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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "Polinomios y Bases de Lag
range en Dos Variables." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 60 "Como \+
hemos visto la base de  Lagrange es aquella que cumple " }{XPPEDIT 18 
0 "L[i](P[k]) = delta[ik];" "6#/-&%\"LG6#%\"iG6#&%\"PG6#%\"kG&%&deltaG
6#%#ikG" }{TEXT -1 4 " .  " }{TEXT 257 125 "Si nos centramos en el cas
o de la interpolaci\363n polinomial cl\341sica  bidimensional y en una
 tabla de tres puntos en el eje X: " }{TEXT 279 11 "x0, x1, x2 " }
{TEXT 280 26 "y dos puntos en el eje Y: " }{TEXT 281 7 "y0, y1 " }
{TEXT 282 234 " tendremos  una tabla con seis valores y, por lo tanto,
 seis polinomios de Lagrange que ahora dependen de las variables x e y
. Estos polinomios vendr\341n dados por el producto de los polinomios \+
de Lagrange en una sola variable (x o y). " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }{TEXT 259 26 "Los puntos xd los tomamos:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 21 "xd:=[x0=0,x1=1,x2=2];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#xdG7%/%#x0G\"\"!/%#x1G\"\"\"/%#x2G\"\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 79 "Con lo que los polinomios
 de Lagrange en una dimensi\363n basados en estos xd son:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "l0x:= x-> eval((x-x1)*(x-x2)/((x0-x
1)*(x0-x2)),xd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$l0xGR6#%\"xG6\"
6$%)operatorG%&arrowGF(-%%evalG6$*&*&,&9$\"\"\"%#x1G!\"\"F3,&F2F3%#x2G
F5F3\"\"\"*&,&%#x0GF3F4F5\"\"\",&F;F3F7F5\"\"\"!\"\"%#xdGF(F(F(" }}}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "l0x(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"!\"\"
F'F',&F&F'!\"#F'F'#F'\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "l1x:= x-> eval((x-x0)*(x-x2)/((x1-x0)*(x1-x2)),xd);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$l1xGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%eval
G6$*&*&,&9$\"\"\"%#x0G!\"\"F3,&F2F3%#x2GF5F3\"\"\"*&,&%#x1GF3F4F5\"\"
\",&F;F3F7F5\"\"\"!\"\"%#xdGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "l1x(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"
\"\",&F%F&!\"#F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "
l2x:= x-> eval((x-x0)*(x-x1)/((x2-x0)*(x2-x1)),xd);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$l2xGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%evalG6$*&
*&,&9$\"\"\"%#x0G!\"\"F3,&F2F3%#x1GF5F3\"\"\"*&,&%#x2GF3F4F5\"\"\",&F;
F3F7F5\"\"\"!\"\"%#xdGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "l2x(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\",&F%F&!
\"\"F&F&#F&\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
64 "En la variable y ocurrir\341 lo mismo. Si tomamos la lista yd como
:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "yd:=[y0
=4,y1=6];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ydG7$/%#y0G\"\"%/%#y1G
\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 
262 154 "Los polinomios en y tendr\341n la misma forma que en x pero c
on las variables cambiadas. Adem\341s, al tener un punto menos, ser
\341n polinomios de un grado menor." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "l0y:= y-> eval((y-y1)/(y0-y1),yd);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$l0yGR6#%\"yG6\"6$%)operatorG%&arrowGF(-%%evalG6$*&
,&9$\"\"\"%#y1G!\"\"\"\"\",&%#y0GF2F3F4!\"\"%#ydGF(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "l0y(y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&%\"yG#!\"\"\"\"#\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "l1y:= y-> eval((y-y0)/(y1-y0),yd);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$l1yGR6#%\"yG6\"6$%)operatorG%&arrowGF(-%%eval
G6$*&,&9$\"\"\"%#y0G!\"\"\"\"\",&%#y1GF2F3F4!\"\"%#ydGF(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 121 "Los polinomios de
 Lagrange soluci\363n de nuestro caso ser\341n la multiplicaci\363n de
 los polinomios en x y los polinomios en y:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 30 "l00xy:=(x,y)-> l0x(x)*l0y(y) ;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&l00xyGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&-%$
l0xG6#9$\"\"\"-%$l0yG6#9%F2F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 264 18 "Este polinomio en " }{TEXT 283 24 "\{1, x, x^2, y,
 yx, yx^2\}" }{TEXT 284 32 " tiene que valer uno en el punto" }{TEXT 
285 8 " (x0,y0)" }{TEXT 286 53 " y cero en todas las dem\341s combinac
iones de valores: " }{TEXT 287 43 "(x0,y1), (x1,y0), (x1,y1), (x2,y0),
 (x2,y1)" }{TEXT 288 37 ". Podemos verlo dibujando su gr\341fica:" }}}
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 106 "Del mismo modo te
ndr\355amos los otros cinco polinomios que formar\355an la base de Lag
range para este problema:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 60 "l10xy:=(x,y)-> l1x(x)*
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}{TEXT 266 83 "Se puede comprobar c\363mo estos polinomios cumplen la \+
regla de valer uno all\355 donde  " }{TEXT 267 4 "i=k " }{TEXT 268 5 "
  en " }{XPPEDIT 18 0 "L[i](P[k]) = delta[ik];" "6#/-&%\"LG6#%\"iG6#&%
\"PG6#%\"kG&%&deltaG6#%#ikG" }{TEXT -1 2 "  " }{TEXT 269 19 "y cero en
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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 78 "Almacenamos e
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 84 "Finalmente podemos
 calcular la interpolaci\363n de una funci\363n cualquiera, por ejempl
o," }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x,y) = y^x;" "6#/-%\"fG6$%\"xG%
\"yG)F(F'" }{TEXT -1 1 " " }{TEXT 273 72 "utilizando los puntos base d
e la tabla. Los valores correspondientes de " }{XPPEDIT 18 0 "f(x,y);
" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 " " }{TEXT 274 4 "son:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "u
nassign('f00','f01','f10','f11','f20','f21');" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 44 "fd:=[f00=1,f01=1,f10=4,f11=6,f20=16,f21=36];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG7(/%$f00G\"\"\"/%$f01GF(/%$f10G
\"\"%/%$f11G\"\"'/%$f20G\"#;/%$f21G\"#O" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 275 140 "Y la soluci\363n del problema de interpo
laci\363n no ser\341 m\341s que la multiplicaci\363n de estos valores \+
por el correspondiente polinomio de Lagrange:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 52 "sol:=(x,y)-> eval(f00*l00xy(x,y) + f01*l01xy(x,y
) + " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 70 "f10*l10xy(x,y)
 + f11*l11xy(x,y) + f20*l20xy(x,y) + f21*l21xy(x,y),fd);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$solGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%
%evalG6$,.*&%$f00G\"\"\"-%&l00xyG6$9$9%F3F3*&%$f01GF3-%&l01xyGF6F3F3*&
%$f10GF3-%&l10xyGF6F3F3*&%$f11GF3-%&l11xyGF6F3F3*&%$f20GF3-%&l20xyGF6F
3F3*&%$f21GF3-%&l21xyGF6F3F3%#fdGF)F)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 276 40 "Podemos dibujar la funci\363n interpolant
e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot3d(sol(x,y),x=0..
2,y=4..6,axes=boxed);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%%GRIDG6%;\"\"!
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