{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Mejor aproximaci\363n: Al gunos ejemplos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 263 66 "Aproximaci\363n \+ minimax, caso discreto (Interpolaci\363n de Tchebychev):" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Se trata de obtene r un polinomio " }{XPPEDIT 18 0 "p[m](x);" "6#-&%\"pG6#%\"mG6#%\"xG" } {TEXT -1 24 " de grado no superior a " }{XPPEDIT 18 0 "m <= n;" "6#1% \"mG%\"nG" }{TEXT -1 14 " que minimice " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " \+ " }{XPPEDIT 18 0 "max[i = 0 .. n];" "6#&%$maxG6#/%\"iG;\"\"!%\"nG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(y[i]-p[m](x[i]));" "6#-%$absG6#,&&% \"yG6#%\"iG\"\"\"-&%\"pG6#%\"mG6#&%\"xG6#F*!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Si " } {XPPEDIT 18 0 "m = n;" "6#/%\"mG%\"nG" }{TEXT -1 96 " el polinomio bus cado interpola, en el sentido cl\341sico, la tabla de valores dada por los puntos " }{XPPEDIT 18 0 "\{x[i], y[i]\};" "6#<$&%\"xG6#%\"iG&%\"y G6#F'" }{TEXT -1 14 ". Supondremos " }{XPPEDIT 18 0 "m = n-1.;" "6#/% \"mG,&%\"nG\"\"\"$\"\"\"\"\"!!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 8 "Teorema:" }{TEXT -1 2 " \+ " }{TEXT 257 4 "Sea " }{XPPEDIT 18 0 "r := inf[p[m]];" "6#>%\"rG&%$inf G6#&%\"pG6#%\"mG" }{XPPEDIT 18 0 "max[i = 0 .. m+1](abs(y[i]-p[m](x[i] )));" "6#-&%$maxG6#/%\"iG;\"\"!,&%\"mG\"\"\"\"\"\"F-6#-%$absG6#,&&%\"y G6#F(F--&%\"pG6#F,6#&%\"xG6#F(!\"\"" }{TEXT -1 2 ". " }{TEXT 264 71 "E ntonces existe un \372nico polinomio soluci\363n. Adem\341s, dicho pol inomio es" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p[m]^s;" "6#)&%\"pG6#%\"mG% \"sG" }{TEXT -1 1 " " }{TEXT 265 19 "si y s\363lo si existe" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 1 " " }{TEXT 266 8 "t al que " }}{PARA 0 "" 0 "" {TEXT -1 47 " \+ " }{XPPEDIT 18 0 "(-1)^i*h+p[m]^s*x[i] = y[i],i = 0 . . m+1;" "6$/,&*&),$\"\"\"!\"\"%\"iG\"\"\"%\"hGF+F+*&)&%\"pG6#%\"mG%\"s GF+&%\"xG6#F*F+F+&%\"yG6#F*/F*;\"\"!,&F2F+\"\"\"F+" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 267 22 "En este caso se tiene " }{XPPEDIT 18 0 " r = abs(h);" "6#/%\"rG-%$absG6#%\"hG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "La demostraci\363n de e ste resultado proporciona el valor de " }{XPPEDIT 18 0 "h;" "6#%\"hG " }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " " } {XPPEDIT 18 0 "h = y[0,1 .. m+1]/s[0,1 .. m+1];" "6#/%\"hG*&&%\"yG6$\" \"!;\"\"\",&%\"mG\"\"\"\"\"\"F.F.&%\"sG6$F);\"\"\",&F-F.\"\"\"F.!\"\" " }{TEXT -1 4 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 49 "donde el numerador indica la diferencia dividida " } {TEXT -1 25 " asociada a los puntos " }{XPPEDIT 18 0 "\{x[i], y[i]\} ,i = 0 .. m+1;" "6$<$&%\"xG6#%\"iG&%\"yG6#F'/F';\"\"!,&%\"mG\"\"\"\"\" \"F0" }{TEXT -1 46 " y el denominador la asociada a los puntos " } {XPPEDIT 18 0 "\{x[i], s(x[i])\},i = 0 .. m+1;" "6$<$&%\"xG6#%\"iG-%\" sG6#&F%6#F'/F';\"\"!,&%\"mG\"\"\"\"\"\"F2" }{TEXT -1 10 " , siendo" } }{PARA 0 "" 0 "" {TEXT -1 48 " \+ " }{XPPEDIT 18 0 "s(x[i]) = (-1)^i,i = 0 .. m+1;" "6$/-%\"sG6# &%\"xG6#%\"iG),$\"\"\"!\"\"F*/F*;\"\"!,&%\"mG\"\"\"\"\"\"F4" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Con vistas a la implementaci\363n unicamente se necesita el valor \+ de " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 20 " , pues la relaci\363 n " }}{PARA 0 "" 0 "" {TEXT -1 30 " " }} {PARA 0 "" 0 "" {TEXT -1 50 " \+ " }{XPPEDIT 18 0 "p[m]^s*x[i] = y[i]-(-1)^i;" "6#/*&)&%\"pG6# %\"mG%\"sG\"\"\"&%\"xG6#%\"iGF+,&&%\"yG6#F/F+),$\"\"\"!\"\"F/F7" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "i = 0 .. m+1;" "6#/%\"iG;\"\"!,&%\"mG \"\"\"\"\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "indica que el polinomio buscado es el que iterpola los va lores " }}{PARA 0 "" 0 "" {TEXT -1 53 " \+ " }{XPPEDIT 18 0 "y[i]-(-1)^i*h,i = 0 .. m+1;" " 6$,&&%\"yG6#%\"iG\"\"\"*&),$\"\"\"!\"\"F'F(%\"hGF(F-/F';\"\"!,&%\"mGF( \"\"\"F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 12 "con soporte " }{XPPEDIT 18 0 "\{x[i], 0 .. m+1\}; " "6#<$&%\"xG6#%\"iG;\"\"!,&%\"mG\"\"\"\"\"\"F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Como apli caci\363n se resolver\341 el siguiente problema:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 258 4 "Sea " }{XPPEDIT 18 0 "f (x) = abs(x);" "6#/-%\"fG6#%\"xG-%$absG6#F'" }{TEXT -1 6 " para " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 4 " en " }{XPPEDIT 18 0 "[-1, 1 ];" "6#7$,$\"\"\"!\"\"\"\"\"" }{TEXT -1 23 ". Obtener el polinomio " } {XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 31 " mejor aproximac i\363n minimax de " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 58 " de acuerdo con el siguiente conjunto de abscisas y grado:" }} {PARA 0 "" 0 "" {TEXT 259 5 "1\272) " }{XPPEDIT 18 0 "\{-1, -1/2, 1/2 , 1\},deg(p) <= 2.;" "6$<&,$\"\"\"!\"\",$*&\"\"\"\"\"\"\"\"#F&F&*&\"\" \"F*\"\"#F&\"\"\"1-%$degG6#%\"pG$\"\"#\"\"!" }{TEXT 260 3 " " }} {PARA 0 "" 0 "" {TEXT 261 4 "2\272) " }{XPPEDIT 18 0 "\{-1, -1/2, 0, 1 /2, 1\},deg(p) <= 3;" "6$<',$\"\"\"!\"\",$*&\"\"\"\"\"\"\"\"#F&F&\"\"! *&\"\"\"F*\"\"#F&\"\"\"1-%$degG6#%\"pG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 9 "Soluci\363n:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "1\272) T abla de diferencias divididas asociadas a " }{XPPEDIT 18 0 "f(x);" "6# -%\"fG6#%\"xG" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 41 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:=4;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x:=[-1,-1/2,1/2,1];" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "for i from 1 to N do y[i]:=abs(x[i]) od:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7&!\"\"#F&\"\"##\"\"\"F(F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 88 "for j from 1 to N-1 do for m from 1 to N-j do \+ y[m]:=(y[m]-y[m+1])/(x[m]-x[m+j]); od; od;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "y[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Por tanto, dado que " }{XPPEDIT 18 0 "h = 0;" "6#/%\"hG\"\"!" }{TEXT -1 16 " , el polinomio " } {XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 31 " buscado es el q ue interpola a " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " en los valores de " }{XPPEDIT 18 0 "x[i];" "6#&%\"xG6#%\"iG" } {TEXT -1 3 " : " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "p:=proc(x::uneva l) interp([-1,-1/2,1/2,1],[1,1/2,1/2,1], x) end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\" \"#\"\"\"#F'\"\"$#\"\"\"F*F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Al comparar al gr\341fica de la funcio n " }{XPPEDIT 18 0 "f(x) = abs(x);" "6#/-%\"fG6#%\"xG-%$absG6#F'" } {TEXT -1 65 " con la del polinomio obtenido se observa lo particular d el caso:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot([abs(t),p(t)],t=-1.1..1.1);" }} {PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7U7$$!1+++++++6!#:$\"1+++++++ 6F*7$$!1LL$3EY?0\"F*$\"1LL$3EY?0\"F*7$$!1n;zo>K55F*$\"1n;zo>K55F*7$$!1 ML$eQ')Rj*!#;$\"1ML$eQ')Rj*F:7$$!1NL$3qU;;*F:$\"1NL$3qU;;*F:7$$!1nm\"H )Ra\"p)F:$\"1nm\"H)Ra\"p)F:7$$!1MLe%H-dD)F:$\"1MLe%H-dD)F:7$$!1++DrMT/ yF:$\"1++DrMT/yF:7$$!1MLeC**oPtF:$\"1MLeC**oPtF:7$$!1++DmJYsoF:$\"1++D mJYsoF:7$$!1nmmh-#RR'F:$\"1nmmh-#RR'F:7$$!1ML$3T>C(fF:$\"1ML$3T>C(fF:7 $$!1+++&G4z\\&F:$\"1+++&G4z\\&F:7$$!1+++v2X@]F:$\"1+++v2X@]F:7$$!1+++: ZHiXF:$\"1+++:ZHiXF:7$$!1MLeCqLXTF:$\"1MLeCqLXTF:7$$!1nmm,S`\\OF:$\"1n mm,S`\\OF:7$$!1mmm1Q_HKF:$\"1mmm1Q_HKF:7$$!1++D,@&4u#F:$\"1++D,@&4u#F: 7$$!1nmmcL[3BF:$\"1nmmcL[3BF:7$$!1++D,m*R$=F:$\"1++D,m*R$=F:7$$!1***\\ P+t@Q\"F:$\"1***\\P+t@Q\"F:7$$!1jmm\"zCu5*!#<$\"1jmm\"zCu5*Fdr7$$!1mm; 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od; od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"%\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Tabl a de diferencias correspondiente a " }{XPPEDIT 18 0 "s(x[i]) = (-1)^i, i = 0 .. 5;" "6$/-%\"sG6#&%\"xG6#%\"iG),$\"\"\"!\"\"F*/F*;\"\"!\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for i from 1 to N do s[i] :=(-1)^(i+1) od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "for j from 1 to N-1 do for m from 1 to N-j do s[m]:=(s[m]-s[m+1])/(x[m]-x[m+j]); od; \+ od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#K\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Coci ente entre ambas diferencias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "h:=y[1]/s[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG#!\"\"\" \")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 " El polinomio buscado es la soluci\363n al problema de interpolaci\363n cl\341sico en los punt os " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ " }{XPPEDIT 18 0 "\{x[i], y[i]-1/(8(-1)^i) \};" "6#<$&%\"xG6#%\"iG,&&%\"yG6#F'\"\"\"*&\"\"\"F,)-\"\")6#,$\"\"\"! \"\"F'F5F5" }{TEXT -1 9 "\000\000\000\000\000\000\000\000\000" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "p:=proc(x::uneval) interp([- 1,-1/2,0,1/2,1],[1+1/8,1/2-1/8,1/8,1/2-1/8,1+1/8],x) end:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "p(x);" }{TEXT -1 38 " \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\" \"\"\"\"#F)\"\")F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "Comparamos las gr\341ficas de " }{XPPEDIT 18 0 "f(x) = abs(x);" "6#/-%\"fG6#%\"xG-%$absG6#F'" }{TEXT -1 112 " , del p olinomio soluci\363n y del polinomio de interpolaci\363n, en sentido c l\341sico, con soporte en los puntos dados:" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "q:=proc(x::uneval) interp([-1,-1/2, 0,1/2,1],[1,1/2,0,1/2,1],x) end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"%\"\"\"#!\"%\" \"$*$)F&\"\"#F(#\"\"(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " plot([abs(t),p(t),q(t)],t=-1.1..1.1, color=[red, green, blue]);" }} {PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7U7$$!1+++++++6!#:$\"1+++++++ 6F*7$$!1LL$3EY?0\"F*$\"1LL$3EY?0\"F*7$$!1n;zo>K55F*$\"1n;zo>K55F*7$$!1 ML$eQ')Rj*!#;$\"1ML$eQ')Rj*F:7$$!1NL$3qU;;*F:$\"1NL$3qU;;*F:7$$!1nm\"H )Ra\"p)F:$\"1nm\"H)Ra\"p)F:7$$!1MLe%H-dD)F:$\"1MLe%H-dD)F:7$$!1++DrMT/ yF:$\"1++DrMT/yF:7$$!1MLeC**oPtF:$\"1MLeC**oPtF:7$$!1++DmJYsoF:$\"1++D mJYsoF:7$$!1nmmh-#RR'F:$\"1nmmh-#RR'F:7$$!1ML$3T>C(fF:$\"1ML$3T>C(fF:7 $$!1+++&G4z\\&F:$\"1+++&G4z\\&F:7$$!1+++v2X@]F:$\"1+++v2X@]F:7$$!1+++: ZHiXF:$\"1+++:ZHiXF:7$$!1MLeCqLXTF:$\"1MLeCqLXTF:7$$!1nmm,S`\\OF:$\"1n mm,S`\\OF:7$$!1mmm1Q_HKF:$\"1mmm1Q_HKF:7$$!1++D,@&4u#F:$\"1++D,@&4u#F: 7$$!1nmmcL[3BF:$\"1nmmcL[3BF:7$$!1++D,m*R$=F:$\"1++D,m*R$=F:7$$!1***\\ P+t@Q\"F:$\"1***\\P+t@Q\"F:7$$!1jmm\"zCu5*!#<$\"1jmm\"zCu5*Fdr7$$!1mm; 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b:=4*Pi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "with(numapprox): #(llamada a minimax)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "r:= proc(x::uneval) minimax(x*sin(x), x=a..b, [5,0]) \+ end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 " El primer argumento de minimax() es la funci\363n que se quiere aproxi mar, el segundo es el rango, y el tercero es el par de grados de numer ador y denominador, respectivamente, de " }{XPPEDIT 18 0 "r(x);" "6#-% \"rG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "En este ca so, " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG" }{TEXT -1 44 " es un po linomio de grado no superior a 5. 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