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1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 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 } } {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 19 "Aula pr\341tica sobre:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 53 "M\351to dos iterativos para solu\347\343o de sistemas lineares\n" }{TEXT -1 31 "C\341lculo Num\351rico, Turma C, UnB." }}{PARA 257 "" 0 "" {TEXT -1 27 "Prof. Mauricio Ayala-Rinc\363n" }}{PARA 257 "" 0 "" {TEXT -1 18 "17 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 -1 143 "Nesta aula iremos a implementa r dois m\351todos iterativos para solu\347\343o de sistemas lineares: \+ o m\351todo de Gauss-Jacobi e o m\351todo de Gauss-Seidel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Estes m\351todos , similarmente aos m\351todos iterativos lineares ou de ponto fixo, pa ra aproxima\347\343o de ra\355zes de fun\347\365es em " }{TEXT 258 1 " R" }{TEXT -1 65 ", est\343o baseados na constru\347\343o de uma fun \347\343o de itera\347\343o tal que:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 43 " \+ A x = b" }{TEXT -1 20 " se, e somente se " }{TEXT 260 11 "x = C x + g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "onde " }{TEXT 261 1 "A" }{TEXT -1 3 " e " }{TEXT 262 1 "b " }{TEXT -1 7 " s\343o a " }{TEXT 263 4 "nxn-" }{TEXT -1 11 "matriz e \+ o " }{TEXT 264 2 "n-" }{TEXT -1 35 "vetor do sistema linear original e " }{TEXT 265 1 "C" }{TEXT -1 3 " e " }{TEXT 266 1 "g" }{TEXT -1 3 " a " }{TEXT 267 4 "nxn-" }{TEXT -1 11 "matriz e o " }{TEXT 268 2 "n-" } {TEXT -1 32 "vetor da \"fun\347\343o de ponto fixo\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Assim, utilizando a fu n\347\343o de ponto fixo e partindo de uma aproxima\347\343o inicial \+ " }{XPPEDIT 18 0 "x^0;" "6#*$%\"xG\"\"!" }{TEXT -1 14 " computa-se \+ " }{XPPEDIT 18 0 "x^(k+1);" "6#)%\"xG,&%\"kG\"\"\"F'F'" }{TEXT -1 7 " \+ como " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 " \+ " }{XPPEDIT 18 0 "x^(k+1) = C*x^k+g;" "6#/)%\"xG,&%\"kG\"\"\"F (F(,&*&%\"CGF()F%F'F(F(%\"gGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Na " }{TEXT 315 4 "k+1-" } {TEXT -1 85 "\351sima itera\347\343o, a maneira de verificar que a apr oxima\347\343o \351 adequada, \351 testando que " }{XPPEDIT 18 0 "x^( k+1)" "6#)%\"xG,&%\"kG\"\"\"F'F'" }{TEXT -1 33 " est\341 suficientemen te pr\363ximo de " }{XPPEDIT 18 0 "x^k;" "6#)%\"xG%\"kG" }{TEXT -1 37 ". \nPara isto \351 definida uma no\347\343o de " }{TEXT 316 18 "dist \342ncia relativa" }{TEXT -1 7 " entre " }{TEXT 317 2 "n-" }{TEXT -1 9 "vetores, " }{TEXT 280 1 "x" }{TEXT -1 3 " e " }{TEXT 281 1 "y" } {TEXT -1 8 ", assim:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 " " }{XPPEDIT 18 0 "`m\341ximo `(abs(x[i]-y[i]))/`m\341ximo`(abs(x[j]));" "6#*&-%'m|\\yximoG6#-%$absG 6#,&&%\"xG6#%\"iG\"\"\"&%\"yG6#F.!\"\"F/-F%6#-F(6#&F,6#%\"jGF3" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "onde " } {TEXT 282 2 "i " }{TEXT -1 2 "e " }{TEXT 283 2 "j " }{TEXT -1 13 "vari am entre " }{TEXT 284 2 "1 " }{TEXT -1 2 "e " }{TEXT 285 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "O primeiro exerc\355cio consiste em \+ implementar um procedimento que compute esta dist\342ncia relativa." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "Um proc edimento para computar a maior componente (em m\363dulo) de um vetor p ode-se definir facilmente como segue:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "maxvect := proc(b,n)\n local max, i;\n max := abs(b[1]);\n for i from 2 to n do\n if abs(b[ i]) > max then max := abs(b[i]); end if;\n end do;\n return max;\nend \+ proc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(maxvectGf*6$%\"bG%\"nG6$% $maxG%\"iG6\"F,C%>8$-%$absG6#&9$6#\"\"\"?(8%\"\"#F69%%%trueG@$2F/-F16# &F46#F8>F/F>OF/F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 357 97 "1. Vefique o funcionamento do procedimento com diferentes veto res permuta\347\365es de \{-2,-1,0,1,2,3\}." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "maxvect([-2,0,2,- 1],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "maxvect([3,2,0,1],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 358 30 "2. Implem ente um procedimento " }{TEXT 359 12 "maxdifvects " }{TEXT 360 164 "qu e permita computar o m\341ximo das dist\342ncias entre as componentes \+ de dois vetores a e b de tamanho n. O procedimento tem como entradas o s 2 vetores e o seu tamanho." }}{PARA 0 "" 0 "" {TEXT 361 46 "Basta um a modifica\347\343o simples do procedimento " }{TEXT 363 8 "maxvect." }{TEXT 364 212 " Este procedimento ser\341 utilizado para computar o \+ numerador da dist\342ncia relativa definida acima. Novamente, teste s eu procedimento com pares de vetores cujas componentes pertencem a \{ -4,-3,-2,-1,0,1,2,3,4,5\}." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 365 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "maxdifvects := proc(a,b,n)\n local max, i;\n max := \+ abs(b[1]-a[1]);\n for i from 2 to n do\n if abs(b[i]-a[i]) > max then max := abs(b[i]-a[i]); end if;\n end do;\n return max;\nend proc; " } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,maxdifvectsGf*6%%\"aG%\"bG%\"nG6$% $maxG%\"iG6\"F-C%>8$-%$absG6#,&&9%6#\"\"\"F8&9$F7!\"\"?(8%\"\"#F89&%%t rueG@$2F0-F26#,&&F66#F=F8&F:FGF;>F0FCOF0F-F-F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "maxdifvects([3,4,5],[-1,-3,4],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Observe a utiliza\347\343o do seu proce dimento para computar a dist\342ncia relativa no procedimento " } {TEXT 369 13 "testeparadaGJ" }{TEXT -1 9 " embaixo." }}{PARA 0 "" 0 " " {TEXT 368 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "testeparadaGJ := proc(x,y,n)\n return maxdifvects(x,y,n)/maxvect(x,n );\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.testeparadaGJGf*6% %\"xG%\"yG%\"nG6\"F*F*O*&-%,maxdifvectsG6%9$9%9&\"\"\"-%(maxvectG6$F0F 2!\"\"F*F*F*" }}}{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 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " O procedimento " }{TEXT 366 0 "" }{TEXT -1 0 "" }{TEXT 367 14 "testepa radaGJ " }{TEXT -1 93 "ser\341 utilizado pelos procedimentos que imple mentam os m\351todos de Gauss-Jacobi e Gauss-Seidel." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{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 15 "O procedimento " }{TEXT 286 11 "GaussJacobi" } {TEXT -1 44 " definido abaixo, assim como o procedimento " }{TEXT 287 11 "GaussSeidel" }{TEXT -1 77 ", utilizam a fun\347\343o de dist\342nc ia relativa, que foi implementada na quest\343o 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 22 "M\351todo de Gauss-Jacob i" }{TEXT -1 17 "\n\nO procedimento " }{TEXT 270 12 "GaussJacobi " } {TEXT -1 94 "aproxima solu\347\365es de sistemas lineares pelo m\351to do de Gauss-Jacobi. Os seus par\342metros s\343o a " }{TEXT 271 3 "nx n" }{TEXT -1 9 "-matriz " }{TEXT 273 1 "A" }{TEXT -1 5 " e o " } {TEXT 272 2 "n-" }{TEXT -1 6 "vetor " }{TEXT 274 1 "b" }{TEXT -1 27 " \+ de constantes e o pr\363prio " }{TEXT 275 1 "n" }{TEXT -1 40 ". Adicio nalmente, deve ser fornecido um " }{TEXT 276 2 "n-" }{TEXT -1 34 "vect or com a aproxima\347\343o inicial, " }{TEXT 277 3 "xin" }{TEXT -1 23 ", a precis\343o desejada, " }{TEXT 278 3 "eps" }{TEXT -1 45 " e um n \372mero m\341ximo de itera\347\365es permitidas, " }{TEXT 279 5 "itma x" }{TEXT -1 97 ". Isto \372ltimo \351 devido ao fato de que o proced imento n\343o realiza nenhum teste de converg\352ncia. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Um exerc\355cio i nteressante \351 o de estender o procedimento, de forma tal que seja v erificado o crit\351rio das linhas para o sistema de entrada e caso es te n\343o se cumpra o sistema seja modificado para que o venha a satis fazer se poss\355vel.\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 733 "GaussJacobi := proc(Ain,bin,xin,eps,n,itmax) \n local A,b,xant,x new,d,it,i,k,j;\n A:=Ain; b:=bin; xant:=xin; xnew:=vector(n); \n pri nt(xant); \n d := 2*eps; # Garante que seja executada a prime ira itera\347\343o.\n for it from 1 while (d > eps and it <= itmax) d o \n print(`itera\347\343o`,it); \n for i from 1 to n do \n \+ xnew[i] := b[i]; \n for k from 1 to n do \n if k <> i the n xnew[i] := xnew[i] - A[i,k]*xant[k]; end if; \n end do; \n \+ xnew[i] := xnew[i]/A[i,i]; \n end do; \n d := testeparadaGJ(x new,xant,n); \n for j from 1 to n do xant[j]:=xnew[j] end do; \n \+ print(`dist\342ncia relativa`,d); print(xant); \n end do; \n print( `Aproxima\347\343o computada `);\n print(evalf(evalm(xnew))); \nen d proc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,GaussJacobiGf*6(%$AinG %$binG%$xinG%$epsG%\"nG%&itmaxG6+%\"AG%\"bG%%xantG%%xnewG%\"dG%#itG%\" iG%\"kG%\"jG6\"F7C+>8$9$>8%9%>8&9&>8'-%'vectorG6#9(-%&printG6#F@>8(,$9 '\"\"#?(8)\"\"\"FRF732FNFL1FQ9)C(-FI6$%)itera|by|^yoGFQ?(8*FRFRFG%%tru eGC%>&FC6#Ffn&F=F[o?(8+FRFRFGFgn@$0F^oFfn>Fjn,&FjnFR*&&F:6$FfnF^oFR&F@ 6#F^oFR!\"\">Fjn*&FjnFR&F:6$FfnFfnFho>FL-%.testeparadaGJG6%FCF@FG?(8,F RFRFGFgn>&F@6#Fbp&FCFep-FI6$%3dist|]yncia~relativaGFLFH-FI6#%:Aproxima |by|^yo~computada~~~~G-FI6#-%&evalfG6#-%&evalmG6#FCF7F7F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 299 76 "3. Forne\347a uma aproxima\347\343o com precis\343o 0.0001 da \+ solu\347\343o do sistema linear " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 " \+ " }{XPPEDIT 18 0 "matrix([[10, 2, 1], [1, 5, 1], [2, 3, 10]])*RTAB LE(3046240,MATRIX([[x1], [x2], [x3]])) = RTABLE(10467288,MATRIX([[7], \+ [-8], [6]]));" "6#/*&-%'matrixG6#7%7%\"#5\"\"#\"\"\"7%F,\"\"&F,7%F+\" \"$F*F,-%'RTABLEG6$\"(Si/$-%'MATRIXG6#7%7#%#x1G7#%#x2G7#%#x3GF,-F26$\" ))Gn/\"-F66#7%7#\"\"(7#,$\"\")!\"\"7#\"\"'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 65 "aplicando o m\351todo de Gauss -Jacobi, partindo do vetor (0,0,0). " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 301 58 "O que acontece se selecionamos ponto s iniciais diferentes?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Teste pontos iniciais como (100,100,100), (1000,1000 ,1000), etc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 302 72 "Verifique que a sua aproxima\347\343o est \341 realmente pr\363xima da solu\347\343o exata." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Neste exerc\355cio limite o n\372mero m\341ximo de itera\347 \365es a 20." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "GaussJacobi([[10, 2, 1], [1, 5, 1], [2, 3, 10]], [7,-8,6],[0,0,0],0.0001,3,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\" \"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"(\"#5#!\")\"\"&#\"\"$F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"#<\"#$*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%#\"#C\"#D#!#$*\"#]#\"#ZF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%3dist|]yncia~relativaG#\"\"#\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7%#\"$*[\"$+&#!#**\"#]#\"$$[F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)it era|by|^yoG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~rela tivaG#\"#\")\"%s\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%(*\\\"%+]# !%V7\"$D'#\"$C'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\" \"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"$p#\" &*)*\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%Pi\"%]i#!&*)*\\\"&+]## \"&>\\#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"$\"))\"'M(* \\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"'f+D\"'++D#!'n)\\#\"'+]7#\" 'r+DF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"$Y\"\"'8+D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"((R*\\#\"(++]##!'8+D\"'+]7#\"'r Zi\"'+]i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"%tb\")\"[)* \\#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"(f.D'\"(++D'#!)\"[)*\\#\") ++]7#\")$*4]7F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\" *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"%M5\")z !*)Q\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"*pf*\\7\"*+++D\"#!*6<+D \"\")++]i#\"*2S*\\7F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxima|by |^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%$ \"+?vn****!#5$!+wt-+?!\"*$\"+g0_****F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 303 60 "4. Aproxime com precis\343o 0.0001 a solu\347\343o do sistema linear " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 44 " \+ " }{XPPMATH 20 "6#/*&-%'matrixG6#7%7%\"\"&\"\"#F+7%\"\"\"\"\"$F-7% \"\"!\"\"'\"\")F--%'RTABLEG6$\"*%[Ax8-%'MATRIXG6#7%7#%#x1G7#%#x2G7#%#x 3GF--F46$\"*K&)oN\"-F86#7%7#F.7#!\"#7#!\"'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 38 "aplicando o m\351todo de Gauss -Jacobi. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 305 72 "Verifique que a sua aproxima\347\343o est\341 r ealmente pr\363xima da solu\347\343o exata." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "GaussJacobi([[5, \+ 2, 2], [1, 3, 1], [0, 6, 8]],[3,-2,-6],[0,0,0],0.0001,3,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yn cia~relativaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"$\"\"&#! \"#F%#!\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"#<\"#N " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"(\"\"'#!#P\"#g#!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"#k\"$v\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#r\"#v#!#N\"#O#!#B\"#!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%3dist|]yncia~relativaG#\"$![\"%()>" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7%#\"%()>\"%+=#!%\">$\"%+O#!\"\"\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|] yncia~relativaG#\"%E:\"&*46" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%L V\"%+X#!&*46\"&+3\"#!$4%\"%+[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)ite ra|by|^yoG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relat ivaG#\"&X9\"\"'xG6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"'xG6\"'+!3 \"#!'$>2#\"'+g@#\"$*H\"&+W\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)iter a|by|^yoG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relati vaG#\"'igA\"(&>6L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"'x;8\"'+]8#! 'RAm\"'+!['#!%2))\"'+!)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by |^yoG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG# \"']\\I\"(>ih'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"(>ih'\"(++['#!) f:s7\")++'H\"#\"&RU\"\"'+S')" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)iter a|by|^yoG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relati vaG#\"(E()z'\"*:,$o>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\")()R(f\" \")++?;#!)BgOR\")++))Q#!'T%Q#\")++G<" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%)itera|by|^yoG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia ~relativaG#\"(tV^*\"*h+*GR" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"*h+ *GR\"*++!))Q#!*xrSq(\"*++gx(#\"'Bg[\")++%=&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %3dist|]yncia~relativaG#\"*Q3g4#\",NBNT<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"),uHg\")++vg#!+n/F[B\"+++!GL##!(BG>(\"+++!o.\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#7" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"*\"H:/I\",vshaM#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"*\"p%=Q*\"*++7L*#!,6BCKk%\",+++cm%#\")n /Z:\"+++S5J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"+9kSLl\"-vxX mAq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\",`E4#4e\",+++?$e#!,6$e14G \",++g$*z##!**odPA\",+++3A'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera |by|^yoG\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativa G#\"+\\(Rq^H\" /^EK$e/T)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"/^EK$e/T)\"/++++3)R) #!0**pq)HUx;\"0++++;'z;#\",nTM*=:\"/+++Su>6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %3dist|]yncia~relativaG#\"-Q*eSw8*\"00qd!Hd-O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"0ZAFH%G(4#\"0++++?&*4##!02y!o?gV]\"0++++[)Q]#!-,IH ,$>#\"0++++)[RA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#= " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\".4VLO5C* \"1P#o)QLsU]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"1P#o)QLsU]\"1++++ ![)Q]#!2P4'*[$H325\"2++++gpx+\"#\"-2y!oSv%\"0++++k%=n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"0=T)\\oY-?\"3&*yE**)[)R7:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"1za&3GIv9$\"1+++++G\\J#!2zN&)z(pzCI\"2++++ !)3L-$#!.j!R5lmo\"2++++!GpV8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)iter a|by|^yoG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativ aG#\"0**)Rm,+$*G\"3$)ewyp?_CI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\" 3$)ewyp?_CI\"3+++++)3L-$#!36Z;4#ynW/'\"3+++++whYg#\"/zN&)z<)[\"\"2++++ Sy5.%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"1M.\"[Hg!pi\"4bsnw K!eDs!*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"380obd43cv\"3+++++?Fev #!4^aLb1;^W\"=\"4+++++G&)R\"=#!0*GN3z\")\\@\"3+++++o:i!)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"1$4FDggo0*\"50%f)Q(e6lV\"=" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"4\")=xZex*Qii>\"6vGvQlj3&oUa" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%#\"5xhT7jrl<#\"7++++++OByw@# \"0B+J#4(R$e\"4++++g*Gv/e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|b y|^yoG\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG# \"4'GD()Ht)pU9'\"8bv,R)p@/=SlK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%# \"7n1(y*fm!=j2s#\"7++++++?z(4s##!76N!y'RV3O!3`'\"7++++++3qMIl#!3T++)Qe Zq5#\"7++++++[kP-H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG \"#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"40%= VTx&)[w))\"88W@j8cCG>2`'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"88W@j 8cCG>2`'\"8++++++!3qMIl#!9d+$f9!GfPM+18\"9++++++g,%pgI\"#\"36N!y'RV+mX \"7++++++W$Hrq)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#F " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"6Q!oJ7<- yaB>\":buS<7;y[dv\"f>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"8P[Ik\"[ ?1%\\1/#\"8+++++++WL2/##!9\"\\\"[VAjv\\6N=R\"9++++++![?3#=R#!4V*pS&)>2 S'f'\"9++++++!)oeUT<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxima|by| ^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%$ \"+=#)e****!#5$!+#[O++\"!\"*$!+?3$zy$!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 318 19 "Crit\351r io das linhas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Considere " }{TEXT 306 7 "A x = b" }{TEXT -1 31 ", um sis tema linear de tamanho " }{TEXT 307 3 "nxn" }{TEXT -1 4 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Defina pa ra " }{XPPEDIT 18 0 "1 <= j;" "6#1\"\"\"%\"jG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j <= n;" "6#1%\"jG%\"nG" }{TEXT -1 5 ", " }}{PARA 0 "" 0 "" {TEXT -1 33 " " }{XPPEDIT 18 0 "alpha[j] = (Sum(abs(A[j,i]),i = 1 .. n)-abs(A[j,j]))/abs(A[j,j]);" "6#/&%&alphaG6#%\"jG*&,&-%$SumG6$-%$absG6#&%\"AG6$F'%\"iG/F3;\"\"\"%\" nGF6-F.6#&F16$F'F'!\"\"F6-F.6#&F16$F'F'F<" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "e " }{XPPEDIT 18 0 "alpha = `m\341ximo`(alpha[j]);" "6#/%&alphaG-%'m|\\yximoG6#&F$6#%\"jG" } {TEXT -1 28 ", para j=1..n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "O " }{TEXT 308 19 "crit\351rio das \+ linhas" }{TEXT -1 17 " garante que se " }{XPPEDIT 18 0 "alpha;" "6#%& alphaG" }{TEXT -1 81 "<1, independentemente da aproxima\347\343o inici al, o m\351todo de Gauss-Jacobi converge. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 309 25 "5. Calcule manualment e o " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 310 124 " dos dois s istemas lineares das quest\365es 3 e 4 e verifique assim, que o crit \351rio das linhas aplica para ambos os sistemas. " }}{PARA 0 "" 0 " " {TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "Exerc\355cio 3 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }{XPPEDIT 18 0 "matrix([[10, 2, 1], [1, 5, 1], [2, 3, 10] ])*RTABLE(3046240,MATRIX([[x1], [x2], [x3]])) = RTABLE(10467288,MATRIX ([[7], [-8], [6]]));" "6#/*&-%'matrixG6#7%7%\"#5\"\"#\"\"\"7%F,\"\"&F, 7%F+\"\"$F*F,-%'RTABLEG6$\"(Si/$-%'MATRIXG6#7%7#%#x1G7#%#x2G7#%#x3GF,- F26$\"))Gn/\"-F66#7%7#\"\"(7#,$\"\")!\"\"7#\"\"'" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&al phaG6#\"\"\"#\"\"$\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alphaG6# \"\"##F'\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alphaG6#\"\"$#\" \"\"\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 5 "Logo " }{XPPEDIT 18 0 "alpha \+ = 1/2;" "6#/%&alphaG*&\"\"\"F&\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Exerc\355cio 4:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " " } {XPPEDIT 18 0 "matrix([[5, 2, 2], [1, 3, 1], [0, 6, 8]])*RTABLE(127924 92,MATRIX([[x1], [x2], [x3]])) = RTABLE(3202472,MATRIX([[3], [-2], [-6 ]]));" "6#/*&-%'matrixG6#7%7%\"\"&\"\"#F+7%\"\"\"\"\"$F-7%\"\"!\"\"'\" \")F--%'RTABLEG6$\"*%[Ax8-%'MATRIXG6#7%7#%#x1G7#%#x2G7#%#x3GF--F46$\"* K&)oN\"-F86#7%7#F.7#,$F+!\"\"7#,$F1FJ" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alphaG6#\"\"\"#\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alphaG6#\"\"##F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alphaG6#\"\"$#F'\"\"%" }}{PARA 0 "" 0 "" {TEXT -1 5 "Logo " }{XPPEDIT 18 0 "alpha = 4/5;" "6#/%&alphaG*&\"\"%\"\"\"\"\"&!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 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 0 "" }{TEXT 311 79 "6. Verifique que o crit\351rio das linhas n\343o vale para a matriz do sistema linear" }}{PARA 0 "" 0 " " {TEXT -1 15 " " }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ " }{XPPEDIT 18 0 "matrix([[1, 3, 5], [4, \+ 1, 2], [2, 6, 3]])*RTABLE(3195348,MATRIX([[x1], [x2], [x3]])) = RTABLE (3195388,MATRIX([[1], [2], [3]]));" "6#/*&-%'matrixG6#7%7%\"\"\"\"\"$ \"\"&7%\"\"%F*\"\"#7%F/\"\"'F+F*-%'RTABLEG6$\"([`>$-%'MATRIXG6#7%7#%#x 1G7#%#x2G7#%#x3GF*-F36$\"()Q&>$-F76#7%7#F*7#F/7#F+" }}{PARA 0 "" 0 "" {TEXT -1 31 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alphaG6#\"\"\"\"\")" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Logo " }{XPPEDIT 18 0 "alpha;" "6#%&a lphaG" }{TEXT -1 5 " > 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Neste caso n\343o temos garantia de que o m\351todo de Gauss-Jacobi venha convergir. E efetivamente o m\351todo diverge. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 312 207 "Reorganize o sistema linear (t roque linhas do sistema) para obter um sistema, onde o crit\351rio das linhas se cumpra e solucione o sistema utilizando o m\351todo de Gaus s-Jacobi. Deve-se alterar a ordem do vetor " }{TEXT 372 1 "b" }{TEXT 371 2 "! " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 " " }{XPPEDIT 18 0 "matrix([[4, 1, 2], [2, 6, 3], [1, 3, 5]])*matrix([[x[1]], [x[2]] , [x[3]]]) = matrix([[2], [3], [1]]);" "6#/*&-%'matrixG6#7%7%\"\"%\"\" \"\"\"#7%F,\"\"'\"\"$7%F+F/\"\"&F+-F&6#7%7#&%\"xG6#F+7#&F76#F,7#&F76#F /F+-F&6#7%7#F,7#F/7#F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Para este sistema: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alphaG6#\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %&alphaG6#\"\"##\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&alph aG6#\"\"$#\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&alphaG#\"\" &\"\"'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " GaussJacobi([[4, 1, 2], [2, 6, 3], [1, 3, 5]],[2, 3, 1],[0,0,0],0.1,3,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"\"\"\"#F$#F%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%3dist|]yncia~relativaG#\"#;\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 %#\"#6\"#S#\"\"(\"#I#!\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)i tera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~rel ativaG#\"#L\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#8\"#C#\"#h\"$ ?\"#\"\"\"\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"$C&\"$* ))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"$*))\"%+C#\"%T6\"%+O#!#;\"# v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"$U$\"%&3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%>:\"%!)G#\"%zM\"%+s#!$d#\"%+S" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"&'))=\"&J#f" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%#\"&J#f\"'+S9#\"&T#>\"&+S&#!&pS\"\"&+?(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"&[]%\"'D(>#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%#\"%*y)\"&!G<#\"%4J\"%]n#!&`I#\"'++C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"(*4k5\"(6,h&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"(P+(=\"(++K%#\"(\"=0\\\")++'H\"#!'`YQ\"(++;# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"(k\\V$\")b1jD" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"(Jh7&\")+!o.\"#\"(h>)G\"(++['#!&n a%\"'++S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\")rY*4%\"*%4N lM" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\")\\evd\"*++gH\"#\"*\\1%[I\" *++gx(#!)>x&H%\"*++?f#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^ yoG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"* Ep`J\"\"+lc53:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"*L6i,$\"*++3A'# \"*h(\\an\"+++?b:#!)\\ur`\"*+++K%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% :Aproxima|by|^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' vectorG6#7%$\"+W2f[[!#5$\"+g(pJM%F)$!+o%fMC\"F)" }}}{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 288 23 "M\351todo de Gauss-Seidel\n" }{TEXT -1 16 "\nO procedimento " } {TEXT 289 11 "GaussSeidel" }{TEXT -1 121 " embaixo \351 uma simples mo difica\347\343o do procedimento implementado para o m\351todo de Gauss -Jacobi. Os seus par\342metros s\343o a " }{TEXT 290 4 "nxn-" }{TEXT -1 8 "matriz " }{TEXT 291 1 "A" }{TEXT -1 5 " e o " }{TEXT 292 2 "n- " }{TEXT -1 6 "vetor " }{TEXT 293 1 "b" }{TEXT -1 27 " de constantes e o pr\363prio " }{TEXT 294 1 "n" }{TEXT -1 40 ". Adicionalmente, deve \+ ser fornecido um " }{TEXT 295 2 "n-" }{TEXT -1 34 "vector com a aproxi ma\347\343o inicial, " }{TEXT 296 3 "xin" }{TEXT -1 23 ", a precis\343 o desejada, " }{TEXT 297 3 "eps" }{TEXT -1 34 " e um n\372mero m\341xi mo de itera\347\365es, " }{TEXT 298 5 "itmax" }{TEXT -1 99 ". Isto \+ \372ltimo \351 devido ao fato de que o procedimento n\343o realiza nen hum teste de converg\352ncia. \n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 687 "GaussSeidel := proc(Ain,bin,xin,eps,n,itmax) \n loc al A,b,xant,xnew,d,it,i,k,j;\n A:=Ain; b:=bin; xant:=xin; xnew:=vecto r(n); print(xant); d := 2*eps; \n for it from 1 while (d > eps and it <= itmax) do \n print(`itera\347\343o`,it); \n for i from 1 to \+ n do \n xnew[i] := b[i]; \n for k from 1 to i-1 do xnew[i] : = xnew[i] - A[i,k]*xnew[k]; end do; \n for k from i+1 to n do xne w[i] := xnew[i] - A[i,k]*xant[k]; end do; \n xnew[i] := xnew[i]/A [i,i]; \n end do; \n d := testeparadaGJ(xnew,xant,n); \n fo r j from 1 to n do xant[j]:=xnew[j] end do; \n print(d); print(xant ); \n end do; \n print(`Aproxima\347\343o computada `);\n print( evalf(evalm(xnew))); \nend proc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%,GaussSeidelGf*6(%$AinG%$binG%$xinG%$epsG%\"nG%&itmaxG6+%\"AG%\"bG%% xantG%%xnewG%\"dG%#itG%\"iG%\"kG%\"jG6\"F7C+>8$9$>8%9%>8&9&>8'-%'vecto rG6#9(-%&printG6#F@>8(,$9'\"\"#?(8)\"\"\"FRF732FNFL1FQ9)C(-FI6$%)itera |by|^yoGFQ?(8*FRFRFG%%trueGC&>&FC6#Ffn&F=F[o?(8+FRFR,&FfnFRFR!\"\"Fgn> Fjn,&FjnFR*&&F:6$FfnF^oFR&FC6#F^oFRF`o?(F^o,&FfnFRFRFRFRFGFgn>Fjn,&Fjn FR*&FdoFR&F@FgoFRF`o>Fjn*&FjnFR&F:6$FfnFfnF`o>FL-%.testeparadaGJG6%FCF @FG?(8,FRFRFGFgn>&F@6#Fgp&FCFjp-FI6#FLFH-FI6#%:Aproxima|by|^yo~computa da~~~~G-FI6#-%&evalfG6#-%&evalmG6#FCF7F7F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 313 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 314 105 "7. Busque aproxima\347oes dos sistemas lineare s dos exerc\355cios 3, 4 e 6 utilizando o m\351todo de Gauss-Seidel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Observ e que o sistema do exerc\355cio 6 tamb\351m dever\341 ser alterado par a que o m\351todo de Gauss-Seidel aplique." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "GaussSeidel([[10, 2, 1], [1, 5, 1], [2, 3, 10]],[7,-8,6],[0,0,0],0.1,3,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"(\"#5#!#()\"#]#\"$\"\\\"$+&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%Xi\"&f'\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%# \"%\\Z\"%+]#!&f'\\\"&+]##\"'([^#\"'++D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"'lfe \")jl+D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"($p\"\\#\"(++]##!)jl+D \")++]7#\"*fF5D\"\"*+++D\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxi ma|by|^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG 6#7%$\"++?xm**!#5$!+S]_+?!\"*$\"+s?#3+\"F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "GaussSeidel([[5, 2, 2], [1, 3, 1], [0, 6, 8]],[3,-2 ,-6],[0,0,0],0.1,3,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\" \"$\"\"&#!#8\"#:#!\"\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera| by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#H\"#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#u\"#v#!$L%\"$]%#!#<\"$+'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$J(\"&)*o#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%$[%\"%+X# !&6n#\"&+q##!$*G\"&+g$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxima|b y|^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7% $\"+AAAi**!#5$!+jH'H*)*F)$!+yxxF!)!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "GaussSeidel([[4, 1, 2], [2, 6, 3], [1, 3, 5]],[2, 3, \+ 1],[0,0,0],0.1,3,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\" \"\"\"\"##F%\"\"$#!\"\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera |by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#6\"#%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"(\"#:#\"#r\"$!=#!#8\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$j\"\"%P]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%z ;\"%+O#\"%BW\"&+3\"#!$R\"\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Ap roxima|by|^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec torG6#7%$\"+*))))Qm%!#5$\"+q.P&4%F)$!++++!R\"F)" }}}{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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 322 22 "Crit\351rio de Sassenfeld" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Considere " }{TEXT 319 7 "A x = b" }{TEXT -1 31 ", um sistema linear de tamanho " }{TEXT 320 3 "nxn" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Defina " }{XPPEDIT 18 0 "beta[1] = Sum(abs(A[1,i]), i = 2 .. n)/abs(A[1,1]);" "6#/&%%betaG6#\"\"\"*&-%$SumG6$-%$absG6#&%\" AG6$F'%\"iG/F2;\"\"#%\"nGF'-F-6#&F06$F'F'!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 8 "e para " }{XPPEDIT 18 0 "2 <= j;" "6#1\"\"#%\"jG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j <= n;" "6#1%\"jG%\"nG" }{TEXT -1 11 ", \+ defina " }}{PARA 0 "" 0 "" {TEXT -1 48 " \+ " }{XPPEDIT 18 0 "beta[j] = (Sum(beta[i]*abs(A[j,i]) ,i = 1 .. j-1)+Sum(abs(A[j,i]),i = j+1 .. n))/abs(A[j,j]);" "6#/&%%bet aG6#%\"jG*&,&-%$SumG6$*&&F%6#%\"iG\"\"\"-%$absG6#&%\"AG6$F'F0F1/F0;F1, &F'F1F1!\"\"F1-F+6$-F36#&F66$F'F0/F0;,&F'F1F1F1%\"nGF1F1-F36#&F66$F'F' F;" }}{PARA 0 "" 0 "" {TEXT -1 12 "e seja " }{XPPEDIT 18 0 "beta \+ = `m\341ximo`(beta[j]);" "6#/%%betaG-%'m|\\yximoG6#&F$6#%\"jG" }{TEXT -1 28 ", para j=1..n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 2 "O " }{TEXT 321 22 "crit\351rio de Sassenf eld" }{TEXT -1 17 " garante que se " }{XPPEDIT 18 0 "beta < 1;" "6#2% %betaG\"\"\"" }{TEXT -1 80 ", independentemente da aproxima\347\343o \+ inicial, o m\351todo de Gauss-Seidel converge. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 323 131 "8. Verifique q ue o crit\351rio de Sassenfeld vale para os sistemas dos exerc\355cios 3 e 4 e a sua modifica\347\343o do sistema do exerc\355cio 6." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Exerc\355 cio 3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "matrix([[10, 2, 1], [1, 5, 1], [2, 3 , 10]])*RTABLE(3046240,MATRIX([[x1], [x2], [x3]])) = RTABLE(10467288,M ATRIX([[7], [-8], [6]]));" "6#/*&-%'matrixG6#7%7%\"#5\"\"#\"\"\"7%F,\" \"&F,7%F+\"\"$F*F,-%'RTABLEG6$\"(Si/$-%'MATRIXG6#7%7#%#x1G7#%#x2G7#%#x 3GF,-F26$\"))Gn/\"-F66#7%7#\"\"(7#,$\"\")!\"\"7#\"\"'" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"\"#\"\"$\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"##\"#8\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"$#\"#p\"$+&" }}{PARA 0 "" 0 "" {TEXT -1 5 "Logo " }{XPPEDIT 18 0 "beta = 3/10;" "6#/%%betaG*&\"\"$\" \"\"\"#5!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Exerc\355cio 4:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "matrix([[5, 2, 2], [1, 3, 1], [0, 6, 8]])*RTABLE(12792492,MATRIX([[x1], [x2], [x3]]) ) = RTABLE(3202472,MATRIX([[3], [-2], [-6]]));" "6#/*&-%'matrixG6#7%7% \"\"&\"\"#F+7%\"\"\"\"\"$F-7%\"\"!\"\"'\"\")F--%'RTABLEG6$\"*%[Ax8-%'M ATRIXG6#7%7#%#x1G7#%#x2G7#%#x3GF--F46$\"*K&)oN\"-F86#7%7#F.7#,$F+!\"\" 7#,$F1FJ" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"\"#\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%%betaG6#\"\"##\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%bet aG6#\"\"$#\"\"*\"#?" }}{PARA 0 "" 0 "" {TEXT -1 5 "Logo " }{XPPEDIT 18 0 "beta = 4/5;" "6#/%%betaG*&\"\"%\"\"\"\"\"&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Modifica\347\343o do e xerc\355cio 6:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "matrix([[4, 1, 2], [2, 6 , 3], [1, 3, 5]])*matrix([[x[1]], [x[2]], [x[3]]]) = matrix([[2], [3], [1]]);" "6#/*&-%'matrixG6#7%7%\"\"%\"\"\"\"\"#7%F,\"\"'\"\"$7%F+F/\" \"&F+-F&6#7%7#&%\"xG6#F+7#&F76#F,7#&F76#F/F+-F&6#7%7#F,7#F/7#F+" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"\"#\"\" $\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"##\"\"$\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"$#F'\"\"&" }}{PARA 0 "" 0 "" {TEXT -1 5 "Logo " }{XPPEDIT 18 0 "beta = 3/4;" "6#/%%betaG* &\"\"$\"\"\"\"\"%!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 324 31 "9. Considere o sistema linear " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 " " }{XPPEDIT 18 0 "matrix([[ 3, 1, 2], [2, 6, 3], [1, 3, 4]])*RTABLE(2888544,MATRIX([[x1], [x2], [x 3]])) = RTABLE(2900064,MATRIX([[1], [2], [3]]));" "6#/*&-%'matrixG6#7% 7%\"\"$\"\"\"\"\"#7%F,\"\"'F*7%F+F*\"\"%F+-%'RTABLEG6$\"(W&))G-%'MATRI XG6#7%7#%#x1G7#%#x2G7#%#x3GF+-F26$\"(k+!H-F66#7%7#F+7#F,7#F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 325 56 "Verifique que \+ o crit\351rio de Sassenfeld n\343o se satisfaz. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "beta[1] = 1;" "6#/&%%be taG6#\"\"\"F'" }{TEXT -1 7 " Logo " }{XPPEDIT 18 0 "beta;" "6#%%betaG " }{TEXT -1 6 " >= 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 160 "Modifique o sistema de forma tal que o crit\351rio d e Sassenfeld se satisfa\347a (verifique-o!) e, ent\343o, compute uma a proxima\347\343o utilizando o m\351todo de Gauss-Seidel." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "A aproxima\347\343 o deve ser dada em termos das vari\341veis " }{TEXT 326 6 "x1, x2" } {TEXT -1 3 " e " }{TEXT 327 2 "x3" }{TEXT -1 55 " originais: observe q ue ao trocar as colunas da matriz " }{TEXT 328 1 "A" }{TEXT -1 34 " a \+ ordem das componentes do vetor " }{TEXT 329 1 "x" }{TEXT -1 56 " tamb \351m muda. Ao trocar linhas, deve-se alterar o vetor " }{TEXT 374 1 " b" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Trocando linhas um e dois e colunas um e dois obtemos o s istema equivalente:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {XPPEDIT 18 0 "matrix([[6, 2, 3], [1, 3, 2], [3, 1, 4]])*matrix([[x[ 2]], [x[1]], [x[3]]]) = matrix([[2], [1], [3]]);" "6#/*&-%'matrixG6#7% 7%\"\"'\"\"#\"\"$7%\"\"\"F,F+7%F,F.\"\"%F.-F&6#7%7#&%\"xG6#F+7#&F66#F. 7#&F66#F,F.-F&6#7%7#F+7#F.7#F," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "para o qual o crit\351rio de Sassenfield vale sempre que: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"\"#\"\"&\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"##\"#<\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%betaG6#\"\"$#\"#J\"#O" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Logo " }{XPPEDIT 18 0 "bet a = 17/18;" "6#/%%betaG*&\"#<\"\"\"\"#=!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "Devido a trocas de colunas a aproxima\347 \343o \351 dada por" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 81 " \+ " }{XPPEDIT 18 0 "RTABLE(2980700,MATRIX([[x2 ], [x3], [x1]])) = RTABLE(2980740,MATRIX([[.4631050146e-4], [.79995600 51], [-.1999861069]]));" "6#/-%'RTABLEG6$\"(+2)H-%'MATRIXG6#7%7#%#x2G7 #%#x3G7#%#x1G-F%6$\"(S2)H-F)6#7%7#-%&FloatG6$\"+Y,0JY!#97#-F:6$\"+^+c* *z!#57#,$-F:6$\"+p5')**>FB!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "O crit\351rio das linhas pode ser verif icado com um simples procedimento " }{TEXT 330 22 "verifCriterioLinhas . " }{TEXT -1 26 "Este \351 uma fun\347\343o booleana" }}{PARA 0 "" 0 "" {TEXT -1 77 "que retorna o valor verdadeiro se e somente se o cr it\351rio das linhas vale. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "verifCriterioLinhas := proc (A,n)\n local i, j, alpha, maxalpha;\n maxalpha := 0;\n for i from \+ 1 to n do\n alpha[i] := -abs(A[i,i]); # Inicia-se substraindo o co mponente da diagonal\n # para evitar o t ratamento da exce\347\343o no seguinte\n \+ # \"for\". \n for j from 1 to n do alpha[i] := alpha[i] + abs(A[i, j]); end do;\n alpha[i] := alpha[i] / abs(A[i,i]); \n # print(a lpha[i]);\n if alpha[i] > maxalpha then maxalpha := alpha[i] end if ;\n end do;\n if maxalpha < 1 then return true\n else return false \n end if;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%4verifCrit erioLinhasGf*6$%\"AG%\"nG6&%\"iG%\"jG%&alphaG%)maxalphaG6\"F.C%>8'\"\" !?(8$\"\"\"F59%%%trueGC&>&8&6#F4,$-%$absG6#&9$6$F4F4!\"\"?(8%F5F5F6F7> F:,&F:F5-F?6#&FB6$F4FFF5>F:*&F:F5F>FD@$2F1F:>F1F:@%2F1F5OF7O%&falseGF. F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 331 92 "10. Verifique o funcionamento deste procediment o com os sistemas dos exerc\355cios 3, 4, 6 e 9." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "verifCriterioLinhas([[10, 2, 1], [1, 5, 1], [ 2, 3, 10]],3);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "verifCriterioLinhas([ [5, 2, 2], [1, 3, 1], [0, 6, 8]],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "verifCriterioLinhas([[1, 3, 5], [4, 1, 2], [2, 6, 3]],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "verifCriterioLinhas([[3, 1, 2], [2, 6, 3], [1, 3, 4]],3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 332 34 "11. Utilizando a fun\347\343o booleana \+ " }{TEXT 333 19 "verifCriterioLinhas" }{TEXT 334 28 ", crie um novo pr ocedimento " }{TEXT 337 13 "CLGaussJacobi" }{TEXT 338 28 " modificando o procedimento " }{TEXT 335 11 "GaussJacobi" }{TEXT 336 310 " de form a tal que para um sistema linear dado, o novo procedimento verifique o crit\351rio das linhas e unicamente compute uma aproxima\347\343o com o m\351todo de Gauss-Jacobi se o crit\351rio das linhas se satisfaz. \+ O novo procedimento n\343o utilizar\341 o par\342metro itmax, que con trola o n\372mero m\341ximo de itera\347\365es permitidas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 341 70 "Verifique a sua \+ implementa\347\343o com os quatro sistemas lineares da aula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Nota: cole o pr ocedimento " }{TEXT 339 11 "GaussJacobi" }{TEXT -1 90 " embaixo e nest a copia inclua um teste do crit\351rio das linhas invocando a fun\347 \343o booleana " }{TEXT 340 19 "verifCriterioLinhas" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 878 "CLGaussJacobi := proc(Ain,bin,xin,eps,n) \n local A,b,xant,xne w,d,it,i,k,j;\n if not verifCriterioLinhas(Ain,n) then print(`O crit \351rio das linhas n\343o \351 v\341lido! Reorganize o seu sistema.`) \n else\n A:=Ain; b:=bin; xant:=xin; xnew:=vector(n); \n print( xant); \n d := 2*eps; # Garante que seja executada a primei ra itera\347\343o.\n for it from 1 while (d > eps) do \n print (`itera\347\343o`,it); \n for i from 1 to n do \n xnew[i] \+ := b[i]; \n for k from 1 to n do \n if k <> i then xne w[i] := xnew[i] - A[i,k]*xant[k]; end if; \n end do; \n \+ xnew[i] := xnew[i]/A[i,i]; \n end do; \n d := testeparadaG J(xnew,xant,n); \n for j from 1 to n do xant[j]:=xnew[j] end do; \+ \n print(`dist\342ncia relativa`,d); print(xant); \n end do; \+ \n print(`Aproxima\347\343o computada `);\n print(evalf(evalm (xnew))); \n end if;\nend proc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%.CLGaussJacobiGf*6'%$AinG%$binG%$xinG%$epsG%\"nG6+%\"AG%\"bG%%xantG% %xnewG%\"dG%#itG%\"iG%\"kG%\"jG6\"F6@%4-%4verifCriterioLinhasG6$9$9(-% &printG6#%hnO~crit|dyrio~das~linhas~n|^yo~|dy~v|\\ylido!~Reorganize~o~ seu~sistema.GC+>8$F<>8%9%>8&9&>8'-%'vectorG6#F=-F?6#FI>8(,$9'\"\"#?(8) \"\"\"FYF62FUFSC(-F?6$%)itera|by|^yoGFX?(8*FYFYF=%%trueGC%>&FL6#Fjn&FF F_o?(8+FYFYF=F[o@$0FboFjn>F^o,&F^oFY*&&FD6$FjnFboFY&FI6#FboFY!\"\">F^o *&F^oFY&FD6$FjnFjnF\\p>FS-%.testeparadaGJG6%FLFIF=?(8,FYFYF=F[o>&FI6#F fp&FLFip-F?6$%3dist|]yncia~relativaGFSFP-F?6#%:Aproxima|by|^yo~computa da~~~~G-F?6#-%&evalfG6#-%&evalmG6#FLF6F6F6" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "CLGau ssJacobi([[10, 2, 1], [1, 5, 1], [2, 3, 10]],[7,-8,6],[0,0,0],0.1,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%3dist|]yncia~relativaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%# \"\"(\"#5#!\")\"\"&#\"\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera |by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativ aG#\"#<\"#$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#C\"#D#!#$*\"#]# \"#ZF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"\"#\"#L" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"$*[\"$+&#!#**\"#]#\"$$[F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxima|by|^yo~computada~~~~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%$\"++++!y*!#5$!++++!)>! \"*$\"++++g'*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "CLGaussJ acobi([[5, 2, 2], [1, 3, 1], [0, 6, 8]],[3,-2,-6],[0,0,0],0.1,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%3dist|]yncia~relativaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%# \"\"$\"\"&#!\"#F%#!\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera |by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativ aG#\"#<\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"(\"\"'#!#P\"#g#! \"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"#k\"$v\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#r\"#v#!#N\"#O#!#B\"#!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"$![\"%()>" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%#\"%()>\"%+=#!%\">$\"%+O#!\"\"\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"%E:\"&*46" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%LV\"%+X#!&*46\"&+3\"#!$4%\"%+[" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%)itera|by|^yoG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3dist|]yncia~relativaG#\"&X9\"\"'xG6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"'xG6\"'+!3\"#!'$>2#\"'+g@#\"$*H\"&+W\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%3dist|]yncia~relativaG#\"'igA\"(&>6L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"'x;8\"'+]8#!'RAm\"'+!['#!%2))\"'+!)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxima|by|^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%$\"+_=&Qv*!#5$!+lP(>-\"!\"*$!+6h)z0$!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "CLGaussJacobi([[1, 3, 5], [4, 1, 2], [2, 6, 3]],[1,2,3],[0,0,0],.000001,3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%hnO~crit|dyrio~das~linhas~n|^yo~|dy~v|\\ylido!~Reorg anize~o~seu~sistema.G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "CL GaussJacobi([[3, 1, 2], [2, 6, 3], [1, 3, 4]],[1,2,3],[0,0,0],.000001, 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnO~crit|dyrio~das~linhas~n|^y o~|dy~v|\\ylido!~Reorganize~o~seu~sistema.G" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 342 33 "12. Con strua uma fun\347\343o booleana " }{TEXT 343 23 "verifCriterioSassenfe ld" }{TEXT 344 27 ", que de maneira similar a " }{TEXT 345 19 "verifCr iterioLinhas" }{TEXT 346 37 ", verifique o crit\351rio de Sassenfeld. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Verif ique-a com as matrizes dos sistemas da aula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 468 "verifCriterioSas senfeld := proc(A,n)\n local i, j, beta, maxbeta;\n maxbeta := 0;\n \+ for i from 1 to n do\n beta[i]:=0;\n for j from 1 to i-1 do bet a[i] := beta[i] + beta[j]*abs(A[i,j]); end do;\n for j from i+1 to \+ n do beta[i] := beta[i] + abs(A[i,j]); end do;\n beta[i] := beta[i] / abs(A[i,i]); \n # print(beta[i]);\n if beta[i] > maxbeta then maxbeta := beta[i] end if;\n end do;\n if maxbeta < 1 then return t rue\n else return false\n end if;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%8verifCriterioSassenfeldGf*6$%\"AG%\"nG6&%\"iG%\"jG%% betaG%(maxbetaG6\"F.C%>8'\"\"!?(8$\"\"\"F59%%%trueGC'>&8&6#F4F2?(8%F5F 5,&F4F5F5!\"\"F7>F:,&F:F5*&&F;6#F>F5-%$absG6#&9$6$F4F>F5F5?(F>,&F4F5F5 F5F5F6F7>F:,&F:F5FFF5>F:*&F:F5-FG6#&FJ6$F4F4F@@$2F1F:>F1F:@%2F1F5OF7O% &falseGF.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{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 0 "" }{TEXT 347 64 "13. Similarmente ao exerc\355cio 10, utilizando a fun\347\343o boole ana " }{TEXT 348 23 "verifCriterioSassenfeld" }{TEXT 349 28 ", crie um novo procedimento " }{TEXT 352 13 "CSGaussSeidel" }{TEXT 353 28 " mod ificando o procedimento " }{TEXT 350 11 "GaussSeidel" }{TEXT 351 316 " de forma tal que para um sistema linear dado, o novo procedimento ver ifique o crit\351rio de Sassenfeld e unicamente compute uma aproxima \347\343o com o m\351todo de Gauss-Seidel se o crit\351rio de Sassenfe ld se satisfaz. O novo procedimento n\343o utilizar\341 o par\342metr o itmax, que controla o n\372mero m\341ximo de itera\347\365es permiti das." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 356 63 " Verifique a sua implementa\347\343o com os sistemas lineares da aula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Nota: cole o procedimento " }{TEXT 354 11 "GaussSeidel" }{TEXT -1 93 " emba ixo e nesta copia inclua um teste do crit\351rio de Sassenfeld invocan do a fun\347\343o booleana " }{TEXT 355 23 "verifCriterioSassenfeld" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 832 "CSGaussSei del := proc(Ain,bin,xin,eps,n) \n local A,b,xant,xnew,d,it,i,k,j;\n \+ if not verifCriterioSassenfeld(Ain,n) then print(`Crit\351rio de Sasse nfeld n\343o se aplica! Reorganize o seu sistema`)\n else\n A:=Ai n; b:=bin; xant:=xin; xnew:=vector(n); print(xant); d := 2*eps; \n \+ for it from 1 while (d > eps) do \n print(`itera\347\343o`,it); \+ \n for i from 1 to n do \n xnew[i] := b[i]; \n for \+ k from 1 to i-1 do xnew[i] := xnew[i] - A[i,k]*xnew[k]; end do; \n \+ for k from i+1 to n do xnew[i] := xnew[i] - A[i,k]*xant[k]; end do ; \n xnew[i] := xnew[i]/A[i,i]; \n end do; \n d := \+ testeparadaGJ(xnew,xant,n); \n for j from 1 to n do xant[j]:=xnew [j] end do; \n print(d); print(xant); \n end do; \n print(` Aproxima\347\343o computada `);\n print(evalf(evalm(xnew)));\n \+ end if; \nend proc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%.CSGaussSei delGf*6'%$AinG%$binG%$xinG%$epsG%\"nG6+%\"AG%\"bG%%xantG%%xnewG%\"dG%# itG%\"iG%\"kG%\"jG6\"F6@%4-%8verifCriterioSassenfeldG6$9$9(-%&printG6# %jnCrit|dyrio~de~Sassenfeld~n|^yo~se~aplica!~~Reorganize~o~seu~sistema GC+>8$F<>8%9%>8&9&>8'-%'vectorG6#F=-F?6#FI>8(,$9'\"\"#?(8)\"\"\"FYF62F UFSC(-F?6$%)itera|by|^yoGFX?(8*FYFYF=%%trueGC&>&FL6#Fjn&FFF_o?(8+FYFY, &FjnFYFY!\"\"F[o>F^o,&F^oFY*&&FD6$FjnFboFY&FL6#FboFYFdo?(Fbo,&FjnFYFYF YFYF=F[o>F^o,&F^oFY*&FhoFY&FIF[pFYFdo>F^o*&F^oFY&FD6$FjnFjnFdo>FS-%.te steparadaGJG6%FLFIF=?(8,FYFYF=F[o>&FI6#F[q&FLF^q-F?6#FSFP-F?6#%:Aproxi ma|by|^yo~computada~~~~G-F?6#-%&evalfG6#-%&evalmG6#FLF6F6F6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "CSGaussSeidel([[10, 2, 1], [ 1, 5, 1], [2, 3, 10]],[7,-8,6],[0,0,0],0.01,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera| by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%#\"\"(\"#5#!#()\"#]#\"$\"\\\"$+&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%Xi\"&f'\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"% \\Z\"%+]#!&f'\\\"&+]##\"'([^#\"'++D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%)itera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"'lfe\")jl+ D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"($p\"\\#\"(++]##!)jl+D\")++] 7#\"*fF5D\"\"*+++D\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yo G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\")0+\"4#\",\"4c5]7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"+,&G+D\"\"++++]7#!,\"4c5]7\"++++] i#\",jK)G]i\",++++D'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxima|by| ^yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%$ \"+,G-+5!\"*$!+v*o,+#F)$\"+Kh/+5F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "CSGaussSeidel([[5, 2, 2], [1, 3, 1], [0, 6, 8]],[3,-2 ,-6],[0,0,0],0.1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by|^yoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"$ \"\"&#!#8\"#:#!\"\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)itera|by| ^yoG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#H\"#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"#u\"#v#!$L%\"$]%#!#<\"$+'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%)itera|by|^yoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$J(\"&)*o#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"%$[%\"%+X#!&6 n#\"&+q##!$*G\"&+g$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Aproxima|by|^ yo~computada~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%$\" +AAAi**!#5$!+jH'H*)*F)$!+yxxF!)!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "CSGaussSeidel([[1, 3, 5], [4, 1, 2], [2, 6, 3]],[1,2, 3],[0,0,0],.000001,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnCrit|dyri o~de~Sassenfeld~n|^yo~se~aplica!~~Reorganize~o~seu~sistemaG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "CSGaussSeidel([[3, 1, 2], [2 , 6, 3], [1, 3, 4]],[1,2,3],[0,0,0],.000001,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnCrit|dyrio~de~Sassenfeld~n|^yo~se~aplica!~~Reorganiz e~o~seu~sistemaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 11 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 137722484 135688532 } {RTABLE M7R0 I6RTABLE_SAVE/137722484X*%)anythingG6"6"[gl!#%!!!"$"$%#x1G%#x2G%#x3G6" } {RTABLE M7R0 I6RTABLE_SAVE/135688532X*%)anythingG6"6"[gl!#%!!!"$"$""$!"#!"'6" }