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-1 20 "Integraci\363n num\351rica" }}{PARA 0 "" 0 "" {TEXT -1 117 "El \+
objetivo de esta pr\341ctica es comparar resultados obtenidos mediante
 integraci\363n simb\363lica e integraci\363n num\351rica. " }}{PARA 
0 "" 0 "" {TEXT -1 135 "Asimismo, se trata de obtener conclusiones ace
rca de la aplicaci\363n pr\341ctica de las f\363rmulas estudiadas en t
eor\355a para estimar el error." }}{PARA 0 "" 0 "" {TEXT -1 135 "Las f
unciones de integraci\363n del Maple utilizan m\351todos no estudiados
 en el curso, por lo que utilizaremos nuestras propias funciones. " }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "F\363rmulas de Newton-Cotes compu
estas " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x->5
+x*sin(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(f(x),x=a..b)=int(
 f(x), x=a..b  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"
6$%)operatorG%&arrowGF(,&\"\"&\"\"\"*&9$F.-%$sinG6#F0F.F.F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,&\"\"&\"\"\"*&%\"xGF)-%$sin
G6#F+F)F)/F+;%\"aG%\"bG,.F2F(-F-6#F2F)*&F2F)-%$cosGF5F)!\"\"F1!\"&-F-6
#F1F9*&F1F)-F8F<F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Defnimos \+
los l\355mites de integraci\363n " }{XPPEDIT 18 0 "a;" "6#%\"aG" }
{TEXT -1 3 " y " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 34 ". La inte
gral de la funci\363n entre " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 
3 " y " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 38 " indica el \341rea
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{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 154 "Vamos \+
a aproximar la integral mediante una f\363rmula de Newton-Cotes con do
s puntos de soporte (trapecio) compuesta. En primer lugar determinamos
 el n\372mero " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 107 " de subin
tervalos as\355 como los puntos de la partici\363n que vamos a utiliza
r para aplicar la f\363rmula simple: " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=7;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "for k from 1 to n+1 do x[k]:=a+(k-1)*(b-a)/n od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq(x[k],k=1..n+1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"nG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"
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0 "" {TEXT -1 91 "En segundo lugar, se sabe que la f\363rmula del trap
ecio simple, aplicada a cada subintervalo " }{XPPEDIT 18 0 "[x[i-1], x
[i]],i = 1 .. n;" "6$7$&%\"xG6#,&%\"iG\"\"\"\"\"\"!\"\"&F%6#F(/F(;\"\"
\"%\"nG" }{TEXT -1 29 ", proporciona la proximaci\363n:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(f(x),x=x[i
-1]..x[i])= 1/2*(x[i]-x[i-1])*(f(x[i-1])+f(x[i]))+R[i](f(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,&\"\"&\"\"\"*&%\"xGF)-%$sin
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{PARA 0 "" 0 "" {TEXT -1 7 "siendo " }{XPPEDIT 18 0 "x[i]-x[i-1] = (b-
a)/n;" "6#/,&&%\"xG6#%\"iG\"\"\"&F&6#,&F(F)\"\"\"!\"\"F.*&,&%\"bGF)%\"
aGF.F)%\"nGF." }{TEXT -1 5 "   y " }{XPPEDIT 18 0 "R[i];" "6#&%\"RG6#%
\"iG" }{TEXT -1 53 " el correspondiente resto de integraci\363n. Por t
anto, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(f(x),x=a..b)=Sum('Int
(f(x),x=x[i-1]..x[i])','i'=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%$IntG6$,&\"\"&\"\"\"*&%\"xGF)-%$sinG6#F+F)F)/F+;\"\"!,$%#PiG\"\"#-%$
SumG6$-F%6$-%\"fGF./F+;&F+6#,&%\"iGF)!\"\"F)&F+6#FA/FA;F)\"\"(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "y, aplicando la f\363rmula simple \+
a cada subintervalo," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=0:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for j from 1 to n do y[j]:=1/2*(x[j
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{MPLTEXT 1 0 31 "Int(f(x),x=a..b)=evalf(c)+R(f);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "d:=evalf(c): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$
IntG6$,&\"\"&\"\"\"*&%\"xGF)-%$sinG6#F+F)F)/F+;\"\"!,$%#PiG\"\"#,&$\"+
$4Pgb#!\")F)-%\"RG6#%\"fGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 214 "Comparemos el valor aproximado obtenido
 con el dado por el c\341lculo simb\363lico de la integral. Lo que est
amos haciendo, realmente, al aplicar la f\363rmula del trapecio en cad
a subintervalo, es aproximar la integral de " }{XPPEDIT 18 0 "f;" "6#%
\"fG" }{TEXT -1 51 "   por la de una poligonal definida por los puntos
 " }{XPPEDIT 18 0 "\{x[i], f(x[i])\};" "6#<$&%\"xG6#%\"iG-%\"fG6#&F%6#
F'" }{TEXT -1 79 " sobre la curva. Dibujamos la gr\341fica de las func
iones que estamos integrando: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "readlib(spline):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "S:=proc
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+F+F+F+F+F+F+F+F+F+F+-%+AXESLABELSG6%%\"xG%\"yG%!G-%+PROJECTIONG6%$!#!
*\"\"!F9\"\"\"" 3 400 300 300 2 0 1 0 2 1 0 1 2 1.000000 0.000000 
-90.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16368 0 0 0 0 
0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 9 "El error " }{XPPEDIT 18 0 "R(f);" "6#-%\"RG6#%\"fG" }
{TEXT -1 78 " puede expresarse como la diferencia, en valor absoluto, \+
entre dichos valores:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Int(f(x),x=a..b)=evalf(int(f
(x),x=a..b));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e:=evalf(int(f(x),
x=a..b)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "R(f):=abs(d-e); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,&\"\"&\"\"\"*&%\"xGF)-%$sin
G6#F+F)F)/F+;\"\"!,$%#PiG\"\"#$\"+BTF8D!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"RG6#%\"fG$\")qHwU!\")" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 147 " De otro modo, podr\355a calcularse el error cometido en
 la f\363rmula compuesta a partir de los errores cometidos en cada una
 de las simples, como suma:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "r:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "for s from
 1 to n do A[s]:=int(f(x),x=x[s]..x[s+1])-y[s]; r:=r+A[s]; od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "r:=abs(r):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "R(f):=evalf(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-
%\"RG6#%\"fG$\"*m'HwU!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 327 "Nat
uralmente, cabe la posibilidad de que no se pueda o no interese obtene
r simb\363licamente la integral (realmente estos motivos ser\341n los \+
que nos animen a utilizar integraci\363n num\351rica). En este caso, s
eg\372n sabemos, una cota del error cometido (para funciones suficient
emente regulares) est\341 dada, para cada f\363rmula simple, por " }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "abs(R(f)) = sup[t](abs(D^(n+1)*f(x)))
*int(abs((x-a)*(x-b)),x = a .. b);" "6#/-%$absG6#-%\"RG6#%\"fG*&-&%$su
pG6#%\"tG6#-F%6#*&)%\"DG,&%\"nG\"\"\"\"\"\"F9F9-F*6#%\"xGF9F9-%$intG6$
-F%6#*&,&F=F9%\"aG!\"\"F9,&F=F9%\"bGFFF9/F=;FEFHF9" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "donde " }{XPPEDIT 18 0 "I = [a, b];" "6#/%
\"IG7$%\"aG%\"bG" }{TEXT -1 149 " es el intervalo de integraci\363n. C
on vistas a obtener una cota de este error se podr\355a buscar el supr
emo o una cota de la derivada. En nuestro caso: " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "g:=diff(f(x),x$n+1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG,&-%$cosG6#%\"xG!\")*&F)\"\"\"-%$sinGF(F,F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Por tanto, una cota es:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "G
:=evalf(abs(-8*1)+(b*1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG$\"
+J&=$G9!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "y como cota del er
ror tenemos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "Int(abs((x-a)*(x-b)),x=a..b)=evalf(int(abs((x-a)*(x-b
)),x=a..b));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(G*int(abs((x-
a)*(x-b)),x=a..b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$ab
sG6#*&%\"xG\"\"\",&F+F,%#PiG!\"#F,/F+;\"\"!,$F.\"\"#$\"+C-<MT!\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V>\"\\!f!\"(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 119 "A la vista de las estimaciones anteriores observa
mos inmediatamente la poca conveniencia de la cota superior obtenida. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Por o
tra parte, derivadas de ordenes superiores pueden resultar complicadas
. Concretamente, si " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 
-1 182 " es una funci\363n racional tendremos, como derivadas, funcion
es racionales de grado relativamente elevado (se indica al alumno la c
onveniencia de probar este hecho con otras funciones " }{XPPEDIT 18 0 
"f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " diferentes de la dada en este
 ejemplo). " }}{PARA 0 "" 0 "" {TEXT -1 218 "Maximizar estas funciones
 resulta tedioso con el m\351todo empleado por el Maple. Sin embargo, \+
para particiones suficientemente finas dicho m\341ximo podr\355a aprox
imarse por el valor de la funci\363n en cualquiera de sus puntos:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "rr:=
0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for i from 1 to n do M[i]:=a
bs(f(x[i])); rr:=rr+M[i]*abs(int((x-x[i])*(x-x[i+1]),x=x[i]..x[i+1]));
  od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(rr);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#rrG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+\"4bAV$!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 " Como puede obser
varse, la cota de error obtenida es un valor superior al error " }
{XPPEDIT 18 0 "R(f);" "6#-%\"RG6#%\"fG" }{TEXT -1 302 " obtenido anter
iormente. Sin embargo, es de esperar que cuando el n\372mero de puntos
 de la partici\363n aumente ambos errores se reduzcan y que la cota da
da sea suficiente para garantizar que estamos, en cada caso, dentro de
 los l\355mites impuestos al problema. Se plantea al alumno las siguie
ntes cuestiones:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 34 "\277Qu\351 ocurre al aumentar el n\372mero " }{XPPEDIT 
18 0 "n;" "6#%\"nG" }{TEXT -1 272 " de intervalos de la partici\363n? \+
\277Mejora el error? \277Y la cota obtenida? \277Podr\341 utilizarse e
ste procedimiento de acotar el error para determinar, por ejemplo, la \+
cantidad necesaria de puntos en la partici\363n con vistas a que el er
ror cometido est\351 dentro de ciertos l\355mites? " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "Por \372ltimo, se prop
one al alumno variar la funci\363n, as\355 como los l\355mites de inte
graci\363n, y que obtenga sus propias  conclusiones.   " }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 18 " F\363rmulas de Gauss" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "examples" {TEXT -1 574 "Con las f\363rmu
las de Gauss se puede aspirar, como sabemos, a exactitud para polinomi
os de mayor grado que con f\363rmulas de Newton-Cotes. Para este nuevo
 tipo de f\363rmulas el soporte est\341 formado todas las por ra\355ce
s de un polinomio de la familia de los ortogonales asociados a la inte
gral. Nuestro primer objetivo es, por tanto, la obtenci\363n de dichas
 ra\355ces. Aunque se conoce un m\351todo, basado en la resoluci\363n \+
de uin sistema lineal, que proporciona directamente el soporte, aqu
\355 vamos a obtenerlo del polinomio ortogonal, y dichos polinomios se
 van a obtener recurrentemente:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "p[2]:=1 ; p[1]:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "n:=10; a:=-1 ; b=1;" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "g[1]:=0; h[1]:= int(x,x=a..b)/int(1,x=a..b)
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "for i from 1 to n do p[i+2]:=
(x-g[i+1])*p[i+1]-h[i+1]^2*p[i]; g[i+1]:=int(x*p[i+3]^2,x=a..b)/int(p[
i+1]^2) od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"#\"\"\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"\"\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
aG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"gG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"hG6#\"\"\"*&,&*$)%\"bG\"\"#\"\"\"#F'F-#!\"\"F-F'F.
,&F,F'F'F'!\"\"" }}{PARA 8 "" 1 "" {TEXT -1 51 "Error, (in int) wrong \+
number (or type) of arguments" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 129 "Ahora se definir\341 una famil
ia de polinomios ortogonales a partir de su recurrencia a tres t\351rm
inos. Dicha recurrencia es del tipo" }}{PARA 256 "" 0 "" {XPPEDIT 18 
0 "P[n+1](x) = \{x-b[n+1]\}*P[n](x)-a[n+1]^2*P[n-1](x);" "6#/-&%\"PG6#
,&%\"nG\"\"\"\"\"\"F*6#%\"xG,&*&<#,&F-F*&%\"bG6#,&F)F*\"\"\"F*!\"\"F*-
&F&6#F)6#F-F*F**&&%\"aG6#,&F)F*\"\"\"F*\"\"#-&F&6#,&F)F*\"\"\"F76#F-F*
F7" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 7 "siendo " }{XPPEDIT 
18 0 "b[n],a[n];" "6$&%\"bG6#%\"nG&%\"aG6#F&" }{TEXT -1 44 " n\372mero
s reales,  con condiciones iniciales " }}{EXCHG {PARA 257 "" 0 "" 
{XPPEDIT 18 0 "P[-1](x) = 0,P[0](x) = 1;" "6$/-&%\"PG6#,$\"\"\"!\"\"6#
%\"xG\"\"!/-&F&6#F-6#F,\"\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 44 "Definimos las sucesiones de la recurrencia: " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=10;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "a:=[seq( 2+(1/n^2) ,n=1..N)];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "b:=(seq(1,n=1..N));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"NG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7,\"\"$#\"\"*\"
\"%#\"#>F(#\"#L\"#;#\"#^\"#D#\"#t\"#O#\"#**\"#\\#\"$H\"\"#k#\"$j\"\"#
\")#\"$,#\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG6,\"\"\"F&F&F
&F&F&F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 51 "Definimos los polinomios utilizando la recurrencia:
" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Q[1]:=0;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Q[2]:=1;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"QG6#\"\"\"\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"QG6#\"\"#\"\"\"" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 1 " " }}}{MARK "1 5 11 6" 0 }{VIEWOPTS 1 1 0 1 1 1803 
}