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{SECT 0 {PARA 257 "" 0 "" {TEXT 257 55 "Pr\341ctica de An\341lisis Num
\351rico: Splines e Interpolaci\363n." }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 258 0 "" }
{TEXT 259 305 "Objetivo: Se trata de mostrar algunos de los problemas \+
que presenta la interpolacion polinomial cl\341sica y las mejoras que \+
se pueden obtener utilizando splines en su lugar, en particular veremo
s el fen\363meno de oscilaci\363n polinomial y el problema de la conve
rgencia seg\372n sea la elecci\363n de los puntos base." }}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 55 " Los Splines y el Problema de la Oscilaci
\363n Polinomial." }}{PARA 256 "" 0 "" {TEXT -1 811 "Uno de los proble
mas que ten\355amos cuando interpolabamos un grupo de puntos base con \+
un solo polinomio es que este oscilaba mucho. Especialmente cuando se \+
le hac\355a pasar por puntos que correspond\355an a una funci\363n que
 tuviese un comportamiento muy distinto del que pudiese esperarse de u
n polinomio. Esto lo pretend\355amos solucionar dividiendo el interval
o total en el que ten\355amos los puntos base en otros m\341s peque
\361os y haciendo pasar un polinomio de grado menor por cada uno de es
tos intervalos. En el caso que hiciesemos pasar un polinomio de grado \+
tres por dos puntos base consecutivos e impusiesemos continuidad en la
 primera y segunda derivada obten\355amos (a\361adiendo un par de cond
iciones m\341s, normalmente que la segunda derivada en los extremos se
a cero -splines naturales-) lo que llamabamos spline c\372bico." }}
{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Veamos m
\341s detalladamente este problema de oscilaci\363n." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Primero cargamos alguna
s funciones de librer\355a que necesitaremos:" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "readlib(spline):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 511 "Utiliz
aremos  ahora una funci\363n que sabemos que tiene un comportamiento m
uy distinto del que ser\355a capaz de mostrar un polinomio. La funci
\363n que definimos, en el l\355mite en el que el n\372mero que multip
lica a la variable x tiende a infinito -en este ejemplo 100.- se compo
rta como una funci\363n escal\363n, tambi\351n llamada de Heaviside. L
a funci\363n as\355 definida, sin embargo, es continua y derivable. De
 modo que podemos aplicar todo lo que sabemos sobre el comportamiento \+
del error de interpolaci\363n polinomial cl\341sica. " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 66 "f := proc (x) options operator, arrow; 1*
(1+tanh(100.*x))/2. end;\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&$
\"+++++]!#5\"\"\"-%%tanhG6#,$9$$\"$+\"\"\"!F-F(F(F(" }}}{PARA 0 "" 0 "
" {TEXT -1 31 " La gr\341fica de esta funci\363n es:" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "escalon:= plot(f(x),x = -1..1,thickness=2
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(escalon);" }}
{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6%7io7$$!\"\"\"\"!F*7$$!1nmm;p0
k&*!#;F*7$$!1LL$3<XZ=*F.F*7$$!1nmmT%p\"e()F.F*7$$!1mmm\"4m(G$)F.F*7$$!
1ML$3i.9!zF.F*7$$!1nm;/R=0vF.F*7$$!1++]P8#\\4(F.F*7$$!1nm;/siqmF.F*7$$
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YJFhq$\"1(HD&>USBlF.7$$\"1(H$3x\"yY_%Fhq$\"1WdJ-si>rF.7$$\"1)Ge*[og!G(
Fhq$\"1(z,[J(Q4\")F.7$$\"1GL3_Nl.5F^p$\"1zJA[9i:))F.7$$\"1E$ekGR[b\"F^
p$\"1]Bbf@!Hd*F.7$$\"1CL$3-Dg5#F^p$\"1p6LTM+a)*F.7$$\"1B3-))yh\"Q#F^p$
\"1fVo9MM:**F.7$$\"1A$3_v5sl#F^p$\"1>oF\\'R5&**F.7$$\"1@eRAO!G$HF^p$\"
1]I,arsr**F.7$$\"1@Le*['R3KF^p$\"1F#4[x(o$)**F.7$$\"1>$eRA#efPF^p$\"1C
\\/&3xX***F.7$$\"1<LLezw5VF^p$\"1bjk\"\\)>)***F.7$$\"1`mmmJ+IiF^p$\"1k
$))H@h*****F.7$$\"1!****\\PQ#\\\")F^p$\"07[];*******!#:7$$\"1KLLe\"*[H
7F.$\"0p!z**********Fez7$$\"1*******pvxl\"F.$\"0'**************Fez7$$
\"1)****\\_qn2#F.$\"\"\"F*7$$\"1)***\\i&p@[#F.Fc[l7$$\"1)****\\2'HKHF.
Fc[l7$$\"1lmmmZvOLF.Fc[l7$$\"1+++]2goPF.Fc[l7$$\"1KL$eR<*fTF.Fc[l7$$\"
1+++])Hxe%F.Fc[l7$$\"1lm;H!o-*\\F.Fc[l7$$\"1****\\7k.6aF.Fc[l7$$\"1mmm
;WTAeF.Fc[l7$$\"1****\\i!*3`iF.Fc[l7$$\"1MLLL*zym'F.Fc[l7$$\"1LLL3N1#4
(F.Fc[l7$$\"1mm;HYt7vF.Fc[l7$$\"1*******p(G**yF.Fc[l7$$\"1mmmT6KU$)F.F
c[l7$$\"1LLLLbdQ()F.Fc[l7$$\"1++]i`1h\"*F.Fc[l7$$\"1++]P?Wl&*F.Fc[l7$F
c[lFc[l-%'COLOURG6&%$RGBG$\"#5F)F*F*-%*THICKNESSG6#\"\"#-%+AXESLABELSG
6$Q\"x6\"%!G-%%VIEWG6$;F(Fc[l%(DEFAULTG" 2 395 395 395 2 0 1 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 195 "Creamos ahora
 una funci\363n para obtener una lista de 2n+1 puntos base equiespacia
dos en el intervalo [-1,1], zona que recoge la parte de comportamiento
 \"an\363malo\" de la funci\363n que hemos definido. " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 57 "puntosbase := proc (n) local i; seq(i/n,i
 = -n .. n) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+puntosbaseGR6#%
\"nG6#%\"iG6\"F*-%$seqG6$*&8$\"\"\"9$!\"\"/F/;,$F1!\"\"F1F*F*F*" }}}
{PARA 0 "" 0 "" {TEXT -1 72 "As\355, para obtener los cinco primeros p
untos equiespaciados, basta hacer:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "xd := [puntosbase(2)];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#xdG7'!\"\"#F&\"\"#\"\"!#\"\"\"F(F+" }}}{PARA 0 "" 0 "" {TEXT 
-1 81 "Con lo cual la lista de los pares de datos (x(i),y(i)) se puede
 obtener haciendo:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "datos \+
:= [seq([xd[i], f(xd[i])],i = 1 .. nops(xd))];" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&datosG7'7$!\"\"\"\"!7$#F'\"\"#F(7$F
($\"+++++]!#57$#\"\"\"F+$\"+++++5!\"*7$F2F3" }}}{PARA 0 "" 0 "" {TEXT 
-1 82 "Dibujamos la funci\363n original junto con el muestreo correspo
ndiente a estos datos:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "pu
ntos:=pointplot(datos,symbol=cross):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "display([escalon, puntos]);" }}{PARA 13 "" 1 "" 
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!1nm;/R=0vF.F*7$$!1++]P8#\\4(F.F*7$$!1nm;/siqmF.F*7$$!1++](y$pZiF.F*7$
$!1LLL$yaE\"eF.F*7$$!1nmm\">s%HaF.F*7$$!1+++]$*4)*\\F.F*7$$!1+++]_&\\c
%F.F*7$$!1+++]1aZTF.F*7$$!1nm;/#)[oPF.F*7$$!1MLL$=exJ$F.F*7$$!1MLLL2$f
$HF.F*7$$!1++]PYx\"\\#F.F*7$$!1MLLL7i)4#F.F*7$$!1++]P'psm\"F.$\"17UkAz
:vK!#I7$$!1++]74_c7F.$\"1gDBBg)*=7!#E7$$!1JLL$3x%z#)!#<$\"10hB8q$[V'!#
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xY*)Fhq$\"1i6N4v^J9F.7$$!1<+D\"GyNH'Fhq$\"1FTuS1&>@#F.7$$!1%=/E]yp'\\F
hq$\"1xM8_)=Cq#F.7$$!1^$eRsy.k$Fhq$\"1pSKR8<cKF.7$$!1=DJX*yPJ#Fhq$\"1K
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$H&>&F.7$$\"11$3_]\\(o<Fhq$\"1'G=#QiEveF.7$$\"1,e9TQrYJFhq$\"1(HD&>USB
lF.7$$\"1(H$3x\"yY_%Fhq$\"1WdJ-si>rF.7$$\"1)Ge*[og!G(Fhq$\"1(z,[J(Q4\"
)F.7$$\"1GL3_Nl.5F^p$\"1zJA[9i:))F.7$$\"1E$ekGR[b\"F^p$\"1]Bbf@!Hd*F.7
$$\"1CL$3-Dg5#F^p$\"1p6LTM+a)*F.7$$\"1B3-))yh\"Q#F^p$\"1fVo9MM:**F.7$$
\"1A$3_v5sl#F^p$\"1>oF\\'R5&**F.7$$\"1@eRAO!G$HF^p$\"1]I,arsr**F.7$$\"
1@Le*['R3KF^p$\"1F#4[x(o$)**F.7$$\"1>$eRA#efPF^p$\"1C\\/&3xX***F.7$$\"
1<LLezw5VF^p$\"1bjk\"\\)>)***F.7$$\"1`mmmJ+IiF^p$\"1k$))H@h*****F.7$$
\"1!****\\PQ#\\\")F^p$\"07[];*******!#:7$$\"1KLLe\"*[H7F.$\"0p!z******
****Fez7$$\"1*******pvxl\"F.$\"0'**************Fez7$$\"1)****\\_qn2#F.
$\"\"\"F*7$$\"1)***\\i&p@[#F.Fc[l7$$\"1)****\\2'HKHF.Fc[l7$$\"1lmmmZvO
LF.Fc[l7$$\"1+++]2goPF.Fc[l7$$\"1KL$eR<*fTF.Fc[l7$$\"1+++])Hxe%F.Fc[l7
$$\"1lm;H!o-*\\F.Fc[l7$$\"1****\\7k.6aF.Fc[l7$$\"1mmm;WTAeF.Fc[l7$$\"1
****\\i!*3`iF.Fc[l7$$\"1MLLL*zym'F.Fc[l7$$\"1LLL3N1#4(F.Fc[l7$$\"1mm;H
Yt7vF.Fc[l7$$\"1*******p(G**yF.Fc[l7$$\"1mmmT6KU$)F.Fc[l7$$\"1LLLLbdQ(
)F.Fc[l7$$\"1++]i`1h\"*F.Fc[l7$$\"1++]P?Wl&*F.Fc[l7$Fc[lFc[l-%'COLOURG
6&%$RGBG$\"#5F)F*F*-%*THICKNESSG6#\"\"#-%'POINTSG6(F'7$$!+++++]!#5F*7$
F*$\"+++++]F\\`l7$F^`l$\"+++++5!\"*7$Fc[lFa`l-%'SYMBOLG6#%&CROSSG-%+AX
ESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fc[l%(DEFAULTG" 2 394 394 394 2 0 1 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 276 "Veamo
s ahora el aspecto del polinomio  de interpolaci\363n que pasa por est
os puntos. La funci\363n que realiza la interpolaci\363n polinomial re
quiere los datos (x(i),y(i)) en dos listas separadas, puesto que ya te
nemos la x(i) en lista aparte, s\363lo tenemos que formar la de las y(
i):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "yd:=[seq(datos[i][2],
i = 1 .. nops(datos))];" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#ydG7'\"\"!F&$\"+++++]!#5$\"+++++5!\"*F*" }}}{PARA 0 "" 0 "" 
{TEXT -1 15 "e interpolarla:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "polint:=interp(xd,yd,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'pol
intG,**$)%\"xG\"\"$\"\"\"$!+nmmmm!#5*$)F(\"\"#F*$!\"\"F-F($\"+nmmm6!\"
*$\"+++++]F-\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 129 "El aspecto que ti
ene, a pesar de pasar por todos los puntos, no es especialmente bueno \+
cuando se compara con la funci\363n original:" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "interpola:=plot(polint,x=-1..1,color=blue):" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display([es
calon, puntos, interpola]);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
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$$\"1lm;H!o-*\\F.Fc[l7$$\"1****\\7k.6aF.Fc[l7$$\"1mmm;WTAeF.Fc[l7$$\"1
****\\i!*3`iF.Fc[l7$$\"1MLLL*zym'F.Fc[l7$$\"1LLL3N1#4(F.Fc[l7$$\"1mm;H
Yt7vF.Fc[l7$$\"1*******p(G**yF.Fc[l7$$\"1mmmT6KU$)F.Fc[l7$$\"1LLLLbdQ(
)F.Fc[l7$$\"1++]i`1h\"*F.Fc[l7$$\"1++]P?Wl&*F.Fc[l7$Fc[lFc[l-%'COLOURG
6&%$RGBG$\"#5F)F*F*-%*THICKNESSG6#\"\"#-%'POINTSG6(F'7$$!+++++]!#5F*7$
F*$\"+++++]F\\`l7$F^`l$\"+++++5!\"*7$Fc[lFa`l-%'SYMBOLG6#%&CROSSG-F$6%
7S7$F($!14X=T9++S!#D7$F,$!1K'*\\:'*GeKF^p7$F0$!1GaEGGj+bF^p7$F3$!1AO#4
yt>R(F^p7$F6$!1V@`2+4_')F^p7$F9$!1([g!*\\_iH*F^p7$F<$!1R'H2n>rP*F^p7$F
?$!1enDBa^k*)F^p7$FB$!1vz:e>nN!)F^p7$FE$!1s*>\"\\)H<j'F^p7$FH$!1J&3r(=
^@ZF^p7$FK$!1*Rz#H1UtEF^p7$FN$\"1uR/F$fuE\"F]q7$FQ$\"1/Y6KI2%3$F^p7$FT
$\"1tkY*=^%ojF^p7$FW$\"1!**3n7%=-'*F^p7$FZ$\"1xv?,2vs8F.7$Fgn$\"14,nZ&
fMu\"F.7$Fjn$\"1)QpTYrg>#F.7$F]o$\"1J](H.FKh#F.7$F`o$\"12C.(z\\d3$F.7$
Ffo$\"19xt6YGZNF.7$F\\p$\"1x/Q8Z%y.%F.7$Fip$\"1]'3dTlP\\%F.7$Fau$\"1aC
8;H[))\\F.7$Fgy$\"1Qk(**)))Q-bF.7$Faz$\"1,5?Ll8ZfF.7$Fgz$\"1!4y!*o8?U'
F.7$F\\[l$\"1SXHl()p.pF.7$Fa[l$\"1'e#=1_=jtF.7$Ff[l$\"1F8$*p2\"Rz(F.7$
Fi[l$\"1!yYX$e#HD)F.7$F\\\\l$\"1cMl#)e?X')F.7$F_\\l$\"1l?;B4))R!*F.7$F
b\\l$\"1VuQzNKt$*F.7$Fe\\l$\"15o([rB'3(*F.7$Fh\\l$\"1Lm$eb-N***F.7$F[]
l$\"1P]m@mmD5Fez7$F^]l$\"1b,1QJpZ5Fez7$Fa]l$\"1#))z%)=Dl1\"Fez7$Fd]l$
\"1[>(ff!G!3\"Fez7$Fg]l$\"1f89R$*f*3\"Fez7$Fj]l$\"1?i>F1!Q4\"Fez7$F]^l
$\"1*e(y*zzH4\"Fez7$F`^l$\"1rCPx'=i3\"Fez7$Fc^l$\"1iz7%)Hju5Fez7$Ff^l$
\"1nR$ziGi0\"Fez7$Fi^l$\"1Y_Tf5\\K5Fez7$Fc[l$\"1++?++++5Fez-F]_l6&F__l
F*F*$\"*++++\"!\")-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6$Q\"x6\"%!G-%%VIE
WG6$;F(Fc[l%(DEFAULTG" 2 400 300 300 2 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 
0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 66 "Veamos ahora qu\351 aspecto \+
tiene el spline que pasa por esos puntos:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "scubico:=spline(xd,yd
,x,cubic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(scubicoG-%*PIECEWISEG
6&7$,*$\"+,+++v!#5\"\"\"%\"xG$\"++++]F!\"**$)F.\"\"#\"\"\"$\"+++++IF1*
$)F.\"\"$F5$\"+++++5F12F.#!\"\"F47$,*$\"+++++]F,F-F.$\"++++]7F1F2$F1F,
F8$!+++++5F12F.\"\"!7$,*FBF-F.FDF2$!+++++!*!#>F8$!+')********F,2F.#F-F
47$,*$\"+0+++DF,F-F.$\"+)*****\\FF1F2$!+)*******HF1F8$\"+&*********F,%
*otherwiseG" }}}{PARA 0 "" 0 "" {TEXT -1 16 "Su gr\341fica ser\341:" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "splinecubico:=plot(scubico,
x=-1..1,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "di
splay([escalon, puntos, interpola, splinecubico]);" }}{PARA 13 "" 1 "
" {INLPLOT "6(-%'CURVESG6%7io7$$!\"\"\"\"!F*7$$!1nmm;p0k&*!#;F*7$$!1LL
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$$\"1@Le*['R3KF^p$\"1F#4[x(o$)**F.7$$\"1>$eRA#efPF^p$\"1C\\/&3xX***F.7
$$\"1<LLezw5VF^p$\"1bjk\"\\)>)***F.7$$\"1`mmmJ+IiF^p$\"1k$))H@h*****F.
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[l7$$\"1lm;H!o-*\\F.Fc[l7$$\"1****\\7k.6aF.Fc[l7$$\"1mmm;WTAeF.Fc[l7$$
\"1****\\i!*3`iF.Fc[l7$$\"1MLLL*zym'F.Fc[l7$$\"1LLL3N1#4(F.Fc[l7$$\"1m
m;HYt7vF.Fc[l7$$\"1*******p(G**yF.Fc[l7$$\"1mmmT6KU$)F.Fc[l7$$\"1LLLLb
dQ()F.Fc[l7$$\"1++]i`1h\"*F.Fc[l7$$\"1++]P?Wl&*F.Fc[l7$Fc[lFc[l-%'COLO
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1aC8;H[))\\F.7$Fgy$\"1Qk(**)))Q-bF.7$Faz$\"1,5?Ll8ZfF.7$Fgz$\"1!4y!*o8
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$F,$!1B))>eFd\"3\"F^p7$F0$!1MnOI>&R)>F^p7$F3$!1P3sQ$oI\"HF^p7$F6$!1=Gx
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y=F^p7$FN$\"1X)Hh`X'3&*!#?7$FQ$\"1R>*[!e)3X#F^p7$FT$\"1>\"z;/!Q!H&F^p7
$FW$\"1X0#)=srX#)F^p7$FZ$\"1)3$HW^+=7F.7$Fgn$\"1m1)*)oaJe\"F.7$Fjn$\"1
klx:X**R?F.7$F]o$\"1(eVo;^\"pCF.7$F`o$\"1W9#\\^fA'HF.7$Ffo$\"1FaOut=\\
MF.7$F\\p$\"1h'*fC4uqRF.7$Fip$\"1UE=%)*RyX%F.7$Fau$\"1Vp.q-m()\\F.7$Fg
y$\"1x'>o$\\/QbF.7$Faz$\"1!3-1)GC8gF.7$Fgz$\"1,$\\P&fF=lF.7$F\\[l$\"1o
G(pHgm-(F.7$Fa[l$\"1+>aXDR1vF.7$Ff[l$\"1nb;s8y\\zF.7$Fi[l$\"1MZd72C8%)
F.7$F\\\\l$\"1A@oQ9V*z)F.7$F_\\l$\"1,(pv+@b<*F.7$Fb\\l$\"1![I)om-![*F.
7$Fe\\l$\"16H%[kq!p(*F.7$Fh\\l$\"152h;)>^***F.7$F[]l$\"1$H[:+(3=5Fez7$
F^]l$\"1pFP\"\\J:.\"Fez7$Fa]l$\"1$R\\3ho5/\"Fez7$Fd]l$\"1&)3/#R1j/\"Fe
z7$Fg]l$\"1i9-E'3\"[5Fez7$Fj]l$\"10jE%>%zY5Fez7$F]^l$\"1ke\\!QZK/\"Fez
7$F`^l$\"1jr)p$o)o.\"Fez7$Fc^l$\"12)*[V%G&H5Fez7$Ff^l$\"1B(*>9HQ?5Fez7
$Fi^l$\"1)[Xo)=y55FezF[_lF\\_l-Fc_l6#\"\"%-Ffjl6#Fd[l-%+AXESLABELSG6$Q
\"x6\"%!G-%%VIEWG6$;F(Fc[l%(DEFAULTG" 2 400 300 300 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 
0 0 276 15144 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 329 "Aparentemen
te no se ha ganado gran cosa  pero, como sabemos, los problemas de osc
ilaci\363n polinomial se agravan con el n\372mero de puntos base (al m
enos con elecciones no \363ptimas de \351stos). Ve\341moslo tomando nu
eve puntos base, con lo que obtendremos un polinomio de interpolaci
\363n de grado ocho, y repitiendo los c\341lculos anteriores: " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xd:=[puntosbase(4)];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xdG7+!\"\"#!\"$\"\"%#F&\"\"##F&F)\"
\"!#\"\"\"F)#F/F+#\"\"$F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "datos := [seq([xd[i], f(xd[i])],i = 1 .. nops(xd))];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&datosG7+7$!\"\"\"\"!7$#!\"$\"\"%F(7$#F'\"\"#F
(7$#F'F,F(7$F($\"+++++]!#57$#\"\"\"F,$\"+++++5!\"*7$#F8F/F97$#\"\"$F,F
97$F8F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "puntos:=pointplo
t(datos,symbol=cross):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "y
d:=[seq(datos[i][2],i = 1 .. nops(datos))];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#ydG7+\"\"!F&F&F&$\"+++++]!#5$\"+++++5!\"*F*F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "polint:=interp(xd,yd,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'polintG,4*$)%\"xG\"\")\"\"\"$!+3l$z
y\"!#<*$)F(\"\"(F*$!+XT)p7)!\"**$)F(\"\"'F*$\"#FF3*$)F(\"\"&F*$\"+\\WW
k:!\")*$)F(\"\"%F*$\"#AF3*$)F(\"\"$F*$!+\\bbb&*F3*$)F(\"\"#F*$FAF3F($
\"+S_4QDF3$\"+.+++]!#5\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "interpola:=plot(polint,x=-1..1,color=blue):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "scubico:=spline(xd,yd,x,cubic);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%(scubicoG-%*PIECEWISEG6*7$,*$\"*\"H9d`!\"*\"
\"\"%\"xG$\"+W9dy;F,*$)F.\"\"#\"\"\"$\"+IdG9<F,*$)F.\"\"$F4$\"+odG9d!#
52F.#!\"$\"\"%7$,*$!+HG92\"*F<F-F.$!+\\G92TF,F1$!+#*******fF,F7$!+bG9d
GF,2F.#!\"\"F37$,*$\"+sUrN!)F<F-F.$\"+<9dyhF,F1$\"+aG9d9!\")F7$\"+$G9d
3\"FV2F.#FMF@7$,*$\"+++++]F<F-F.$\"+%G9d`#F,F1$\"+++++?!#=F7$!+_&G9d)F
,2F.\"\"!7$,*FgnF-F.FinF1$F3F,F7$!+u&G9d)F,2F.#F-F@7$,*$\"+3dGk>F<F-F.
$\"+G9dyhF,F1$!+dG9d9FVF7$\"+'G9d3\"FV2F.#F-F37$,*$\"+%G92\">F,F-F.$!+
_G92TF,F1$\"+'*******fF,F7$!+dG9dGF,2F.#F9F@7$,*$\"*5dGk%F,F-F.F/F1$!+
JdG9<F,F7$\"+rdG9dF<%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "splinecubico:=plot(scubico,x=-1..1,thickness=4):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display([escalon, puntos, in
terpola, splinecubico]);" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6%7
io7$$!\"\"\"\"!F*7$$!1nmm;p0k&*!#;F*7$$!1LL$3<XZ=*F.F*7$$!1nmmT%p\"e()
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!1nm;/#)[oPF.F*7$$!1MLL$=exJ$F.F*7$$!1MLLL2$f$HF.F*7$$!1++]PYx\"\\#F.F
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Fhq7$$!1N$eRZ<>v#F^p$\"1L8P2?iaSFhq7$$!1o\"H#=vf'[#F^p$\"1%yd\\#)GM(oF
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Rsy.k$Fhq$\"1pSKR8<cKF.7$$!1=DJX*yPJ#Fhq$\"1K<^!yAL'QF.7$$!1^omm;zr)*F
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zJA[9i:))F.7$$\"1E$ekGR[b\"F^p$\"1]Bbf@!Hd*F.7$$\"1CL$3-Dg5#F^p$\"1p6L
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$F]^l$\"1h:Qd^?-5Fez7$F`^l$\"1)RPzM<L+\"Fez7$Fc^l$\"1%o&QM\"eL+\"Fez7$
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c_l6#\"\"$-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fc[l%(DEFAULTG" 2 
400 300 300 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 12336 11056 0 0 0 0 0 0 }}}{PARA 0 "
" 0 "" {TEXT -1 327 "Vemos c\363mo ahora la diferencia es m\341s notab
le. En particular resulta evidente que en los extremos de los interval
os el polinomio de interpolaci\363n oscila mucho mientras que el splin
e se comporta bien. Estos problemas, en general, se agravan cuanto may
or sea el n\372mero de puntos base, a menos que escojamos \351stos cui
dadosamente. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xd:=[puntos
base(8)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xdG73!\"\"#!\"(\"\")#!
\"$\"\"%#!\"&F)#F&\"\"##F+F)#F&F,#F&F)\"\"!#\"\"\"F)#F6F,#\"\"$F)#F6F0
#\"\"&F)#F9F,#\"\"(F)F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "
datos := [seq([xd[i], f(xd[i])],i = 1 .. nops(xd))];" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%&datosG737$!\"\"\"\"!7$#!\"(\"\")F(7$#!\"$\"\"%F(7
$#!\"&F,F(7$#F'\"\"#F(7$#F/F,F(7$#F'F0F(7$#F'F,F(7$F($\"+++++]!#57$#\"
\"\"F,$\"+++++5!\"*7$#FCF0FD7$#\"\"$F,FD7$#FCF6FD7$#\"\"&F,FD7$#FKF0FD
7$#\"\"(F,FD7$FCFD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "punto
s:=pointplot(datos,symbol=cross):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "yd:=[seq(datos[i][2],i = 1 .. nops(datos))];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG73\"\"!F&F&F&F&F&F&F&$\"+++++]!#
5$\"+++++5!\"*F*F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "polint:=interp(xd,yd,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'p
olintG,D%\"xG$\"+KZ(HI&!\"**$)F&\"\"#\"\"\"$!$b\"F)*$)F&\"\"$F-$!+%G'4
k(*!\")*$)F&\"\"%F-$\"%Q6F5*$)F&\"\"&F-$\"+@pv+**!\"(*$)F&\"\"'F-$!%GJ
F@*$)F&\"\"(F-$!+q*[EA&!\"'*$)F&\"\")F-$\"%7EFK*$)F&\"#7F-$\"%_@!\"&*$
)F&\"#8F-$\"+<j;X=FV*$)F&\"\"*F-$\"+/fL'\\\"FV*$)F&\"#;F-$\"+BH%3D(!#7
*$)F&\"#:F-$!+qmMrdFK$\"+/+++]!#5\"\"\"*$)F&\"#9F-$!&B.#FK*$)F&\"#5F-$
!%/6FV*$)F&\"#6F-$!+bk#=L#FV" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "interpola:=plot(polint,x=-1..1,color=blue):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "scubico:=spline(xd,yd,x,cubic);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%(scubicoG-%*PIECEWISEG627$,*$\"+2ve>B!#6\"\"
\"%\"xG$\"+B-SKqF,*$)F.\"\"#\"\"\"$\"+u!>#pqF,*$)F.\"\"$F4$\"+ejScBF,2
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,%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "splinecubi
co:=plot(scubico,x=-1..1,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "display([escalon, puntos, interpola, splinecubico]);
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0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "scubi
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49 "splinecubico2:=plot(scubico,x=-1..1,thickness=3):" }{TEXT -1 0 "" 
}}}{PARA 0 "" 0 "" {TEXT -1 184 "El comportamiento de los splines corr
espondientes a los dos conjuntos de puntos base es, sin embargo, muy b
ueno, siendo pr\341cticamente indistinguibles entre si y de la funci
\363n original:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "display([
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}}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
92 " Mejora de la Interpolaci\363n Polinomial Cl\341sica a trav\351s d
e la Elecci\363n de los Puntos Base.  " }}{PARA 0 "" 0 "" {TEXT -1 
708 "Uno de los problemas de la interpolaci\363n es que no se puede as
egurar la convergencia del polinomio de interpolaci\363n a la funci
\363n conforme se aumenta el n\372mero de puntos base (y por tanto el \+
grado del polinomio)  para una elecci\363n cualquiera de estos. Lament
ablemente, esto es cierto en particular para el caso de puntos base eq
uiespaciados, elecci\363n obvia y muchas veces la \372nica disponible.
 En cambio, si es posible seleccionar los puntos base de manera arbitr
aria, siempre podemos elegirlos de modo que las propiedades de converg
encia sean mejores.  En concreto,  si utilizamos las denominadas absci
sas de Chebyshev (ceros de los polinomios de Chebyshev de primera clas
e) que vienen dados por la expresi\363n " }{XPPEDIT 18 0 "cos((i+1/2)*
Pi/(n+1)),i = 0 .. n;" "6$-%$cosG6#*(,&%\"iG\"\"\"*&\"\"\"F)\"\"#!\"\"
F)F)%#PiGF),&%\"nGF)\"\"\"F)F-/F(;\"\"!F0" }{TEXT -1 108 " podemos gar
antizar (aunque no es la elecci\363n \363ptima de puntos base) que, si
 la funci\363n que interpolamos es " }{TEXT -1 1 " " }{XPPEDIT 18 0 "C
^`1`;" "6#)%\"CG%\"1G" }{TEXT -1 57 " en [a,b], el intervalo en el que
 interpolamos, entonces:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {XPPEDIT 18 0 "Limit(p[n](x),n = infinity) = f(x);" "6#/-%&Limi
tG6$-&%\"pG6#%\"nG6#%\"xG/F+%)infinityG-%\"fG6#F-" }{TEXT -1 31 "  par
a x perteneciente a [a,b]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 149 " Veamos como se comporta el mismo problema de in
terpolaci\363n anterior si utilizamos las abscisas de Chebyshev en lug
ar de  puntos base equiespaciados." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Primero d
efinimos los nodos de Chebyshev mediante una funci\363n:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "chebynodes:= proc (n) local i; seq(
-cos((i+1/2)/(n+1)*Pi),i = 0 .. n) end;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+chebynodesGR6#%\"nG6#%\"iG6\"F*-%$seqG6$,$-%$cosG6#*&*&,&8$\"
\"\"#F6\"\"#F6F6%#PiGF6\"\"\",&9$F6F6F6!\"\"!\"\"/F5;\"\"!F<F*F*F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d:=[chebynodes(8)];" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG7+,$-%$cosG6#,$%
#PiG#\"\"\"\"#=!\"\",$*$-%%sqrtG6#\"\"$\"\"\"#F/\"\"#,$-F(6#,$F+#\"\"&
F.F/,$-F(6#,$F+#\"\"(F.F/\"\"!F@F:,$F1#F-F8F'" }}}{PARA 0 "" 0 "" 
{TEXT -1 172 "Utilizaremos la evaluaci\363n num\351rica de los nodos d
e Chebyshev para evitar que Maple los trate como cantidades exactas y \+
use para ello muchos m\341s recursos de los necesarios." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xd:=evalf(d);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#xdG7+$!+Iv2[)*!#5$!+SSDg')F($!+&4wyU'F($!+G9??MF(\"
\"!$\"+G9??MF($\"+&4wyU'F($\"+SSDg')F($\"+Iv2[)*F(" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 59 "datos := [seq([xd[i], evalf(f(xd[i]))],i = \+
1 .. nops(xd))];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&datosG7+7$$!+Iv
2[)*!#5\"\"!7$$!+SSDg')F)F*7$$!+&4wyU'F)F*7$$!+G9??MF)F*7$F*$\"+++++]F
)7$$\"+G9??MF)$\"+++++5!\"*7$$\"+&4wyU'F)F:7$$\"+SSDg')F)F:7$$\"+Iv2[)
*F)F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "puntos:=pointplot(
datos,symbol=cross):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "yd:
=[seq(datos[i][2],i = 1 .. nops(datos))];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#ydG7+\"\"!F&F&F&$\"+++++]!#5$\"+++++5!\"*F*F*F*" }}}
{PARA 0 "" 0 "" {TEXT -1 40 "El polinomio de interpolaci\363n nos qued
a:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "polint:=interp(xd,yd,x
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'polintG,4*$)%\"xG\"\")\"\"\"$
!+Ej0:p!#<*$)F(\"\"(F*$!+TGB)e#!\"**$)F(\"\"'F*$\"$3\"F3*$)F(\"\"&F*$
\"+T,9@gF3*$)F(\"\"%F*$!#UF3*$)F(\"\"$F*$!+J*p\"**[F3*$)F(\"\"#F*$F@F3
F($\"+@ruc>F3$\"+,+++]!#5\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "interpola:=plot(polint,x=-1..1, color=blue):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "display([interpola]);" }}{PARA 13 "" 1 "
" {INLPLOT "6%-%'CURVESG6$7[o7$$!\"\"\"\"!$\"1ntV\\4^:&*!#=7$$!1nm;HU,
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5!#<7$$!1LL$3<XZ=*F1$!1uDMj/\"=r*F-7$$!1***\\iId9(*)F1$!1#)ePS_:WlF-7$
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1%o2qr#fReF17$$\"1!****\\PQ#\\\")FH$\"1g0Ti<IolF17$$\"1KLLe\"*[H7F1$\"
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\"1QIp6\\CG'*F17$$\"1lmmmZvOLF1$\"1)o*4rEAY**F17$$\"1+++]2goPF1$\"1!z+
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g)*F17$$\"1mmmT6KU$)F1$\"1-LU66pI**F17$$\"1LLLLbdQ()F1$\"15AOB'[<+\"Fc
x7$$\"1nm\"zW?)\\*)F1$\"1B$=MLTh+\"Fcx7$$\"1++]i`1h\"*F1$\"1$RSg)yW45F
cx7$$\"1++++PDj$*F1$\"1S]85nt55Fcx7$$\"1++]P?Wl&*F1$\"1Ml[+#)=45Fcx7$$
\"1+]7G:3u'*F1$\"1OPGQ=y15Fcx7$$\"1++v=5s#y*F1$\"1a:b.Y,.5Fcx7$$\"1+]P
40O\"*)*F1$\"1ar'onGm(**F17$$\"\"\"F*$\"1qV\\4\\%[!**F1-%'COLOURG6&%$R
GBGF*F*$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F^_l%(DEFA
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" 0 "" {TEXT -1 261 "Que, si lo comparamos con el polinomio de grado o
cho obtenido con abcisas equiespaciadas es una mejora notable, vi\351n
dose, en particular c\363mo comienza a desaparecer el problema de osci
lacion en los extremos. Si lo comparamos con el spline c\372bico corre
spondiente:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "scubico:=spli
ne(xd,yd,x,cubic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(scubicoG-%*PI
ECEWISEG6*7$,*$\"*J[_x*!\"*\"\"\"%\"xG$\"+3772IF,*$)F.\"\"#\"\"\"$\"+m
/RoIF,*$)F.\"\"$F4$\"+6^dQ5F,2F.$!+SSDg')!#57$,*$!+X!>e:\"F,F-F.$!+g()
*HQ%F,F1$!+P])\\Y&F,F7$!+^E#fC#F,2F.$!+&4wyU'F?7$,*$\"+hq#oc)F?F-F.$\"
+A?p4]F,F1$\"+?yWZ\"*F,F7$\"+2SrJ`F,2F.$!+G9??MF?7$,*$\"+++++]F?F-F.$
\"+913\")=F,F1$F3F?F7$!+p?S$e$F,2F.\"\"!7$,*FfnF-F.FhnF1$\"+++++?!#>F7
F[o2F.$\"+G9??MF?7$,*$\"+PH<L9F?F-F.FQF1$!+?yWZ\"*F,F7$\"+3SrJ`F,2F.$
\"+&4wyU'F?7$,*$\"+X!>e:#F,F-F.FDF1$\"+Q])\\Y&F,F7$!+_E#fC#F,2F.$\"+SS
Dg')F?7$,*$\")o^ZAF,F-F.$\"+4772IF,F1$!+n/RoIF,F7$\"+7^dQ5F,%*otherwis
eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "splinecubico:=plot(sc
ubico,x=-1..1,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "display([escalon, puntos, interpola, splinecubico]);" }}{PARA 13 "
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}}}{PARA 0 "" 0 "" {TEXT -1 393 "Vemos c\363mo pr\341cticamente se sup
erponen el uno con el otro. Esto es una expresi\363n directa de la con
vergencia a la funci\363n interpolada de ambos metodos, ya que ahora t
anto el spline como el polinomio de interpolaci\363n, a medida que se \+
aumenta el n\372mero de puntos base tienen que dar como resultado una \+
aproximaci\363n cada vez mejor a la funci\363n original. Si volvemos a
 aumentar el n\372mero de puntos:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "xd:=[chebynodes(16)];" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%#xdG73,$-%$cosG6#,$%#PiG#\"\"\"\"#M!\"\",$-F(6#,$F+#\"\"$F.F/,
$-F(6#,$F+#\"\"&F.F/,$-F(6#,$F+#\"\"(F.F/,$-F(6#,$F+#\"\"*F.F/,$-F(6#,
$F+#\"#6F.F/,$-F(6#,$F+#\"#8F.F/,$-F(6#,$F+#\"#:F.F/\"\"!FUFOFIFCF=F7F
1F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "xd:=evalf(xd);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#xdG73$!+j<Md**!#5$!+KkD='*F($!+9Hj^
*)F($!+tA<!)zF($!+Nk&pt'F($!+J;Kk_F($!+jmT7OF($!+y^\\P=F(\"\"!$\"+y^\\
P=F($\"+jmT7OF($\"+J;Kk_F($\"+Nk&pt'F($\"+tA<!)zF($\"+9Hj^*)F($\"+KkD=
'*F($\"+j<Md**F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "datos :
= [seq([xd[i], evalf(f(xd[i]))],i = 1 .. nops(xd))];" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%&datosG737$$!+j<Md**!#5\"\"!7$$!+KkD='*F)F*7$$!+9H
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puntos:=pointplot(datos,symbol=cross):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "yd:=[seq(datos[i][2],i = 1 .. nops(datos))];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG73\"\"!F&F&F&F&F&F&F&$\"+++++]!#
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0 24 "polint:=interp(xd,yd,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'p
olintG,D%\"xG$\"+ZCUNP!\"**$)F&\"\"$\"\"\"$!+(>H_k$!\")*$)F&\"\"*F-$\"
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1 0 44 "interpola:=plot(polint,x=-1..1, color=blue):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "plot(polint,x=-1..1);" }}{PARA 13 "" 1 "
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2 9 1 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 77 "Vemos de
 nuevo como las oscilaciones en los extremos disminuyen cada vez m\341
s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "scubico:=spline(xd,yd,
x,cubic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(scubicoG-%*PIECEWISEG6
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4+\"Fez7$F^]l$\"17+y\"4&)>+\"Fez7$Fa]l$\"1ymnNXG,5Fez7$Fd]l$\"11`QVy9+
5Fez7$Fg]l$\"1AiEoU6'***F.7$Fj]l$\"1K#=Mn:q***F.7$F]^l$\"1rlNH(z&****F
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$Fi^l$\"0q]B@()*****Fez7$Fc[l$\"0++^z&******FezF\\_l-Fc_l6#\"\"$-%+AXE
SLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fc[l%(DEFAULTG" 2 400 300 300 2 0 1 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 52 "Que, d
e nuevo, vuelven a ser virtualmente id\351nticos." }}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 21 "El ejemplo de Runge. " }}{PARA 0 "" 0 "" {TEXT -1 
170 "Podemos realizar la misma comparaci\363n con el ejemplo de Runge \+
y ver c\363mo se comporta en este caso la interpolaci\363n con puntos \+
base de Chebyshev. Puesto que la funci\363n es " }{XPPEDIT 18 0 "C^`1`
;" "6#)%\"CG%\"1G" }{TEXT -1 86 ", tenemos garantizada la convergencia
, de modo que esperamos que las cosas vayan bien." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "g:=x-> 1/(1+25*x*x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&\"
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{MPLTEXT 1 0 19 "d:=[chebynodes(8)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"dG7+,$-%$cosG6#,$%#PiG#\"\"\"\"#=!\"\",$*$-%%sqrtG6#\"\"$\"\"\"#
F/\"\"#,$-F(6#,$F+#\"\"&F.F/,$-F(6#,$F+#\"\"(F.F/\"\"!F@F:,$F1#F-F8F'
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11 "" 1 "" {XPPMATH 20 "6#>%#xdG7+$!+Iv2[)*!#5$!+SSDg')F($!+&4wyU'F($!
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "datos := [seq([xd[i], evalf(
g(xd[i]))],i = 1 .. nops(xd))];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&
datosG7+7$$!+Iv2[)*!#5$\"+x()*4'R!#67$$!+SSDg')F)$\"+P6Hj]F,7$$!+&4wyU
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "yd:=[seq(datos[i][2],i = 1 .
. nops(datos))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG7+$\"+x()*4'
R!#6$\"+P6Hj]F($\"+'\\%fE))F($\"+l88[D!#5$\"\"\"\"\"!F-F+F)F&" }}}
{PARA 0 "" 0 "" {TEXT -1 40 "El polinomio de interpolaci\363n nos qued
a:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "polint:=interp(xd,yd,x
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'polintG,4*$)%\"xG\"\"(\"\"\"$
!\"%!\")*$)F(\"\"'F*$!++D/NSF-*$)F(\"\")F*$\"+\"HF?w\"F-*$)F(\"\"&F*$F
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44 "interpola:=plot(polint,x=-1..1, color=blue):" }}}{EXCHG {PARA 0 ">
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