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1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 18 "Aula Pr\341tica sobre" }} {PARA 0 "" 0 "" {TEXT 326 13 "Interpola\347\343o " }{TEXT 319 0 "" } {TEXT 318 10 "Polinomial" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 30 "C\341lculo Num\351rico, Turma A, UnB" }} {PARA 257 "" 0 "" {TEXT -1 27 "Prof. Mauricio Ayala-Rinc\363n" }} {PARA 257 "" 0 "" {TEXT -1 18 "31 de maio de 2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 25 "Nome: \+ " }}{PARA 256 "" 0 "" {TEXT -1 10 "Matr\355cula:" }}{PARA 256 "" 0 " " {TEXT -1 6 "Nome: " }}{PARA 256 "" 0 "" {TEXT -1 10 "Matr\355cula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Aspetos t e\363ricos devem ser consultados na bibliografia da disciplina." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 48 "I. Interpola\347ao via solu\347\343o de sistem as lineares" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 333 22 "Defini\347\343o do problema:" }}{PARA 0 "" 0 "" {TEXT -1 45 "D ada uma tabela com os valores de uma fun\347\343o " }{TEXT 327 2 "f " }{TEXT -1 3 "em " }{TEXT 328 3 "n+1" }{TEXT -1 21 " pontos diferentes: " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 7 ", ..., " } {XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 3 " e " }{TEXT 329 2 "f(" }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT 413 11 "), ... , f(" }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 414 1 ")" }{TEXT -1 35 ", deseja-se computar um polin\364mio " } {XPPEDIT 18 0 "P[n];" "6#&%\"PG6#%\"nG" }{TEXT -1 28 " de grau menor o u igual que " }{TEXT 330 1 "n" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "que ajuste a fun\347\343o " }{TEXT 331 2 "f " }{TEXT -1 32 "em ditos pontos, denominados de " }{TEXT 332 19 "n\363s de interpo la\347\343o" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "O problema pode ser observado como o de solucio nar um sistema linear de " }{TEXT 334 3 "n+1" }{TEXT -1 19 " equa\347 \365es da forma:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " " }{XPPEDIT 18 0 "a[0]+a[1]*x[j]+a[2]*x[j]^2; " "6#,(&%\"aG6#\"\"!\"\"\"*&&F%6#F(F(&%\"xG6#%\"jGF(F(*&&F%6#\"\"#F(*$ &F-6#F/F3F(F(" }{TEXT -1 10 "+ ... + " }{XPPEDIT 18 0 "a[n]*x[j]^n; " "6#*&&%\"aG6#%\"nG\"\"\")&%\"xG6#%\"jGF'F(" }{TEXT -1 7 " = " } {TEXT 415 3 "f( " }{XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" }{TEXT 416 12 " ) , " }{TEXT -1 5 "para " }{TEXT 335 9 "j = 0..n." }{TEXT -1 11 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Para " }{TEXT 336 5 "n = 4" }{TEXT -1 47 ", p. ex., teri amos o seguinte sistema linear:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "matrix([[1, x[0], x[0]^2, x[0]^3], [1, \+ x[1], x[1]^2, x[1]^3], [1, x[2], x[2]^2, x[2]^3], [1, x[3], x[3]^2, x[ 3]^3]])*matrix([[a[0]], [a[1]], [a[2]], [a[3]]]) = matrix([[f(x[0])], \+ [f(x[1])], [f(x[2])], [f(x[3])]]);" "6#/*&-%'matrixG6#7&7&\"\"\"&%\"xG 6#\"\"!*$&F,6#F.\"\"#*$&F,6#F.\"\"$7&F*&F,6#F**$&F,6#F*F2*$&F,6#F*F67& F*&F,6#F2*$&F,6#F2F2*$&F,6#F2F67&F*&F,6#F6*$&F,6#F6F2*$&F,6#F6F6F*-F&6 #7&7#&%\"aG6#F.7#&FW6#F*7#&FW6#F27#&FW6#F6F*-F&6#7&7#-%\"fG6#&F,6#F.7# -Fao6#&F,6#F*7#-Fao6#&F,6#F27#-Fao6#&F,6#F6" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "onde a primeira m atriz \351 denominada matriz de Vandermonde dos n\363s de interpola \347\343o " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 7 ", .. ., " }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 25 " e o vetor de inc\363gnitas " }}{PARA 0 "" 0 "" {TEXT -1 50 "corresponde aos coe ficientes do polin\364mio buscado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "Desta forma, aplicando algum dos m\351to dos diretos ou iterativos conhecidos para solucionar o problema linear correspondente, " }}{PARA 0 "" 0 "" {TEXT -1 30 "podemos obter os coe ficientes " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT 337 2 ", \+ " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT 417 8 ", ... , " } {XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 27 " que descrevem o polin\364mio " }{XPPEDIT 18 0 "P[n];" "6#&%\"PG6#%\"nG" }{TEXT -1 10 " buscado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "Implementar-se-\341 um m\351todo para interpola\347\343o polin omial baseado no m\351todo de elimina\347\343o de Gauss para solu\347 \343o de sistemas lineares." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Embaixo est\341 incluido o procedimento " } {TEXT 338 9 "elimGauss" }{TEXT -1 85 ", que foi implementado para solu cionar sistemas lineares. O m\351todo de interpola\347\343o," }} {PARA 0 "" 0 "" {TEXT -1 125 "tamb\351m pode-se basear em outros m\351 todos de solu\347\343o de sistemas lineares como fatora\347\343o LU, G auss-Jacobi e Gauss Seidel. Todos" }}{PARA 0 "" 0 "" {TEXT -1 73 "este s, estudados nas aulas pr\341ticas sobre solu\347\343o de sistemas lin eares. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 698 "elimGauss := proc (Ain,bin,n) \n local x,A,b,i,j,mult,k;\n x := vector(n); A := Ain; b := bin; \n print(evalm(A),convert(b,Vector[column]));\n for i from \+ 1 to n-1 do \n for j from i+1 to n do \n mult := A[j,i]/A[i,i ]; A[j,i] := 0; \n for k from i+1 to n do A[j,k]:= A[j,k]-mult * \+ A[i,k]; end do; \n b[j]:=b[j]-b[i]*mult; \n #print(evalm(A) , convert(b,Vector[column])); \n end do; \n end do; \n # print(ev alm(A),convert(b,Vector[column])); \n for i from n by -1 to 1 do \n \+ x[i]:= b[i]; \n for k from i+1 to n do x[i] := x[i] - x[k]*A[i,k ]; end do; \n x[i] := x[i]/A[i,i]; \n # print(convert(x,Vector[co lumn])); \n end do; \n print(convert(x,Vector[column])); \nend proc ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*elimGaussGf*6%%$AinG%$binG%\"n G6)%\"xG%\"AG%\"bG%\"iG%\"jG%%multG%\"kG6\"F2C)>8$-%'vectorG6#9&>8%9$> 8&9%-%&printG6$-%&evalmG6#F;-%(convertG6$F>&%'VectorG6#%'columnG?(8'\" \"\"FO,&F9FOFO!\"\"%%trueG?(8(,&FNFOFOFOFOF9FRC&>8)*&&F;6$FTFNFO&F;6$F NFNFQ>FZ\"\"!?(8*FUFOF9FR>&F;6$FTF[o,&F]oFO*&FXFO&F;6$FNF[oFOFQ>&F>6#F T,&FdoFO*&&F>6#FNFOFXFOFQ?(FNF9FQFOFRC%>&F5FioFho?(F[oFUFOF9FR>F]p,&F] pFO*&&F56#F[oFOFaoFOFQ>F]p*&F]pFOFfnFQ-FA6#-FG6$F5FIF2F2F2" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 341 0 "" }}{PARA 256 "" 0 "" {TEXT -1 32 "1. Dados os n\363s de interpola\347\343o:" }{TEXT 258 26 " x0 = -1, x1=0, x2=1, x3=2" }{TEXT -1 28 " e os valores de uma fun \347\343o " }{TEXT 339 2 "f " }{TEXT -1 15 "nesses pontos: " }{TEXT 340 43 "f(x0) = 4, f(x1) = 1, f(x2) = -1, f(x3) = 2" }{TEXT -1 68 ". U tilize o m\351todo de elimina\347\343o de Gauss para computar o polin \364mio " }{TEXT 342 3 "P3 " }{TEXT -1 16 "de interpola\347\343o." }} {PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 126 "De ver\341 estabelecer o sistema linear correspondente a este problema de interpola\347\343o e fornecer-lo como entrada do procedimento " } {TEXT 343 9 "elimGauss" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 37 "Obtida a resposta, defina uma fun \347\343o " }{TEXT 399 2 "P3" }{TEXT -1 74 " para o polin\364mio e ver ifique que interpola mesmo a tabela de pontos dada." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "elimGauss ([[1,-1,1,-1],[1,0,0,0],[1,1,1,1],[1,2,4,8]],[4,1,-1,2],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7&7&\"\"\"!\"\"F(F)7&F(\"\"!F+F+ 7&F(F(F(F(7&F(\"\"#\"\"%\"\")-%'RTABLEG6$\")_Cz7-%'MATRIXG6#7&7#F/7#F( 7#F)7#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")#\\#z7-%'MA TRIXG6#7&7#\"\"\"7##!#>\"\"'7##F,\"\"#7##F3\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "P3 := x -> 1 - 19/6 * x + 1/2 * x^2 + 2/3 * x ^3;\nP3(-1); P3(0); P3(1); P3(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#P3Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,*\"\"\"F-*&#\"#>\"\"'F-9$F-! \"\"*&#F-\"\"#F-)F2F6F-F-*&#F6\"\"$F-)F2F:F-F-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 220 "A utiliza\347\343o de m\351todos de pivotamento p ara evitar piv\364s nulos \351 desnecess\341ria para as matrizes de Va ndermonde que resultam do sistema linear correspondente a um problema \+ de interpola\347\343o; i.e., matrizes da forma, p. ex.:" }{TEXT 344 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 1 " " }{XPPEDIT 18 0 "matrix([[1, x[0], x[0]^2, x[0]^3], [1, x[1], x[1]^2, x [1]^3], [1, x[2], x[2]^2, x[2]^3], [1, x[3], x[3]^2, x[3]^3]]);" "6#-% 'matrixG6#7&7&\"\"\"&%\"xG6#\"\"!*$&F*6#F,\"\"#*$&F*6#F,\"\"$7&F(&F*6# F(*$&F*6#F(F0*$&F*6#F(F47&F(&F*6#F0*$&F*6#F0F0*$&F*6#F0F47&F(&F*6#F4*$ &F*6#F4F0*$&F*6#F4F4" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 5 "onde " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6# \"\"!" }{TEXT 260 2 ", " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" } {TEXT 418 2 ", " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT 419 3 " e " }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 104 " s\343 o os n\363s de interpola\347\343o do problema, al\351m de serem n\343o singulares, n\343o precisam de mudan\347as de piv\364s" }{TEXT 257 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 354 50 "Na quest\343o 2 iremos a int erpolar pontos da fun\347\343o " }{TEXT 355 4 "ln, " }{TEXT 356 66 "pa ra obter um polin\364mio que ajuste esta fun\347\343o no intervalo [1, 5]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 88 " 2.1. Interpole agora os pontos x = 1 2 3 \+ 4 5" }}{PARA 256 "" 0 "" {TEXT -1 104 " \+ ln(x) = ln(1) ln(2) ln( 3) ln(4) ln(5)" }}{PARA 0 "" 0 "" {TEXT 261 4 " " }}{PARA 0 "" 0 "" {TEXT 347 4 " " }{TEXT -1 48 "Para isto aplique novamente elim ina\347\343o de Gauss." }{TEXT 358 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 357 25 "Defina ent\343o o polin\364mio " } {TEXT 345 2 "P4" }{TEXT 346 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "elimGauss([[1,1,1,1,1],[1,2,4,8,16 ],[1,3,9,27,81],[1,4,16,64,256],[1,5,25,125,625]],[evalf(ln(1)),evalf( ln(2)),evalf(ln(3)),evalf(ln(4)),evalf(ln(5))],5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$-%'matrixG6#7'7'\"\"\"F(F(F(F(7'F(\"\"#\"\"%\"\")\"#; 7'F(\"\"$\"\"*\"#F\"#\")7'F(F+F-\"#k\"$c#7'F(\"\"&\"#D\"$D\"\"$D'-%'RT ABLEG6$\")KDz7-%'MATRIXG6#7'7#$\"\"!FE7#$\"+1=ZJp!#57#$\"+*G7')4\"!\"* 7#$\"+hVH'Q\"FM7#$\"+7zV4;FM" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA BLEG6$\")sDz7-%'MATRIXG6#7'7#$!+1GQn7!\"*7#$\"+*4#=z;F.7#$!+JChQ[!#57# $\"++bD#p(!#67#$!+<\\gg[!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "P4 := x -> -1.267382806 + 1.679182099 * x - .4838612431 * x^2 + . 076922555 * x^3 - .004860604917 * x^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P4Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,,$!+1GQn7!\"*\"\"\"*&$ \"+*4#=z;F/F09$F0F0*&$\"+JChQ[!#5F0)F4\"\"#F0!\"\"*&$\")bD#p(F/F0)F4\" \"$F0F0*&$\"+<\\gg[!#7F0)F4\"\"%F0F;F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "Quando dispomos das derivadas da fun\347\343o se ndo interpolada, como \351 o caso desta quest\343o, podemos estimar o erro utilizando um corol\341rio:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 " " } {XPPEDIT 18 0 "abs(E[n](x)) <= M[n+1]*abs(product(x-x[i],i = 0 .. n))/ (n+1)!;" "6#1-%$absG6#-&%\"EG6#%\"nG6#%\"xG*(&%\"MG6#,&F+\"\"\"F3F3F3- F%6#-%(productG6$,&F-F3&F-6#%\"iG!\"\"/F<;\"\"!F+F3-%*factorialG6#,&F+ F3F3F3F=" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "onde " }{XPPEDIT 18 0 "M[n+1];" "6#&%\"MG6#,&%\"nG\"\"\"F(F(" } {TEXT -1 17 " \351 o m\341ximo de " }{TEXT 359 1 "|" }{XPPEDIT 18 0 "f^(n+1);" "6#)%\"fG,&%\"nG\"\"\"F'F'" }{TEXT 420 6 "(x)| " }{TEXT -1 5 "para " }{TEXT 360 1 "x" }{TEXT -1 30 " no intervalo de interpola \347\343o." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 361 31 "2.2 Estime o erro do polin\364mio " } {TEXT 362 2 "p4" }{TEXT 363 21 " no ponto x = 3/2 " }{TEXT -1 72 "( Verifique que computo o polin\364mio corretamente: p4(3/2) = 0.3926057 206)" }{TEXT 364 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Dever\341 utilizar o anterior " }{TEXT 365 9 "corol \341rio" }{TEXT -1 66 " e maximizar a quinta derivada de ln no interva lo de interpola\347\343o." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(ln(x),x$5); evalf(24 / 5! * pr oduct(abs(3/2-i), i=1..5)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\" \"\"F%*$)%\"xG\"\"&F%!\"\"\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ ++]il!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Observe \+ que existe uma pequena diferen\347a entre P4(1) e ln(1); P4(2) e ln(2) , etc. Teoricamente, o polin\364mio interpolador deveria passar " }} {PARA 0 "" 0 "" {TEXT -1 77 "exatamente pelos pontos de interpola\347 \343o (xj, f(xj)) ! Mas isto n\343o acontece." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 348 39 "2.3. Expl ique o motivo desta diferen\347a." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 130 "No pr\363prio c\341lculo de ln(x) comete mos erros de truncamento, que s\343o propagados ao m\351todo de solu \347\343o de sistemas lineares de Gauss." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 349 80 "2.4. Finalmente, gere utilizando \"plot\" simultaneament e os gr\341ficos das fun\347\365es " }{TEXT 350 3 "ln " }{TEXT 351 2 " e " }{TEXT 352 2 "p4" }{TEXT 353 25 " para observar o ajuste. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{ln(x),P4(x)\}, x=0.1..6);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$\"3/+++++++5!#=$!3YXS*H4&e -B!#<7$$\"3AL$eR-3:K\"F*$!3`'z$4m:\"Q-#F-7$$\"3nmm\"z/;Ik\"F*$!3sd)Ql[ ^g!=F-7$$\"37+](=2CX'>F*$!31_nhz]LF;F-7$$\"3GLL$e4KgG#F*$!3?Fza([ndZ\" F-7$$\"3I+v$4#p^XGF*$!3CE.RZ.%oD\"F-7$$\"30n;/Y<+0MF*$!35M^gS'Rt2\"F-7 $$\"31LL3Z,SjYF*$!3^cN@cESGwF*7$$\"3(RL$eH+9IfF*$!3BG[0lEPD_F*7$$\"3%p mT&=$f3>(F*$!3G[#4h7WxH$F*7$$\"3'GL3F[2(f$)F*$!3QAE;mlh\"z\"F*7$$\"3G+ ]Pa?)*p&*F*$!3oqlosiP&R%!#>7$$\"3]L3xu\\;#3\"F-$\"3K([Pj3kj*yF_o7$$\"3 0+voK.$p?\"F-$\"3W+I_5A!3)=F*7$$\"3'om;*QoEN8F-$\"3wlK<7:J\"*GF*7$$\"3 aL$eMq0$[9F-$\"38$oM&GR%Rq$F*7$$\"3C++v\"pgbd\"F-$\"3!p+%)o.7ha%F*7$$ \"3C++D,#QLq\"F-$\"3X@8?M(**eK&F*7$$\"3Q++D3bZE=F-$\"3?@]Z!e\")Q-'F*7$ $\"3XL3xzfHQ>F-$\"3c^NZhA4=mF*7$$\"3smm\"fLh72#F-$\"3)3ipsix:G(F*7$$\" 3'pmmOV+R=#F-$\"3/c)H'yY76yF*7$$\"3R+v$>[E\\J#F-$\"3[g]r(HzPR)F*7$$\"3 `mm;On!4V#F-$\"3'pUv?IVE)))F*7$$\"3Y+v$pXb\"eDF-$\"3IF0bTZ'GR*F*7$$\"3 g+D\"3LE$zEF-$\"3CXWCQRlb)*F*7$$\"3Gn;/Eav0GF-$\"3=zH\"o\"GnJ5F-7$$\"3 6nTN-o&=#HF-$\"31>US0$>A2\"F-7$$\"3oL$e9#y3ZIF-$\"3xd(e=K'=96F-7$$\"3S L3xaw;xJF-$\"3y9aID,*f:\"F-7$$\"35+D1KDS!H$F-$\"3S\"3ku!*45>\"F-7$$\"3 !QL3'=)H\"F-7$$\"3W+v$4-SAy$F-$\"3puhCKkJI8F-7$$ \"39+]7Ut-:RF-$\"3C`$eLJA[O\"F-7$$\"3Vmm;cEMMSF-$\"37C!fzPV[R\"F-7$$\" 37++D@sthTF-$\"3!>0k$)eKfU\"F-7$$\"3sK3xJc8m\" F-7$$\"3[++]r)*G!Q&F-$\"3g8K=_Au#o\"F-7$$\"3Xm;zOZ)4^&F-$\"3FawIDLu1G9Dv&F-$\"3%)o+mDqj\\$!3o#o*fh+(*y!*F*7$FH$!3a9gE%p'[([(F*7$FM$!3Ku+\"zN%o>eF*7$FR$!3Ib ,g'Q6KE%F*7$FW$!3SBP)o+9!GGF*7$Ffn$!3K__KS?:#f\"F*7$F[o$!3Kx4lJHz?SF_o 7$Fao$\"3!p[eebrYR(F_o7$Ffo$\"3S_h\\`Gm$z\"F*7$F[p$\"3C(*4]z5d(z#F*7$F `p$\"3e(*Rd3$[%>OF*7$Fep$\"3)\\LKUB9/[%F*7$Fjp$\"3k+C#H'=6#G&F*7$F_q$ \"3Kk4D&H;/+'F*7$Fdq$\"3OcRjXqh5mF*7$Fiq$\"3Ycr*\\C()))G(F*7$F^r$\"3hq Bj&[/q#yF*7$Fcr$\"3$**3>l5b^T)F*7$Fhr$\"3fP7k`2E0*)F*7$F]s$\"33n?xwJf8 %*F*7$Fbs$\"3A!p'ylhQR6\"F-7$Fft$\"3IO4/7@7b6F-7$F[u$\"3I&ejiBA(*=\"F-7$F`u$\" 3E=x_.A$fA\"F-7$Feu$\"39'4e2[4AE\"F-7$Fju$\"35,VVeYp'H\"F-7$F_v$\"3Qj) R\">$*=H8F-7$Fdv$\"3%)y(4+k@VO\"F-7$Fiv$\"3!z%[;TE1&R\"F-7$F^w$\"3(HIN .Cyk:F-7$F\\y$\"3yzNzD6!**e\"F-7$ Fay$\"3mv-\"4*eZ7;F-7$Ffy$\"3[N#y^k%pL;F-7$F[z$\"3'471Q!>i_;F-7$F`z$\" 3q@2P8x\"zm\"F-7$Fez$\"3E/0a=`m#o\"F-7$Fjz$\"3u%[C)3U.$p\"F-7$F_[l$\"3 %Gt=P#Q!3q\"F-7$Fd[l$\"3+imq&=5Zq\"F-7$Fi[l$\"3!3+oR%Hj/ " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "O nosso seguinte objetivo \351 o de definir um novo procedimento para interpola\347\343o polinomial baseado no m\351todo de elimina\347\343 o de Gauss. " }}{PARA 0 "" 0 "" {TEXT -1 81 "Isto poder-se-\341 esten der para os outros m\351todos de solu\347\343o de sistemas lineares. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 39 "3. \+ Para isto, definina um procedimento " }{TEXT 366 0 "" }{TEXT 288 4 "Va nd" }{TEXT -1 57 " que gera a matriz de Vandermonde associada com um v etor " }{TEXT 289 1 "X" }{TEXT -1 4 " de " }{TEXT 290 1 "n" }{TEXT -1 21 " n\363s de interpola\347\343o:" }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ " }{TEXT 368 32 "Vand := proc(X,n) ... end proc; " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 367 92 "Verifique o funcion amento do seu procedimento com os n\363s de interpola\347\343o das que st\365es 1 e 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 168 "Vand := proc(X,n) \n local A, i, j;\n A := \+ matrix(n,n); \n for i to n do \n for j to n do \n A[i,j] := X [i]^(j-1); \n end do; \n end do; \n evalm(A); \nend proc;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VandGf*6$%\"XG%\"nG6%%\"AG%\"iG%\"j G6\"F-C%>8$-%'matrixG6$9%F4?(8%\"\"\"F7F4%%trueG?(8&F7F7F4F8>&F06$F6F: )&9$6#F6,&F:F7F7!\"\"-%&evalmG6#F0F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Vand([-1,0,1,2],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'matrixG6#7&7&\"\"\"!\"\"F(F)7&F(\"\"!F+F+7&F(F(F(F(7&F(\"\"#\"\"% \"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Vand([1,2,3,4,5],5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"\"F(F(F(F(7' F(\"\"#\"\"%\"\")\"#;7'F(\"\"$\"\"*\"#F\"#\")7'F(F+F-\"#k\"$c#7'F(\"\" &\"#D\"$D\"\"$D'" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 115 "Agora, \351 poss\355vel especificar um proce dimento de interpola\347\343o baseado no m\351todo de elimina\347\343o de Gauss, denominado " }{TEXT 262 13 "interpGauss, " }{TEXT -1 56 "qu e tem como par\342metros: o vetor de n\363s de interpola\347\343o " } {TEXT 263 4 "Xin," }{TEXT -1 38 " o vetor com os valores desses ponto s" }{TEXT 264 3 " Fx" }{TEXT -1 22 " e o n\372mero de pontos " }{TEXT 265 1 "n" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 2 " " }}{PARA 258 "" 0 "" {TEXT -1 41 "Dito procedimento utiliza o procedimento " } {TEXT 266 6 "Vand, " }{TEXT -1 105 "implementado na quest\343o precede nte, para fornecer a matriz de Vandermonde como argumento do procedime nto " }{TEXT 267 11 "elimGauss, " }{TEXT -1 33 "permitindo assim o uso direto de " }{TEXT 369 9 "elimGauss" }{TEXT -1 59 " para computar os \+ coeficientes do polin\364mio de interpola\347\343o" }{TEXT 268 4 ". \+ " }{TEXT -1 2 "\n\n" }{TEXT 371 5 "Nota:" }{TEXT -1 51 " o procediment o embaixo somente funcionar\341, quando " }{TEXT 370 4 "Vand" }{TEXT -1 19 " seja implementado." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "interpGauss := proc(Xin,Fx,n ) \n elimGauss(Vand(Xin,n),Fx,n); \nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,interpGaussGf*6%%$XinG%#FxG%\"nG6\"F*F*-%*elimGaussG 6%-%%VandG6$9$9&9%F2F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 269 26 "4. Aplique o procedimento " }{TEXT 291 11 "interpGauss" }{TEXT 292 59 " para resolver diretamente os problema s do exerc\355cio 1 e 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "interpGauss([-1,0,1,2],[4,1,-1,2],4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7&7&\"\"\"!\"\"F(F)7 &F(\"\"!F+F+7&F(F(F(F(7&F(\"\"#\"\"%\"\")-%'RTABLEG6$\")7Ez7-%'MATRIXG 6#7&7#F/7#F(7#F)7#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\" )#p#z7-%'MATRIXG6#7&7#\"\"\"7##!#>\"\"'7##F,\"\"#7##F3\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "interpGauss([1,2,3,4,5],[eva lf(ln(1)),evalf(ln(2)),evalf(ln(3)),evalf(ln(4)),evalf(ln(5))],5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7'7'\"\"\"F(F(F(F(7'F(\"\" #\"\"%\"\")\"#;7'F(\"\"$\"\"*\"#F\"#\")7'F(F+F-\"#k\"$c#7'F(\"\"&\"#D \"$D\"\"$D'-%'RTABLEG6$\")sFz7-%'MATRIXG6#7'7#$\"\"!FE7#$\"+1=ZJp!#57# $\"+*G7')4\"!\"*7#$\"+hVH'Q\"FM7#$\"+7zV4;FM" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")_Gz7-%'MATRIXG6#7'7#$!+1GQn7!\"*7#$\"+*4 #=z;F.7#$!+JChQ[!#57#$\"++bD#p(!#67#$!+<\\gg[!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 230 "Como men cionado anteriormente, este mecanismo pode ser aplicado tamb\351m para definir m\351todos de interpola\347\343o baseados em outros m\351todo s de solu\347\343o de sistemas lineares como fatora\347\343o LU e os m \351todos de Gauss-Jacobi e Gauss-Seidel. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 47 "II. Forma de Lagrange do polin\364mio interpolador" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Na forma \+ de Lagrange o polin\364mio de interpola\347\343o para " }{TEXT 372 3 " n+1" }{TEXT -1 50 " n\363s de interpola\347\343o computa-se segundo a \+ f\363rmula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 373 5 " " }{XPPEDIT 18 0 "p[n](x) = f(x[0])*L[0](x)+f(x[1])*L[1](x );" "6#/-&%\"pG6#%\"nG6#%\"xG,&*&-%\"fG6#&F*6#\"\"!\"\"\"-&%\"LG6#F26# F*F3F3*&-F.6#&F*6#F3F3-&F66#F36#F*F3F3" }{TEXT 374 9 " + ... + " } {XPPEDIT 18 0 "f(x[n])*L[n](x);" "6#*&-%\"fG6#&%\"xG6#%\"nG\"\"\"-&%\" LG6#F*6#F(F+" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "onde " }{XPPEDIT 18 0 "L[k](x) = product(x -x[i],i = 0 .. k-1)*product(x-x[i],i = k+1 .. n)/product(x[k]-x[i],i = 0 .. k-1)/product(x[k]-x[i],i = k+1 .. n);" "6#/-&%\"LG6#%\"kG6#%\"xG **-%(productG6$,&F*\"\"\"&F*6#%\"iG!\"\"/F3;\"\"!,&F(F0F0F4F0-F-6$,&F* F0&F*6#F3F4/F3;,&F(F0F0F0%\"nGF0-F-6$,&&F*6#F(F0&F*6#F3F4/F3;F7,&F(F0F 0F4F4-F-6$,&&F*6#F(F0&F*6#F3F4/F3;,&F(F0F0F0FAF4" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Desta for ma, garante-se que " }{XPPEDIT 18 0 "p[n](x[k]) = f(x[k]);" "6#/-&%\"p G6#%\"nG6#&%\"xG6#%\"kG-%\"fG6#&F+6#F-" }{TEXT -1 12 ", para todo " } {TEXT 375 10 "k =0,...,n" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "O seguinte procedimento computa " }{XPPEDIT 18 0 "L[k](x);" "6#-&%\"LG6#%\"kG6#%\"xG" }{TEXT -1 9 " p ara um " }{TEXT 376 2 "n-" }{TEXT -1 6 "vetor " }{TEXT 296 1 "X" } {TEXT -1 37 " de n\363s de interpola\347\343o. O par\342metro " } {TEXT 297 1 "x" }{TEXT -1 55 " \351 o nome da vari\341vel em termos da qual ser\341 computado " }{XPPEDIT 18 0 "L[k](x);" "6#-&%\"LG6#%\"kG6 #%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "Lagr := proc (X, k, n, x) # Aqui k varia entre 1 e n+1 e n\343o entre 0 e n.\n local num, den, i; \+ # A vari\341vel num para numerador e den para denominador. \n num:= 1 ; den:= 1; \n for i to k-1 do \n num := num*(x-X[i]); \n den := den*(X[k]-X[i]); \n end do; \n for i from k+1 to n do \n num := \+ num*(x-X[i]); \n den := den*(X[k]-X[i]); \n end do; \n num/den; \+ \nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%LagrGf*6&%\"XG%\"kG% \"nG%\"xG6%%$numG%$denG%\"iG6\"F/C'>8$\"\"\">8%F3?(8&F3F3,&9%F3F3!\"\" %%trueGC$>F2*&F2F3,&9'F3&9$6#F7F:F3>F5*&F5F3,&&FB6#F9F3FAF:F3?(F7,&F9F 3F3F3F39&F;C$>F2F>>F5FE*&F2F3F5F:F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 11 "5. Compute " }{XPPEDIT 18 0 "L[k](z);" "6#-&%\"LG6#%\"kG6#%\"zG" }{TEXT 377 7 ", para " } {TEXT 378 6 "k=0..3" }{TEXT 379 49 ", para os n\363s de interpola\347 \343o do exerc\355cio 1: X " }{TEXT -1 2 "= " }{TEXT 380 35 "[-1, 0, 1 , 2] (Fx = [4, 1, -1, 2])." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Observe que em " }{TEXT 381 4 "Lang" }{TEXT -1 3 " , " }{TEXT 382 1 "k" }{TEXT -1 13 " varia entre " }{TEXT 383 1 "1" } {TEXT -1 3 " e " }{TEXT 384 1 "4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Lagr([-1, 0 , 1, 2],1,4,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"zG\"\"\",&F% F&F&!\"\"F&,&F%F&\"\"#F(F&#F(\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Lagr([-1, 0, 1, 2],2,4,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"zG\"\"\"F'F'F',&F&F'F'!\"\"F',&F&F'\"\"#F)F'#F 'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Lagr([-1, 0, 1, 2],3 ,4,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"zG\"\"\"F'F'F'F&F', &F&F'\"\"#!\"\"F'#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L agr([-1, 0, 1, 2],4,4,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\" zG\"\"\"F'F'F'F&F',&F&F'F'!\"\"F'#F'\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 66 "6. Verifique que o polin \364mio assim obtido (na forma de Lagrange: " }{XPPEDIT 18 0 "p3(x) = f(x[0])*L[0](x)+f(x[1])*L[1](x)+f(x[2])*L[2](x)+f(x[3])*L[3](x);" "6# /-%#p3G6#%\"xG,**&-%\"fG6#&F'6#\"\"!\"\"\"-&%\"LG6#F/6#F'F0F0*&-F+6#&F '6#F0F0-&F36#F06#F'F0F0*&-F+6#&F'6#\"\"#F0-&F36#FD6#F'F0F0*&-F+6#&F'6# \"\"$F0-&F36#FN6#F'F0F0" }{TEXT 385 43 " ) coincide com o computado no exerc\355cio 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Em Maple pode-se verificar isto da seguinte maneira:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "O polin \364mio obtido no exerc\355cio 1 \351:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "P3(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&#\"#>\"\"'F$%\"zGF$!\"\"*&#F$\"\"#F$)F)F -F$F$*&#F-\"\"$F$)F)F1F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "O polin\364mio obtido na Forma de Lagrang e \351:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "p3 := z -> 4 * (-1/6)*z*(z-1)*(z-2) + 1 * 1/2*(z+ 1)*(z-1)*(z-2)+\n (-1)* (-1/2)*(z+1)*z*(z-2) + 2 * 1/6*(z+1) *z*(z-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3Gf*6#%\"zG6\"6$%)ope ratorG%&arrowGF(,**(9$\"\"\",&F.F/F/!\"\"F/,&F.F/\"\"#F1F/#!\"#\"\"$** #F/F3F/,&F.F/F/F/F/F0F/F2F/F/**F8F/F9F/F.F/F2F/F/**#F/F6F/F9F/F.F/F0F/ F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Observe que tais polin\364mios coincidem:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simpli fy(p3(x)) = P3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"xG\"\"$ \"\"\"#\"\"#F(*&#F)F+F))F'F+F)F)*&#\"#>\"\"'F)F'F)!\"\"F)F)F$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 25 "7 . Usando o procedimento " }{TEXT 294 4 "Lagr" }{TEXT 295 29 " defina u m novo procedimento " }{TEXT 274 16 "interpLagrange, " }{TEXT 277 14 " com par\342metros" }{TEXT 278 10 " X, Fx, n " }{TEXT 279 1 "e" }{TEXT 280 3 " x," }{TEXT 275 51 " que compute o polin\364mio na forma de Lag range para " }{TEXT 386 1 "n" }{TEXT 387 11 " pontos. \n" }}{PARA 0 " " 0 "" {TEXT -1 43 "Os par\342metros deste novo procedimento s\343o: \+ " }{TEXT 281 1 "X" }{TEXT -1 3 " e " }{TEXT 282 2 "Fx" }{TEXT -1 41 ", os n\363s de interpola\347\343o e os valores de " }{TEXT 388 1 "f" } {TEXT -1 2 "; " }{TEXT 284 1 "n" }{TEXT -1 24 ", a quantidade de n\363 s e " }{TEXT 286 2 "x," }{TEXT -1 73 " uma vari\341vel em termos da qu al ser\341 especificado o polin\364mio resultante." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "interpLagr ange := proc(X,Fx,n,x)\n local pol, i; \n pol := 0; \n for i from 1 to n do \n pol := pol + Fx[i]*Lagr(X,i,n,x); \n end do; \n pol \+ \nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/interpLagrangeGf*6&% \"XG%#FxG%\"nG%\"xG6$%$polG%\"iG6\"F.C%>8$\"\"!?(8%\"\"\"F59&%%trueG>F 1,&F1F5*&&9%6#F4F5-%%LagrG6&9$F4F69'F5F5F1F.F.F." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 95 "8. Verifique o fu ncionamento da sua implementa\347\343o aplicando o seu procedimento ao exerc\355cio 1: " }{TEXT 283 18 "X = [-1, 0, 1, 2] " }{TEXT 287 1 "e " }{TEXT 389 19 " Fx = [4, 1, -1, 2]" }{TEXT 285 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "simplify (interpLagrange([-1, 0, 1, 2],[4, 1, -1, 2],4,z)); P3(z);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&# \"#>\"\"'F$%\"zGF$!\"\"*&#F$\"\"#F$)F)F-F$F$*&#F-\"\"$F$)F)F1F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&#\"#>\"\"'F$%\"zGF$!\"\"*& #F$\"\"#F$)F)F-F$F$*&#F-\"\"$F$)F)F1F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 298 45 "II. Forma de Newton do polin\364mio int erpolador" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Na forma de Newton o polin\364mio de interpola\347\343o para " } {TEXT 390 3 "n+1" }{TEXT -1 50 " n\363s de interpola\347\343o computa- se segundo a f\363rmula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 83 "pn(x) = f[x0] + (x-x0) f[x0,x1]+ (x-x0)(x-x1) f [x0,x1,x2] + ... + (x-x0)...(x-x" }{TEXT 299 3 "n-1" }{TEXT -1 17 ") f[x0,x1,...,xn]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "onde " }{TEXT 391 11 "f[x0,...xk]" }{TEXT -1 5 " \351 o " }{TEXT 300 44 "operador de diferen\347\343s divididas de ordem k " } {TEXT -1 27 "definido indutivamente por:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 392 16 "f[x0] := f(x0)" }{TEXT -1 78 " \+ \+ ordem 0" }}{PARA 0 "" 0 "" {TEXT 393 33 "f[x0,x1] := (f[x1]-f[x0])/(x1 -x0)" }{TEXT -1 52 " ordem 1" }}{PARA 0 "" 0 "" {TEXT -1 3 "..." }}{PARA 0 "" 0 "" {TEXT 394 40 "f[x0,...,xn] := (f[x1,...,xn]-f[x0,...,x" }{TEXT 301 3 "n-1" }{TEXT 395 10 "])/(xn-x0)" }{TEXT -1 30 " ordem n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "O procedi mento " }{TEXT 302 8 "vectdd, " }{TEXT -1 59 "a seguir, computa estas \+ diferen\347\343s divididas para vectores " }{TEXT 303 1 "X" }{TEXT -1 3 " e " }{TEXT 305 4 "FXin" }{TEXT -1 4 " de " }{TEXT 304 1 "n" } {TEXT -1 33 " n\363s e valores de interpola\347\343o. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 182 "\311 importante ano tar que neste procedimento s\343o computadas todas as diferen\347as di vididas necess\341rias na ordem adequada para poder realizar os comput os sobre um \372nico vetor de tamanho " }{TEXT 396 1 "n" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "vectdd := proc(X,FXin,n) \n local FX, i, j;\n FX \+ := FXin; \n for i from 1 to n-1 do \n for j from n by -1 to i+1 do \n FX[j] := (FX[j] - FX[j-1])/(X[j]-X[j-i]); \n end do; \n e nd do; \n evalm(FX); \nend proc; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'vectddGf*6%%\"XG%%FXinG%\"nG6%%#FXG%\"iG%\"jG6\"F.C%>8$9%?(8%\"\"\" F5,&9&F5F5!\"\"%%trueG?(8&F7F8,&F4F5F5F5F9>&F16#F;*&,&F>F5&F16#,&F;F5F 5F8F8F5,&&9$F?F5&FG6#,&F;F5F4F8F8F8-%&evalmG6#F1F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 430 "Observe \+ que as \372nicas diferen\347as divididas necess\341rias para escrever \+ o polin\364mio na Forma de Newton s\343o da forma f[x0,...,xk]. Assim , na primeira itera\347\343o o \372nico valor a preservar \351 f[x0]; \+ na segunda f[x0,x1]; na terceira f[x0,x1,x2]; e assim subsequentemente . Desta forma podem-se realizar todos os c\341lculos necess\341rios n um \372nico vetor de comprimento n. Ao fim da computa\347\343o, terem os todas as difere\347as divididas neste vetor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT 306 107 "Tendo computado as difere\347as divididas necess \341rias na forma de Newton, pode ser especificado um algoritmo, " } {TEXT 397 12 "interpNewton" }{TEXT 398 103 ", para constroir o polin \364mio de interpola\347\343o nesta forma. Os par\342metros deste no vo procedimento s\343o: " }{TEXT 307 1 "X" }{TEXT 308 3 " e " }{TEXT 309 2 "Fx" }{TEXT 310 44 ", os n\363s de interpola\347\343o e os valor es de f; " }{TEXT 311 1 "n" }{TEXT 312 24 ", a quantidade de n\363s e \+ " }{TEXT 313 2 "x," }{TEXT 314 73 " uma vari\341vel em termos da qual \+ ser\341 especificado o polin\364mio resultante." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "interpNewt on := proc(X,FX,n,x) \n local DD, pol, mult, i;\n DD:=vectdd(X,FX,n) ; \n pol:=DD[1]; \n mult:=1; \n for i from 2 to n do \n mult:=m ult*(x-X[i-1]); \n pol:=pol+mult*DD[i]; \n end do; \n return pol; \nend proc; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-interpNewtonGf*6& %\"XG%#FXG%\"nG%\"xG6&%#DDG%$polG%%multG%\"iG6\"F0C'>8$-%'vectddG6%9$9 %9&>8%&F36#\"\"\">8&F>?(8'\"\"#F>F9%%trueGC$>F@*&F@F>,&9'F>&F76#,&FBF> F>!\"\"FMF>>F;,&F;F>*&F@F>&F36#FBF>F>OF;F0F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 103 "9. Aplique agora e ste procedimento para gerar a Forma de Newton do polin\364mio interpol ador da quest\343o 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "interpNewton([-1, 0, 1, 2],[4, 1, - 1, 2],4,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&\"\"$F$%\"z GF$!\"\"*(#F$\"\"#F$,&F'F$F$F$F$F'F$F$**#F+F&F$F,F$F'F$,&F'F$F$F(F$F$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 157 "10. Verifique que \+ as suas respostas coincidem com as obitidas via solu\347\343o de siste mas lineares com o m\351todo de elimina\347\343o de Gauss e com a form a de Lagrange." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 86 "Para isto precisar\341 utilizar algum mecanismo de simpl ifica\347\343o como o comando simplify." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Para isto basta verificar q ue a simplifica\347\343o dos polin\364mios na forma de Lagrange e na f orma de Newton coincidem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "simplify(interpNewton([-1, 0, 1, 2],[4, 1, -1, 2],4,z))=simplify( interpLagrange([-1, 0, 1, 2],[4, 1, -1, 2],4,z)); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,*\"\"\"F%*&#\"#>\"\"'F%%\"zGF%!\"\"*&#F%\"\"#F%)F*F .F%F%*&#F.\"\"$F%)F*F2F%F%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 421 148 "11. Exerc\355cio opcional para entregar ap\363s aula. \+ Defina um procedimento interpNewtonEstErro, que al\351m de fornecer a Forma de Newton do polin\364mio de" }}{PARA 0 "" 0 "" {TEXT 422 44 "i nterpola\347\343o forneca a estimativa de erro. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Ajuda:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "Para estimar o erro da i nterpola\347\343o, ser\341 importante dispor da tabela completa de dif eren\347as divididas. Esta pode ser visualizada com uma simples modif ica\347\343o do procedimento " }{TEXT 320 6 "vectdd" }{TEXT -1 64 ", o nde em cada itera\347\343o escrevemos as diferen\347as de cada ordem. \+ " }{TEXT 321 1 " " }{TEXT -1 23 "Observe o procedimento " }{TEXT 322 8 "tabeladd" }{TEXT -1 10 " a seguir." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "tabeladd := proc(X,FXi n,n) \n local FX, i, j;\n FX := FXin; \n for i from 1 to n-1 do \n print(FX[i..n],` ordem`,i-1); \n for j from n by -1 to i+1 \+ do \n FX[j] := (FX[j] - FX[j-1])/(X[j]-X[j-i]); \n end do; \n \+ end do; \n print(FX[n..n],` ordem`,i-1); \n evalm(FX); \nend pr oc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)tabeladdGf*6%%\"XG%%FXinG% \"nG6%%#FXG%\"iG%\"jG6\"F.C&>8$9%?(8%\"\"\"F5,&9&F5F5!\"\"%%trueGC$-%& printG6%&F16#;F4F7%+~~~~~ordemG,&F4F5F5F8?(8&F7F8,&F4F5F5F5F9>&F16#FD* &,&FGF5&F16#,&FDF5F5F8F8F5,&&9$FHF5&FP6#,&FDF5F4F8F8F8-F<6%&F16#;F7F7F AFB-%&evalmG6#F1F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 132 "A tabela de diferen\347as divididas para os n\363s de interpola\347\343o -1, 0, 1, 2 e correspondentes valores 4, 1, -1, 2 \351 computada a seguir:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "tabeladd([-1, 0, 1, 2], [4, 1, -1, 2],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7&\"\"%\"\"\"!\" \"\"\"#%+~~~~~ordemG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%!\"$!\" #\"\"$%+~~~~~ordemG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7$#\"\"\" \"\"##\"\"&F&%+~~~~~ordemGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7##\" \"#\"\"$%+~~~~~ordemGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6 #7&\"\"%!\"$#\"\"\"\"\"##F+\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 21 "III Fen\364meno de Runge" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "N\343o in clui quest\365es. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 115 "Distribui\347\343o uniforme dos n\363s de interpola \347\343o n\343o \351, em geral, uma boa estrat\352gia. Considere por exemplo a fun\347\343o:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fr := x -> 1 / (1 + 25 * x^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#frGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&\"\"\"F-,&F-F-*&\"#DF-)9$\"\"#F-F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "e interpolemos di ta fun\347\343o no interv\342lo [-1,1] nos pontos " }{TEXT 400 24 "xi \+ = -1 + 2i/n, i = 0..n" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "Para gerar ditos n\363s de interpola \347\343o e os correspondentes valores, em qualquer intervalo [linf, l sup], podemos definir dois procedimento simples, a seguir." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "ge rarNos := proc(linf,lsup,n) \n local A, i;\n A := vector(n+1); \n f or i from 0 to n do A[i+1] := linf + i * (lsup-linf)/n; end do; \n re turn evalm(A); \nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)gerar NosGf*6%%%linfG%%lsupG%\"nG6$%\"AG%\"iG6\"F-C%>8$-%'vectorG6#,&9&\"\" \"F6F6?(8%\"\"!F6F5%%trueG>&F06#,&F8F6F6F6,&9$F6*(F8F6,&9%F6F@!\"\"F6F 5FDF6O-%&evalmG6#F0F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "gerarValores := proc(linf,lsup,f,n)\n local A, i;\n A := vector (n+1); \n for i from 0 to n do A[i+1] := evalf(f(linf + i * (lsup-lin f)/n)); end do; \n return evalm(A); \nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-gerarValoresGf*6&%%linfG%%lsupG%\"fG%\"nG6$%\"AG%\"i G6\"F.C%>8$-%'vectorG6#,&9'\"\"\"F7F7?(8%\"\"!F7F6%%trueG>&F16#,&F9F7F 7F7-%&evalfG6#-9&6#,&9$F7*(F9F7,&9%F7FG!\"\"F7F6FKF7O-%&evalmG6#F1F.F. F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Selecionemos " }{TEXT 401 4 "n=10" }{TEXT -1 63 ", p.ex. Obser ve o resultado dos anteriores procedimentos para " }{TEXT 402 4 "n=10 " }{TEXT -1 16 " e nossa fun\347\343o " }{TEXT 403 2 "fr" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "gerarNos(-1,1,10);gerarValores(-1,1,fr,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7-!\"\"#!\"%\"\"&#!\"$F*#!\"#F*# F'F*\"\"!#\"\"\"F*#\"\"#F*#\"\"$F*#\"\"%F*F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7-$\"+YQ:YQ!#6$\"+THN#)eF)$\"+++++5!#5$\"++ +++?F.$\"+++++]F.$\"\"\"\"\"!F1F/F,F*F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Utilizando ditos procedim entos podemos agora interpolar a nossa fun\347\343o " }{TEXT 404 2 "fr " }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "pfr := z -> simplify(interpN ewton(gerarNos(-1,1,10),gerarValores(-1,1,fr,10),11,z)); \npfr(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pfrGf*6#%\"zG6\"6$%)operatorG%&arro wGF(-%)simplifyG6#-%-interpNewtonG6&-%)gerarNosG6%!\"\"\"\"\"\"#5-%-ge rarValoresG6&F5F6%#frGF7\"#69$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,6*$)%\"zG\"\")\"\"\"$\"+B]4\\\\!\"(*&$\"++++c))!#F()F&\"\"$F(F(*&$\"+++++CF/F()F&\"\"(F(F(*&$\"+++'\\Q(FNF(F&F(F<" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 " Para visualizar o ajuste realizado com dito polin\364mio podemos grafi car simultaneamente " }{TEXT 405 2 "fr" }{TEXT -1 15 " e o polin\364mi o " }{TEXT 407 4 "pfr " }{TEXT -1 29 "de interpola\347\343o assim obti do." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{fr(x),pfr(x)\},x=-1..1, color=[red,blue]);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7ct7 $$!\"\"\"\"!$\"33)z!pRn:YQ!#>7$$!3-n;HdNvs**!#=$\"3m$[xee*HeCF17$$!3/M Le9r]X**F1$\"3Y])f!ytByVF17$$!3/,](=ng#=**F1$\"3i1>RyAH^hF17$$!3%pmm\" HU,\"*)*F1$\"3)R#=8'p3Ty(F17$$!3'GLekynP')*F1$\"39[HeMu6$G*F17$$!3()** *\\PM@l$)*F1$\"3/%R_53aa1\"!#<7$$!3)omT5!\\F4)*F1$\"3W6T;o(R/>\"FM7$$! 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Os n\363s de Chebyshev s\343o definidos por: " }} {PARA 0 "" 0 "" {TEXT 410 29 "xi = cos((2i+1)/(n+1) Pi/2), " }{TEXT -1 4 "para" }{TEXT 412 10 " i=0,..,n," }{TEXT -1 6 " onde " }{TEXT 411 1 "n" }{TEXT -1 34 " \351 o grau do polin\364mio desejado. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Para nos so exemplo podemos selecionar os 11 n\363s de Chebyshev e os seus resp ectivos valores utilizando os seguintes procedimentos:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "gera rNosCheby := proc(n)\n local A, i;\n A := vector(n+1); \n for i fro m 0 to n do A[i+1] := evalf(cos((2*i+1)/(n+1)*Pi/2)); end do; \n retu rn evalm(A); \nend proc;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "gerarV aloresCheby := proc(f,n) \n local A, i;\n A := vector(n+1); \n for \+ i from 0 to n do A[i+1] := evalf(f(cos((2*i+1)/(n+1)*Pi/2))); end do; \+ \n return evalm(A); \nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% .gerarNosChebyGf*6#%\"nG6$%\"AG%\"iG6\"F+C%>8$-%'vectorG6#,&9$\"\"\"F4 F4?(8%\"\"!F4F3%%trueG>&F.6#,&F6F4F4F4-%&evalfG6#-%$cosG6#,$*(,&F6\"\" #F4F4F4F2!\"\"%#PiGF4#F4FFO-%&evalmG6#F.F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2gerarValoresChebyGf*6$%\"fG%\"nG6$%\"AG%\"iG6\"F,C%> 8$-%'vectorG6#,&9%\"\"\"F5F5?(8%\"\"!F5F4%%trueG>&F/6#,&F7F5F5F5-%&eva lfG6#-9$6#-%$cosG6#,$*(,&F7\"\"#F5F5F5F3!\"\"%#PiGF5#F5FJO-%&evalmG6#F /F,F,F," }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 1 " " }}{PARA 12 "" 0 "" {TEXT -1 72 "Agora podemos computar o polin\364mio de interpola\347 \343o nestes n\363s e valores." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "pfrCheby := z -> simplify(i nterpNewton(gerarNosCheby(10),gerarValoresCheby(fr,10),11,z));\npfrChe by(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)pfrChebyGf*6#%\"zG6\"6$%) operatorG%&arrowGF(-%)simplifyG6#-%-interpNewtonG6&-%.gerarNosChebyG6# \"#5-%2gerarValoresChebyG6$%#frGF5\"#69$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8*$)%\"zG\"\")\"\"\"$\"+)Qe5I\"!\"(*&$\"+o-'\\*R!#['!#=F()F&\"\"*F(F2$\"+)*********! #5F(*&$\"+`6lZ7F6F()F&\"\"#F(F2*&$\"+5iB0KF/F()F&\"\"$F(F(*&$\"+(*[e " 0 " " {MPLTEXT 1 0 54 "plot(\{fr(x), pfrCheby(x)\},x=-1..1, color=[red,gre en]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV ESG6$7]s7$$!\"\"\"\"!$!3<(oeP(z)oK&!#?7$$!3%pmm\"HU,\"*)*!#=$\"3_l')z9 l*)pT!#>7$$!3!RLL$e%G?y*F1$\"3Wh'>#yR\"H*pF47$$!3#om;HdNvs*F1$\"3!\\%p )*o[NUyF47$$!3u****\\(oUIn*F1$\"3#>i+I@QSQ)F47$$!3xLL3-)\\&='*F1$\"3Yw )GnO*)Gm)F47$$!3ommm;p0k&*F1$\"3DlP@Ta*)>()F47$$!3E++vV5Su$*F1$\"3%yN* pC)e%*o(F47$$!3wKL$36KF47$$!3mmmmT%p\"e()F1$\"3j\\&)H[q:;8F47$$!3qlm;/'=3l)F1$\"3s(HO@ vDxj@F-7$$!33mmT5:W#Q)F1$\"3'e(=.xbx aDF-7$$!3:mmm\"4m(G$)F1$\"35XPU!>PTb$F-7$$!3)****\\i&[3:\")F1$\"3=1w$f V[aI\"F47$$!3\"QLL3i.9!zF1$\"3?!H:ZRLy(HF47$$!3\"ommT!R=0vF1$\"3\\uT%f '=PDrF47$$!3u****\\P8#\\4(F1$\"3c#zG_ew19\"F17$$!3#RLL3FuF)oF1$\"3&yb& HB7;:8F17$$!3+nm;/siqmF1$\"3cG,%[+\"=T9F17$$!3#)******\\Q*[c'F1$\"3)\\ =U%*o\"*Q[\"F17$$!3uLL$e\\g\"fkF1$\"3Qw&Q:9[G^\"F17$$!3mnmmTrU`jF1$\"3 Qr)RYnk\"G:F17$$!3[++](y$pZiF1$\"3:0b<(Hc-`\"F17$$!3YM$ek.M*QhF1$\"34s ()R6JM>:F17$$!3MnmT&Gu,.'F1$\"3g[qSU3@'\\\"F17$$!3?+]PMXT@fF1$\"39l=W? 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