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Resolução de Sistemas Lineares

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Resolução de Sistemas Lineares

  1. 1. Cálculo Numérico Resolução de Sistemas Lineares Escalonamento de matrizes com uso do SciLab Kleber Jacinto Eng Eletricista – Mestre em Ciência da computação1 23/11/12
  2. 2. Representando matrizes no Scilab Uma Matriz é um conjunto de valores numéricos, organizados em linhas e colunas. No Scilab, uma variável deve ser guardada em uma variável. •Uma matriz é SEMPRE representada entre colchetes “[ ]”; •Os elementos de uma linha devem ser separados por “espaço”; •As linhas são separadas por ponto e vírgula “;” -->L=[115 27 19 565;27 19 3 134.98;19 3 6 90] L = 115. 27. 19. 565. 27. 19. 3. 134.98 19. 3. 6. 90.2 23/11/12
  3. 3. Representando matrizes no Scilab É possível extrair de uma matriz, um elemento individualmente -->L(1,2) //linha 1 coluna 2 ans = 27. -->L(3,3) //linha 3 coluna 3 ans = 6. -->L(2,4) //linha 2 coluna 4 ans = 134.983 23/11/12
  4. 4. Representando matrizes no Scilab É possível extrair de uma matriz, uma linha ou coluna individualmente -->L(1,:) //Toda a linha 1 ans = 115. 27. 19. 565. -->L(3,:) //Toda a linha 3 ans = 19. 3. 6. 90. -->L(:,4) //Toda a coluna 4 ans = 565. 134.98 90.4 23/11/12
  5. 5. Representando matrizes no Scilab É possível extrair de sub-matriz de uma matriz existente -->L(1:2,1:3) //sub matriz das linhas 1 e 2 e colunas 1 a 3 ans = 115. 27. 19. 27. 19. 3. -->L(2:3,2:4) //sub matriz das linhas 2 e 3 e colunas 2 a 4 ans = 19. 3. 134.98 3. 6. 90.5 23/11/12
  6. 6. Representando matrizes no Scilab É possível realizar operação com toda a matriz ou ou um elemento, ou uma linha ou coluna -->L*3 ans = 345. 81. 57. 1695. 81. 57. 9. 404.94 57. 9. 18. 270. -->L(1,1)+2 ans = 117. -->L(1,:)*3 ans = 345. 81. 57. 1695.6 23/11/12
  7. 7. Representando matrizes no Scilab É possível realizar operação com toda a matriz ou ou um elemento, ou uma linha ou coluna -->L*3 ans = 345. 81. 57. 1695. 81. 57. 9. 404.94 57. 9. 18. 270. -->L(1,1)+2 ans = 117. -->L(1,:)*3 ans = 345. 81. 57. 1695.7 23/11/12
  8. 8. Sistema de equações lineares Solução por escalonamento da matriz Sistema: 115.x + 27.y + 19.z = 565 27. x + 19.y + 3.z = 134,98 19.x + 3.y + 6.z = 90 Representação Matricial: -->L=[115 27 19 565;27 19 3 134.98;19 3 6 90] L = 115. 27. 19. 565. 27. 19. 3. 134.98 19. 3. 6. 90.8 23/11/12
  9. 9. Sistema de equações lineares Solução por escalonamento da matriz Primeiro Passo: garantir que o elemento (1,1) tenha valor 1 Dividir toda a linha pelo elemento (1,1) -->L(1,:)=L(1,:)/L(1,1) L = 1. 0.2347826 0.1652174 4.9130435 27. 19. 3. 134.98 19. 3. 6. 90.9 23/11/12
  10. 10. Sistema de equações lineares Solução por escalonamento da matriz Segundo Passo: zerar o elemento (2,1) Substituir a linha 2, por ela mesma menos a linha 1 multiplicada pelo elemento (2,1) -->L(2,:)=L(2,:)-L(1,:)*L(2,1) L = 1. 0.2347826 0.1652174 4.9130435 0. 12.66087 - 1.4608696 2.3278261 19. 3. 6. 90.10 23/11/12
  11. 11. Sistema de equações lineares Solução por escalonamento da matriz Terceiro Passo: zerar o elemento (3,1) Substituir a linha 3, por ela mesma menos a linha 1 multiplicada pelo elemento (3,1) -->L(3,:)=L(3,:)-L(1,:)*L(3,1) L = 1. 0.2347826 0.1652174 4.9130435 0. 12.66087 - 1.4608696 2.3278261 0. - 1.4608696 2.8608696 - 3.347826111 23/11/12
  12. 12. Sistema de equações lineares Solução por escalonamento da matriz Quarto Passo: garantir que o elemento (2,2) tenha valor 1 Dividir toda a linha pelo elemento (2,2) -->L(2,:)=L(2,:)/L(2,2) L = 1. 0.2347826 0.1652174 4.9130435 0. 1. - 0.1153846 0.1838599 0. - 1.4608696 2.8608696 - 3.347826112 23/11/12
  13. 13. Sistema de equações lineares Solução por escalonamento da matriz Quinto Passo: zerar o elemento (1,2) Substituir a linha 1, por ela mesma menos a linha 2 multiplicada pelo elemento (1,2) -->L(1,:)=L(1,:)-L(2,:)*L(1,2) L = 1. 0. 0.1923077 4.8698764 0. 1. - 0.1153846 0.1838599 0. - 1.4608696 2.8608696 - 3.347826113 23/11/12
  14. 14. Sistema de equações lineares Solução por escalonamento da matriz Sexto Passo: zerar o elemento (3,2) Substituir a linha 3, por ela mesma menos a linha 2 multiplicada pelo elemento (3,2) -->L(3,:)=L(3,:)-L(2,:)*L(3,2) L = 1. 0. 0.1923077 4.8698764 0. 1. - 0.1153846 0.1838599 0. 0. 2.6923077 - 3.079230814 23/11/12
  15. 15. Sistema de equações lineares Solução por escalonamento da matriz Sétimo Passo: garantir que o elemento (3,3) tenha valor 1 Dividir toda a linha pelo elemento (3,3) -->L(3,:)=L(3,:)/L(3,3) L = 1. 0. 0.1923077 4.8698764 0. 1. - 0.1153846 0.1838599 0. 0. 1. - 1.143714315 23/11/12
  16. 16. Sistema de equações lineares Solução por escalonamento da matriz Oitavo Passo: zerar o elemento (1,3) Substituir a linha 1, por ela mesma menos a linha 3 multiplicada pelo elemento (1,3) -->L(1,:)=L(1,:)-L(3,:)*L(1,3) L = 1. 0. 0. 5.0898214 0. 1. - 0.1153846 0.1838599 0. 0. 1. - 1.143714316 23/11/12
  17. 17. Sistema de equações lineares Solução por escalonamento da matriz Nono Passo: zerar o elemento (2,3) Substituir a linha 1, por ela mesma menos a linha 3 multiplicada pelo elemento (2,3) -->L(2,:)=L(2,:)-L(3,:)*L(2,3) L = 1. 0. 0. 5.0898214 0. 1. 0. 0.0518929 0. 0. 1. - 1.143714317 23/11/12
  18. 18. Sistema de equações lineares Solução por escalonamento da matriz Algoritmo: 1.Dividir toda a linha pelo elemento (1,1) 2.Substituir a linha 2, por ela mesma menos a linha 1 multiplicada pelo elemento (2,1) 3.Substituir a linha 3, por ela mesma menos a linha 1 multiplicada pelo elemento (3,1) 4.Dividir toda a linha pelo elemento (2,2) 5.Substituir a linha 1, por ela mesma menos a linha 2 multiplicada pelo elemento (1,2) 6.Substituir a linha 3, por ela mesma menos a linha 2 multiplicada pelo elemento (3,2) 7.Dividir toda a linha pelo elemento (3,3) 8.Substituir a linha 1, por ela mesma menos a linha 3 multiplicada pelo elemento (1,3) 9.Substituir a linha 1, por ela mesma menos a linha 3 multiplicada pelo elemento (2,3)18 23/11/12
  19. 19. Sistema de equações lineares Solução por escalonamento da matriz Algoritmo Versão codificada para Scilab 1.L(1,:)=L(1,:)/L(1,1) 2.L(2,:)=L(2,:)-L(1,:)*L(2,1) 3.L(3,:)=L(3,:)-L(1,:)*L(3,1) 4.L(2,:)=L(2,:)/L(2,2) 5.L(1,:)=L(1,:)-L(2,:)*L(1,2) 6.L(3,:)=L(3,:)-L(2,:)*L(3,2) 7.L(3,:)=L(3,:)/L(3,3) 8.L(1,:)=L(1,:)-L(3,:)*L(1,3) 9.L(2,:)=L(2,:)-L(3,:)*L(2,3)19 23/11/12
  20. 20. Sistema de equações lineares Solução por escalonamento da matriz Programa Versão Sistema 3 equações e 3 variáveis i=1 tamanho=3 while i<=tamanho j=1 while j<=tamanho if L(i,i)<>1 then L(i,:)=L(i,:)/L(i,i) end if i<>j then L(j,:)=L(j,:)-(L(j,i)*L(i,:)) end j=j+1 end i=i+1 end20 23/11/12

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