RowBox[{"Plus", ",", 
     RowBox[{"Range", "[", 
      RowBox[{"1", ",", "10"}], "]"}]}], "]"}], "//", "Trace"}], "//", 
  "TableForm"}]], "Input"],

Cell["\<\
Asi vemos que List[1,2,\[Ellipsis],10] se convierte en \
Plus[1,2,\[Ellipsis],10] que se evalua automaticamente a 55. Tambi\[EAcute]n \
Apply tiene una abreviaci\[OAcute]n, esta es @@.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Times", "@@", 
  RowBox[{"Range", "[", 
   RowBox[{"1", ",", "10"}], "]"}]}]], "Input"],

Cell["Que corresponde, naturalmente, al valor del factorial de 10:", "Text"],

Cell[BoxData[
 RowBox[{"10", "!"}]], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Definici\[OAcute]n de funciones puras", "Subsubsection"],

Cell["Hasta ahora hemos definido las funciones usando el patron:", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"g", "[", "x_", "]"}], ":=", 
  RowBox[{"x", "^", "2"}]}]], "Input"],

Cell["\<\
Estas funciones se pueden utilizar con los operadores Apply y Map para crear \
operaciones muy complejas. Sin embargo hay un problema con esta forma de \
definir una funci\[OAcute]n.\
\>", "Text"],

Cell["\<\
Al tratar de combinar distintas funciones nos vemos en problemas. Por \
ejemplo, supongamos que me interesa evaluar la suma de dos funciones \
aplicadas sobre una lista. Las funciones g y f se definen:\
\>", "Text"],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"g", "[", "x_", "]"}], ":=", 
   RowBox[{"x", "^", "2"}]}], ";"}], "\n", 
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", "x_", "]"}], ":=", 
   RowBox[{"Sin", "[", "x", "]"}]}], ";"}]}], "Input"],

Cell["podemos evaluarlas en los valores {1, \[Ellipsis], 10} usando Map:", \
"Text"],

Cell[BoxData[
 RowBox[{"data1", "=", 
  RowBox[{"g", "/@", 
   RowBox[{"Range", "[", 
    RowBox[{"1", ",", "10"}], "]"}]}]}]], "Input"],

Cell[BoxData[
 RowBox[{"data2", "=", 
  RowBox[{"f", "/@", 
   RowBox[{"Range", "[", 
    RowBox[{"1", ",", "10"}], "]"}]}]}]], "Input"],

Cell["La suma puede hacerse directamente:", "Text"],

Cell[BoxData[
 RowBox[{"data1", "+", "data2"}]], "Input"],

Cell["\<\
Pero estamos interesados en recorrer la lista una sola vez (y no tres como lo \
hicimos), de este modo intentamos algo as\[IAcute]:\
\>", "Text"],

Cell[BoxData[
 RowBox[{"data3", "=", 
  RowBox[{
   RowBox[{"(", 
    RowBox[{"g", "+", "f"}], ")"}], "/@", 
   RowBox[{"Range", "[", 
    RowBox[{"1", ",", "10"}], "]"}]}]}]], "Input"],

Cell["\<\
Vemos el problema radicado que mientras los simbolos f y g son funciones por \
separados, el simbolo f+g no es una funci\[OAcute]n.\
\>", "Text"],

Cell["\<\
Existen dos formas alternativas para definir una funci\[OAcute]n que remedian \
este problema. La idea es usar la noci\[OAcute]n abstracta de funci\[OAcute]n \
como un mapeo. La definimos con la estructura Function[argumento, mapeo]:\
\>", "Text"],

Cell[BoxData[
 RowBox[{"h", "=", 
  RowBox[{"Function", "[", 
   RowBox[{"x", ",", 
    SuperscriptBox["x", "2"]}], "]"}]}]], "Input"],

Cell["\<\
Las expresiones cuyo encabezado es Function se conocen como funciones puras.\
\>", "Text"],

Cell["Sus valores son id\[EAcute]nticos a los de g:", "Text"],

Cell[BoxData[
 RowBox[{"g", "/@", 
  RowBox[{"{", 
   RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "x"}], "}"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"h", "/@", 
  RowBox[{"{", 
   RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "x"}], "}"}]}]], "Input"],

Cell["\<\
Ahora sin embargo podemos crear la funci\[OAcute]n pura en el argumento de \
Map directamente y el resultado es lo que queriamos, evaluar la suma de las \
funciones con un solo recorrido de la lista de argumentos.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"data3", "=", 
  RowBox[{
   RowBox[{"Function", "[", 
    RowBox[{"x", ",", 
     RowBox[{
      RowBox[{"f", "[", "x", "]"}], "+", 
      RowBox[{"g", "[", "x", "]"}]}]}], "]"}], "/@", 
   RowBox[{"Range", "[", 
    RowBox[{"1", ",", "10"}], "]"}]}]}]], "Input"],

Cell["\<\
La segunda forma se ve muy complicada o abstracta a primera vista, sin \
embargo es solo una abreviaci\[OAcute]n muy comoda de la definici\[OAcute]n \
de una funci\[OAcute]n pura.\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Function", "[", 
    RowBox[{"x", ",", 
     RowBox[{"loquesea", "[", "x", "]"}]}], "]"}], " ", "\[Congruent]", " ", 
   RowBox[{"loquesea", "[", "#", "]"}]}], "&"}]], "NumberedEquation"],

Cell["\<\
por ejemplo en lugar de la evaluaci\[OAcute]n de data3 podr\[IAcute]amos \
haber escrito la forma mucho m\[AAcute]s corta:\
\>", "Text"],

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  RowBox[{
   RowBox[{
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Cell["\<\
Ahora somos capaces de arreglar el problema con nuestro gr\[AAcute]fico de la \
Tanh en una sola linea. Por un lado tenemos la expresi\[OAcute]n completa:\
\>", "Text"],

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Cell["\<\
por otro vemos la ventaja de las abreviaciones para hacer manipulaciones \
funcionales:\
\>", "Text"],

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Cell["Aplicaci\[OAcute]n repetida de funciones", "Subsubsection"],

Cell["\<\
El primer ejemplo de este tipo es la funci\[OAcute]n Nest, cuyo significado \
es evidente al mirar el ejemplo siguiente:\
\>", "Text"],

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Cell["\<\
Muchas veces queremos anidar una funci\[OAcute]n sin saber de antemano el n\
\[UAcute]mero de veces que lo haremos. Lo hacemos hasta que cierta condici\
\[OAcute]n es satisfecha. Para esto se puede usar el comando NestWhile. Por \
ejemplo esta funci\[OAcute]n entrega el primer primo mayor que 8:\
\>", "Text"],

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Cell["\<\
La funci\[OAcute]n FixedPoint es conveniente al buscar puntos fijos de una \
funci\[OAcute]n. De este modo tenemos una forma muy eficiente de evaluar el m\
\[EAcute]todo de Newton para encontrar la ra\[IAcute]z cuadrada de un n\
\[UAcute]mero. El m\[EAcute]todo de Newton establece la siguiente recurrencia:\
\>", "Text"],

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Cell["\<\
A veces necesitamos la lista de valores adquiridos por la funci\[OAcute]n \
antes de alcanzar la convergencia. En dichas situaciones podemos usar \
FixedPointList. Por ejemplo, ahora estudiamos la convergencia del \
m\[EAcute]todo de Newton y vemos que es bastante r\[AAcute]pida. \
\>", "Text"],

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Cell["\<\
M\[AAcute]s adelante llegaremos a resultados m\[AAcute]s precisos respecto a \
esta convergencia.\
\>", "Text"]
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Cell["Problemas de la secci\[OAcute]n.", "Subsection"],

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 " Implemente el m\[EAcute]todo de Gram-Schmidt para ortonormalizar \t\t\
vectores con respecto a un  producto interno. Debe tomar como dato una lista \
de m vectores n-dimensionales y una funci\[OAcute]n de producto interno. En \
la salida debe entregar una lista de m vectores ortonormales. En aras de la \
simplicidad asuma que los vectores de entrada son linealmente independientes. \
Comente como fallaria su programa en caso de entregarle vectores linealmente \
dependientes."
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 " Determine cuales expresiones satisfacen el patron: g[x_+n_Integer y_.]"
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 CellTags->":bib:Maeder99"],

Cell["\t\t(a) g[u + 3x + 2]", "Text"],

Cell["\t\t(b)\tg[2 u^2 + v]", "Text"],

Cell["\t\t(c)\tg[6]", "Text"],

Cell["\t\t(d)\tg[a u + 6]", "Text"],

Cell["\t\t(e)\tg[u^3 - v]", "Text"],

Cell["\t\t(f)\tg[u^3/2]", "Text"],

Cell[TextData[{
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 " El n-\[EAcute]simo polinomio de Bell ",
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 "es el coeficiente de ",
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 "en la expansi\[CloseCurlyQuote]on de Taylor de ",
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Cell["\<\
\t\t(a) Programe la funci\[OAcute]n BellP[n, x, g] que evalue el \
n-\[EAcute]simo polinomio de Bell de una funci\[OAcute]n g. \
\>", "Text"],

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Cell["Ejemplos", "Section"],

Cell[TextData[{
 "En esta secci\[OAcute]n veremos diversos ejemplos de programaci\[OAcute]n \
en ",
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Cell["Determinar el valor m\[AAcute]ximo de una lista", "Subsection"],

Cell["Ac\[AAcute] suponemos dada una lista de valores:", "Text"],

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Cell[TextData[{
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ubicaci\[OAcute]n en la lista?."
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Cell["\<\
Definiremos la funci\[OAcute]n MaxVal[ll] usando varias formas de programaci\
\[OAcute]n.\
\>", "Text"],

Cell["\<\
La primera estrategia ser\[AAcute] recorrer la lista usando un ciclo Do.\
\>", "Text"],

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Cell["\<\
Tambi\[EAcute]n podemos hacer un programa funcional. En este caso el programa \
destaca por su brevedad:\
\>", "Text"],

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Cell["Conjunto de Mandelbrot", "Subsection"],

Cell["\<\
Sea c un n\[UAcute]mero complejo cualquiera. Con su ayuda podemos definir una \
recurrencia del modo siguiente:\
\>", "Text"],

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Cell["\<\
Si iteramos esta recurrencia, para ciertos valores de c, la recurrencia se \
mantendra acotada. El conjunto de Mandelbrot se define como aquel de los \
valores para los cuales la recurrencia converge.\
\>", "Text"],

Cell["\<\
Podemos estudiar el conjunto mediante ensayo y error. Lo primero es definir \
una funci\[OAcute]n que itere la relaci\[OAcute]n anterior.\
\>", "Text"],

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Cell["\<\
Como vemos la forma funcional del n-\[EAcute]simo t\[EAcute]rmino es simple \
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es muy complejo.\
\>", "Text"],

Cell["\<\
Para estudiar convergencia nos bastara con analizar el modulo de la expresi\
\[OAcute]n:\
\>", "Text"],

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