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\\:;:xiy=>B<>:yqyK:>xyyYyA:::::::::::::Z:B:ry::::::::::::Z:KZ:ryxIJZBJ :ZyuyBJ>BjywYZ:Bvry::::::::::::::Z:B:ry:::::::::::::>B<>ryxI:<>;Bxiy=::::::::::::::::Z::xI:::::::::::::BJ>Bxiy=Z:KZ:yqy;<>vY:::::::: :::::::::Z::xI:::::::::::::BJ>Bxiy=Z:KZryxI:<>vY:::::::::::::::::Z::xI :::::::::::::BJjy;:JZZy=:::::::::::::::::::Z::xI:::::::::::::;\\:kywY: :Z:Bvry:::::::::::::::::::Z:B:ry::::::::::::::::::>:[AyA:::::::::::::: ::::Z:B:ry::::::::::::::::::::::::::::::::::::::Z::xI::::::::::::::::: :::::::::::::::::::::B:Zy=::::::::::::::::::::::::::::::::::::::<:ry:: ::::::::::::::::::::::::::::::::::::Z::xI::::::::::::::::::::::::::::: :::::::::B:Zy=::::::::::::::::::::::::::::::::::::::<:ry:::::::::::::: ::::::::::::::::::::::::Z:J::::::::::::::::::::::::::=ja^GB:;::::::::: N;?B:yyyxI:;Z::::::j;>:c:;::::::::::::vYxI:;Z::::::::::::::::::::yay=J :B:::::::::::::::::::jysy:>:<::::::::::::::::::::::::::::::::::::::::: ::::::3:" }}{PARA 258 "" 0 "" {TEXT 261 56 " C\341lculo Diferencial e \+ Integral: um KIT de sobreviv\352ncia" }}{PARA 257 "" 0 "" {TEXT 260 42 "This woksheet is in Portuguese language." }}{PARA 260 "" 0 "" {TEXT -1 21 "Prof. Doherty Andrade" }}{PARA 256 "" 0 "" {TEXT 259 26 " M\351todo de Gauss-Seidel e " }{TEXT 256 29 "M\351todo das Diferen \347as finitas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "O M\351todo de Ga uss-Seidel" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 256 "O m\351todo de Gauss-Seidel, \351 um m\351todo iterativo para resolver sistema de equa\347\365es lineares e que consiste em ir aproximando-se sucessivamente da solu\347\343o. Dado o sistema Ax=B, \+ onde x e B s\343o vetores colunas de n componentes. Ent\343o o sistema \351 equivalente a " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sum(a[j,i]*x[i] ,i=1..n)=b[j]" "6#/-%$sumG6$*&&%\"aG6$%\"jG%\"iG\"\"\"&%\"xG6#F,F-/F,; \"\"\"%\"nG&%\"bG6#F+" }}{PARA 0 "" 0 "" {TEXT -1 12 "com j=1..n. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Se a[j,j] \351 n\343o nulo, ent\343o podemos escrever" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[j]=(b[j]/a[j,j]- sum((a[j,i]/a[j,j])*x[i],i =1..n))" "6#/&%\"xG6#%\"jG,&*&&%\"bG6#F'\"\"\"&%\"aG6$F'F'!\"\"F--%$su mG6$*(&F/6$F'%\"iGF-&F/6$F'F'F1&F%6#F8F-/F8;\"\"\"%\"nGF1" }}{PARA 0 " " 0 "" {TEXT -1 24 "onde i \351 diferente de j." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Denotando por C e por M a vetor e matriz dados por " }}{PARA 0 "" 0 "" {TEXT -1 4 "C=( " } {XPPEDIT 18 0 "b[1]/a[1,1]" "6#*&&%\"bG6#\"\"\"\"\"\"&%\"aG6$\"\"\"\" \"\"!\"\"" }{TEXT -1 6 ", ...," }{XPPEDIT 18 0 "b[n]/c[n,n]" "6#*&&%\" bG6#%\"nG\"\"\"&%\"cG6$F'F'!\"\"" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 4 "M=( " }{XPPEDIT 18 0 "m[i,j]" "6#&%\"mG6$%\"iG%\"jG" } {TEXT -1 7 "), onde" }{XPPEDIT 18 0 " m[i,j]= a[j,i]/a[j,j]" "6#/&%\"m G6$%\"iG%\"jG*&&%\"aG6$F(F'\"\"\"&F+6$F(F(!\"\"" }{TEXT -1 4 " e " } {XPPEDIT 18 0 "m[i,i]" "6#&%\"mG6$%\"iGF&" }{TEXT -1 2 "=0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Assim, temos o se guinte sistema expresso matricialmente" }}{PARA 0 "" 0 "" {TEXT -1 9 " x= Mx+C e" }}{PARA 0 "" 0 "" {TEXT -1 32 "constru\355mos o seguinte al goritmo" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^0" "6#*$% \"xG\"\"!" }{TEXT -1 2 "=0" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x^(m+1)" "6#)%\"xG,&%\"mG\"\"\"\"\"\"F'" }{TEXT -1 2 "= " }{XPPEDIT 18 0 "M*x^( (m))" "6#*&%\"MG\"\"\")%\"xG%\"mGF%" }{TEXT -1 2 "+C" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Neste algoritmo voc \352 entra com a matriz ampliada M= [A|B] do sistema Ax=B, com a pr ecis\343o desejada e com o n\372mero m\341ximo de itera\347\365es." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "A sintaxe \351 GSeidel(M, prec, nmax)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "GSeidel:=proc(a, prec, nmax)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "local n, xold, xnew, i, j, k, m, erro, soma:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "n:=rowdim(a):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "# chute inicial \351 [0,...,0]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "xold:=vector(n,0): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# Come\347ando as iteradas" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xnew:=vector(n,0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "erro:=1: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for j from 1 to nma x do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " if erro > prec then" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " # Calculando vetor xnew para ite rada j" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " for i from 1 to n do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " soma:=0:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 " for k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " if i<>k then soma:=soma+a[i,k]*xnew[k]: \+ fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " xnew[i]:=(a[i,n+1]-evalf(soma))/a[i,i]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " else break" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " f i: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " # Verificando a precis \343o e atualizando os dados" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " f or k from 1 to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " erro:=0 : " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " m:=abs(xnew[k]-xold[k] ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " if erro < m then erro:=m : fi: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " xold[k]:=xnew[k]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "# Escrevendo a solu\347\343o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "print(xold, ` I tera\347\365es executadas`, j-1):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 34 "Exemplos do m\351todo d e Gauss-Seidel" }}{PARA 0 "" 0 "" {TEXT -1 67 "Resolver usando o m\351 todo de Gauss-Seidel o seguinte sistema linear:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "MATRIX([[4, 1, 1], [1, 4, 1], [1, 1, 4]])" "6#-%'MATRIX G6#7%7%\"\"%\"\"\"\"\"\"7%\"\"\"\"\"%\"\"\"7%\"\"\"\"\"\"\"\"%" } {TEXT -1 1 "." }{XPPEDIT 18 0 "MATRIX([[x], [y], [z]])" "6#-%'MATRIXG6 #7%7#%\"xG7#%\"yG7#%\"zG" }{TEXT -1 2 "= " }{XPPEDIT 18 0 "MATRIX([[6] , [6], [6]])" "6#-%'MATRIXG6#7%7#\"\"'7#\"\"'7#\"\"'" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "M:=matrix ([[4,1,1,6],[1,4,1,6],[1,1,4,6]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "GSeidel(M,.01,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "GSeidel(M,.0000001,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "N:=matrix([[4,.24,-0.08,8],[.09,3,- 0.15,9],[.04,-0.08,4,20]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "GSeidel(N,.00000001,100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "P:=matrix([[1,-1,1 ,2],[-2,3,1,0],[1,-1,2,2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "GSeidel(P,.00000001,100);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Exemplos de diferen\347as finitas " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exemplo 1 " }}{PARA 0 "" 0 "" {TEXT -1 20 "Diferen\347as centradas " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Probl ema de fronteira" }}{PARA 0 "" 0 "" {TEXT -1 11 "y''-y'+xy= " } {XPPEDIT 18 0 "e^(x)*(x^2+1)" "6#*&)%\"eG%\"xG\"\"\",&*$F&\"\"#F'\"\" \"F'F'" }}{PARA 0 "" 0 "" {TEXT -1 16 "y(0)=0 e y(1)= e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(linalg): " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wiht(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=7:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1432 "a:=0: b:=1: \+ p:=x-> -1: q:=x-> x: r:=x-> exp(-x)*(x^2+1): \+ \+ N:=10: h:=(b-a)/N: \+ x[0]:=a: x[N]:=b: \+ y[0]:=0: y[N]:=evalf(exp(1)): \+ \+ M:=array(1..N-1,1..N-1): \+ b:=array(1..N-1): \+ f or i from 1 to N do x[i]:= evalf(x[0]+i*h): od: \+ for i from 1 to N-1 do a[i,i-1]:=evalf(1-p(x[i])*h/2): \+ a[i,i]:=evalf(q(x[i])*h^2-2): a[i,i+1]:=evalf(1+p(x[i])*h/2): \+ od: \+ for i from 2 to N-2 do b[i]:= evalf(r(x[i])*h^2):od: b[1 ]:=evalf(r(x[1])*h^2-(1-p(x[1])*h/2)*y[0]): b[N-1]:=evalf(r(x[N-1])*h^ 2-(1+p(x[N-1])*h/2)*y[N]): \+ for i from 1 to N-1 do for j from \+ 1 to N-1 do M[i,j]:= a[i,j]:if abs(i-j)>1 then M[i,j]:=0: fi od:od;\np rint(M ): print( b ): y:=linsolve(M,b);" }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Exemplo 2" }}{PARA 4 "" 0 "" {TEXT 257 27 "Problema de Sturm-Lio uville" }}{PARA 0 "" 0 "" {TEXT -1 22 "y''+p(x)y'+q(x)y=r(x) " }} {PARA 0 "" 0 "" {TEXT -1 5 "y(a)=" }{XPPEDIT 18 0 "y[0]" "6#&%\"yG6#\" \"!" }{TEXT -1 9 " e y(b)= " }{XPPEDIT 18 0 "y[N]" "6#&%\"yG6#%\"NG" } }{PARA 0 "" 0 "" {TEXT -1 38 "Calculando o vetor b no exemplo padrao" }}{PARA 0 "" 0 "" {TEXT -1 11 "y''-y'+xy= " }{XPPEDIT 18 0 "e^(-x)*(x^ 2+1)" "6#*&)%\"eG,$%\"xG!\"\"\"\"\",&*$F'\"\"#F)\"\"\"F)F)" }}{PARA 0 "" 0 "" {TEXT -1 15 "y(0)=0 e y(1)=2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(linalg): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wiht(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Di gits:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1420 "a:=0: b: =1: p:=x-> -1: q: =x-> x: r:=x-> exp(-x)*(x^2+1): \+ N:= 10: h:=(b-a)/N: x[0] :=a: x[N]:=b: y[0]:= 0: y[N]:=2: \+ \+ M:=array(1..N-1,1..N-1): b:=array(1..N- 1): \+ for i from 1 to N do x[i]:= eva lf(x[0]+i*h): od: for i from 1 to N-1 \+ do a[i,i-1]:=evalf(1-p(x[i])*h/2): a[i,i]:=evalf(q(x[i])*h^ 2-2): a[i,i+1]:=evalf(1+p(x[i])*h/2): \+ od: \+ for i from 2 to N-2 do b[i]:= evalf(r(x[i] )*h^2):od: b[1]:=evalf(r(x[1])*h^2-(1-p(x[1] )*h/2)*y[0]): b[N-1]:=evalf(r(x[N-1])*h^2-(1+p(x[N-1])*h/2)*y[N]): \+ \+ for i from 1 to N-1 do for j from 1 to N-1 do M[i,j]:= a[i,j]:if abs(i-j)>1 then M[i,j]:=0: fi od:od;\nprint(M ): print( b ): y:=linso lve(M,b);" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Exerc\355cios" }} {PARA 0 "" 0 "" {TEXT -1 94 "1-Resolva usando o m\351todo das diferen \347as finitas os seguintes problemas de valor de fronteira" }}{PARA 0 "" 0 "" {TEXT -1 38 "a) y''(x)+ y(x)=0 com y(0)=0 e y(1)=1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "b) y''(x) +xy'(x)=y=2x com y(0)=1 e y(1)=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "c) y''+2y'+y=x com y(0)=0 e y(1)=0" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "2- Use o \+ m\351todo de Gauss-Seidel para obter a solu\347\343o de 3 sistemas li neares Ax=b." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "9" 0 } {VIEWOPTS 1 1 0 1 1 1803 }