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Iterative Techniques in Matrix Algebra Jacobi & Gauss-Seidel Iterative Techniques II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University 2011

  1. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form Re-Writing the Equations To write the Gauss-Seidel method in matrix form, Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 10 / 38

  2. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form Re-Writing the Equations To write the Gauss-Seidel method in matrix form, multiply both sides of   i − 1 n = 1 x ( k ) � ( a ij x ( k ) � ( a ij x ( k − 1 )  − ) − ) + b i  i j j a ii j = 1 j = i + 1 by a ii and collect all k th iterate terms, Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 10 / 38

  3. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form Re-Writing the Equations To write the Gauss-Seidel method in matrix form, multiply both sides of   i − 1 n = 1 x ( k ) � ( a ij x ( k ) � ( a ij x ( k − 1 )  − ) − ) + b i  i j j a ii j = 1 j = i + 1 by a ii and collect all k th iterate terms, to give a i 1 x ( k ) + a i 2 x ( k ) + · · · + a ii x ( k ) = − a i , i + 1 x ( k − 1 ) − · · · − a in x ( k − 1 ) + b i n 1 2 i i + 1 for each i = 1 , 2 , . . . , n . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 10 / 38

  4. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form Re-Writing the Equations (Cont’d) Writing all n equations gives a 11 x ( k ) − a 12 x ( k − 1) − a 13 x ( k − 1) − · · · − a 1 n x ( k − 1) + b 1 = n 1 2 3 a 21 x ( k ) a 22 x ( k ) − a 23 x ( k − 1) − · · · − a 2 n x ( k − 1) + b 2 + = n 1 2 3 . . . a n 1 x ( k ) a n 2 x ( k ) + · · · + a nn x ( k ) b n + = n 1 2 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 11 / 38

  5. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form Re-Writing the Equations (Cont’d) Writing all n equations gives a 11 x ( k ) − a 12 x ( k − 1) − a 13 x ( k − 1) − · · · − a 1 n x ( k − 1) + b 1 = n 1 2 3 a 21 x ( k ) a 22 x ( k ) − a 23 x ( k − 1) − · · · − a 2 n x ( k − 1) + b 2 + = n 1 2 3 . . . a n 1 x ( k ) a n 2 x ( k ) + · · · + a nn x ( k ) b n + = n 1 2 With the definitions of D , L , and U given previously, we have the Gauss-Seidel method represented by ( D − L ) x ( k ) = U x ( k − 1 ) + b Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 11 / 38

  6. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form ( D − L ) x ( k ) = U x ( k − 1 ) + b Re-Writing the Equations (Cont’d) Solving for x ( k ) finally gives x ( k ) = ( D − L ) − 1 U x ( k − 1 ) + ( D − L ) − 1 b , for each k = 1 , 2 , . . . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 12 / 38

  7. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form ( D − L ) x ( k ) = U x ( k − 1 ) + b Re-Writing the Equations (Cont’d) Solving for x ( k ) finally gives x ( k ) = ( D − L ) − 1 U x ( k − 1 ) + ( D − L ) − 1 b , for each k = 1 , 2 , . . . Letting T g = ( D − L ) − 1 U and c g = ( D − L ) − 1 b , gives the Gauss-Seidel technique the form x ( k ) = T g x ( k − 1 ) + c g Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 12 / 38

  8. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation The Gauss-Seidel Method: Matrix Form ( D − L ) x ( k ) = U x ( k − 1 ) + b Re-Writing the Equations (Cont’d) Solving for x ( k ) finally gives x ( k ) = ( D − L ) − 1 U x ( k − 1 ) + ( D − L ) − 1 b , for each k = 1 , 2 , . . . Letting T g = ( D − L ) − 1 U and c g = ( D − L ) − 1 b , gives the Gauss-Seidel technique the form x ( k ) = T g x ( k − 1 ) + c g For the lower-triangular matrix D − L to be nonsingular, it is necessary and sufficient that a ii � = 0, for each i = 1 , 2 , . . . , n . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 12 / 38

  9. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Outline The Gauss-Seidel Method 1 The Gauss-Seidel Algorithm 2 Convergence Results for General Iteration Methods 3 Application to the Jacobi & Gauss-Seidel Methods 4 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 13 / 38

  10. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (1/2) To solve A x = b given an initial approximation x ( 0 ) : Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 14 / 38

  11. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (1/2) To solve A x = b given an initial approximation x ( 0 ) : INPUT the number of equations and unknowns n ; the entries a ij , 1 ≤ i , j ≤ n of the matrix A ; the entries b i , 1 ≤ i ≤ n of b ; the entries XO i , 1 ≤ i ≤ n of XO = x ( 0 ) ; tolerance TOL ; maximum number of iterations N . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 14 / 38

  12. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (1/2) To solve A x = b given an initial approximation x ( 0 ) : INPUT the number of equations and unknowns n ; the entries a ij , 1 ≤ i , j ≤ n of the matrix A ; the entries b i , 1 ≤ i ≤ n of b ; the entries XO i , 1 ≤ i ≤ n of XO = x ( 0 ) ; tolerance TOL ; maximum number of iterations N . the approximate solution x 1 , . . . , x n or a message OUTPUT that the number of iterations was exceeded. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 14 / 38

  13. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (2/2) Step 1 Set k = 1 While ( k ≤ N ) do Steps 3–6: Step 2 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38

  14. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (2/2) Step 1 Set k = 1 While ( k ≤ N ) do Steps 3–6: Step 2 Step 3 For i = 1 , . . . , n   i − 1 n set x i = 1 � �  − a ij x j − a ij XO j + b i  a ii j = 1 j = i + 1 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38

  15. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (2/2) Step 1 Set k = 1 While ( k ≤ N ) do Steps 3–6: Step 2 Step 3 For i = 1 , . . . , n   i − 1 n set x i = 1 � �  − a ij x j − a ij XO j + b i  a ii j = 1 j = i + 1 If || x − XO || < TOL then OUTPU T ( x 1 , . . . , x n ) Step 4 ( The procedure was successful ) STOP Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38

  16. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (2/2) Step 1 Set k = 1 While ( k ≤ N ) do Steps 3–6: Step 2 Step 3 For i = 1 , . . . , n   i − 1 n set x i = 1 � �  − a ij x j − a ij XO j + b i  a ii j = 1 j = i + 1 If || x − XO || < TOL then OUTPU T ( x 1 , . . . , x n ) Step 4 ( The procedure was successful ) STOP Step 5 Set k = k + 1 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38

  17. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (2/2) Step 1 Set k = 1 While ( k ≤ N ) do Steps 3–6: Step 2 Step 3 For i = 1 , . . . , n   i − 1 n set x i = 1 � �  − a ij x j − a ij XO j + b i  a ii j = 1 j = i + 1 If || x − XO || < TOL then OUTPU T ( x 1 , . . . , x n ) Step 4 ( The procedure was successful ) STOP Step 5 Set k = k + 1 Step 6 For i = 1 , . . . , n set XO i = x i Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38

  18. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm (2/2) Step 1 Set k = 1 While ( k ≤ N ) do Steps 3–6: Step 2 Step 3 For i = 1 , . . . , n   i − 1 n set x i = 1 � �  − a ij x j − a ij XO j + b i  a ii j = 1 j = i + 1 If || x − XO || < TOL then OUTPU T ( x 1 , . . . , x n ) Step 4 ( The procedure was successful ) STOP Step 5 Set k = k + 1 Step 6 For i = 1 , . . . , n set XO i = x i Step 7 OUTPUT (‘Maximum number of iterations exceeded’) ( The procedure was unsuccessful ) STOP Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38

  19. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm Comments on the Algorithm Step 3 of the algorithm requires that a ii � = 0, for each i = 1 , 2 , . . . , n . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38

  20. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm Comments on the Algorithm Step 3 of the algorithm requires that a ii � = 0, for each i = 1 , 2 , . . . , n . If one of the a ii entries is 0 and the system is nonsingular, a reordering of the equations can be performed so that no a ii = 0. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38

  21. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm Comments on the Algorithm Step 3 of the algorithm requires that a ii � = 0, for each i = 1 , 2 , . . . , n . If one of the a ii entries is 0 and the system is nonsingular, a reordering of the equations can be performed so that no a ii = 0. To speed convergence, the equations should be arranged so that a ii is as large as possible. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38

  22. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm Comments on the Algorithm Step 3 of the algorithm requires that a ii � = 0, for each i = 1 , 2 , . . . , n . If one of the a ii entries is 0 and the system is nonsingular, a reordering of the equations can be performed so that no a ii = 0. To speed convergence, the equations should be arranged so that a ii is as large as possible. Another possible stopping criterion in Step 4 is to iterate until � x ( k ) − x ( k − 1 ) � � x ( k ) � is smaller than some prescribed tolerance. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38

  23. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Gauss-Seidel Iterative Algorithm Comments on the Algorithm Step 3 of the algorithm requires that a ii � = 0, for each i = 1 , 2 , . . . , n . If one of the a ii entries is 0 and the system is nonsingular, a reordering of the equations can be performed so that no a ii = 0. To speed convergence, the equations should be arranged so that a ii is as large as possible. Another possible stopping criterion in Step 4 is to iterate until � x ( k ) − x ( k − 1 ) � � x ( k ) � is smaller than some prescribed tolerance. For this purpose, any convenient norm can be used, the usual being the l ∞ norm. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38

  24. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Outline The Gauss-Seidel Method 1 The Gauss-Seidel Algorithm 2 Convergence Results for General Iteration Methods 3 Application to the Jacobi & Gauss-Seidel Methods 4 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 17 / 38

  25. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Introduction To study the convergence of general iteration techniques, we need to analyze the formula x ( k ) = T x ( k − 1 ) + c , for each k = 1 , 2 , . . . where x ( 0 ) is arbitrary. Theorem on convergent The following lemma and the earlier matrices provide the key for this study. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 18 / 38

  26. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Lemma If the spectral radius satisfies ρ ( T ) < 1, then ( I − T ) − 1 exists, and ∞ ( I − T ) − 1 = I + T + T 2 + · · · = � T j j = 0 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 19 / 38

  27. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Lemma If the spectral radius satisfies ρ ( T ) < 1, then ( I − T ) − 1 exists, and ∞ ( I − T ) − 1 = I + T + T 2 + · · · = � T j j = 0 Proof (1/2) Because T x = λ x is true precisely when ( I − T ) x = ( 1 − λ ) x , we have λ as an eigenvalue of T precisely when 1 − λ is an eigenvalue of I − T . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 19 / 38

  28. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Lemma If the spectral radius satisfies ρ ( T ) < 1, then ( I − T ) − 1 exists, and ∞ ( I − T ) − 1 = I + T + T 2 + · · · = � T j j = 0 Proof (1/2) Because T x = λ x is true precisely when ( I − T ) x = ( 1 − λ ) x , we have λ as an eigenvalue of T precisely when 1 − λ is an eigenvalue of I − T . But | λ | ≤ ρ ( T ) < 1, so λ = 1 is not an eigenvalue of T , and 0 cannot be an eigenvalue of I − T . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 19 / 38

  29. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Lemma If the spectral radius satisfies ρ ( T ) < 1, then ( I − T ) − 1 exists, and ∞ ( I − T ) − 1 = I + T + T 2 + · · · = � T j j = 0 Proof (1/2) Because T x = λ x is true precisely when ( I − T ) x = ( 1 − λ ) x , we have λ as an eigenvalue of T precisely when 1 − λ is an eigenvalue of I − T . But | λ | ≤ ρ ( T ) < 1, so λ = 1 is not an eigenvalue of T , and 0 cannot be an eigenvalue of I − T . Hence, ( I − T ) − 1 exists. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 19 / 38

  30. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/2) Let S m = I + T + T 2 + · · · + T m Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 20 / 38

  31. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/2) Let S m = I + T + T 2 + · · · + T m Then ( I − T ) S m = ( 1 + T + T 2 + · · · + T m ) − ( T + T 2 + · · · + T m + 1 ) = I − T m + 1 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 20 / 38

  32. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/2) Let S m = I + T + T 2 + · · · + T m Then ( I − T ) S m = ( 1 + T + T 2 + · · · + T m ) − ( T + T 2 + · · · + T m + 1 ) = I − T m + 1 Theorem on convergent matrices and, since T is convergent, the implies that m →∞ ( I − T m + 1 ) = I m →∞ ( I − T ) S m = lim lim Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 20 / 38

  33. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/2) Let S m = I + T + T 2 + · · · + T m Then ( I − T ) S m = ( 1 + T + T 2 + · · · + T m ) − ( T + T 2 + · · · + T m + 1 ) = I − T m + 1 Theorem on convergent matrices and, since T is convergent, the implies that m →∞ ( I − T m + 1 ) = I m →∞ ( I − T ) S m = lim lim Thus, ( I − T ) − 1 = lim m →∞ S m = I + T + T 2 + · · · = � ∞ j = 0 T j Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 20 / 38

  34. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Theorem For any x ( 0 ) ∈ I R n , the sequence { x ( k ) } ∞ k = 0 defined by x ( k ) = T x ( k − 1 ) + c , for each k ≥ 1 converges to the unique solution of x = T x + c if and only if ρ ( T ) < 1. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 21 / 38

  35. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (1/5) First assume that ρ ( T ) < 1. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 22 / 38

  36. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (1/5) First assume that ρ ( T ) < 1. Then, T x ( k − 1 ) + c x ( k ) = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 22 / 38

  37. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (1/5) First assume that ρ ( T ) < 1. Then, T x ( k − 1 ) + c x ( k ) = T ( T x ( k − 2 ) + c ) + c = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 22 / 38

  38. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (1/5) First assume that ρ ( T ) < 1. Then, T x ( k − 1 ) + c x ( k ) = T ( T x ( k − 2 ) + c ) + c = T 2 x ( k − 2 ) + ( T + I ) c = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 22 / 38

  39. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (1/5) First assume that ρ ( T ) < 1. Then, T x ( k − 1 ) + c x ( k ) = T ( T x ( k − 2 ) + c ) + c = T 2 x ( k − 2 ) + ( T + I ) c = . . . T k x ( 0 ) + ( T k − 1 + · · · + T + I ) c = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 22 / 38

  40. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (1/5) First assume that ρ ( T ) < 1. Then, T x ( k − 1 ) + c x ( k ) = T ( T x ( k − 2 ) + c ) + c = T 2 x ( k − 2 ) + ( T + I ) c = . . . T k x ( 0 ) + ( T k − 1 + · · · + T + I ) c = Theorem on convergent matrices implies that T Because ρ ( T ) < 1, the is convergent, and k →∞ T k x ( 0 ) = 0 lim Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 22 / 38

  41. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/5) The previous lemma implies that   ∞ k →∞ T k x ( 0 ) + k →∞ x ( k ) � T j  c lim = lim  j = 0 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 23 / 38

  42. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/5) The previous lemma implies that   ∞ k →∞ T k x ( 0 ) + k →∞ x ( k ) � T j  c lim = lim  j = 0 0 + ( I − T ) − 1 c = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 23 / 38

  43. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/5) The previous lemma implies that   ∞ k →∞ T k x ( 0 ) + k →∞ x ( k ) � T j  c lim = lim  j = 0 0 + ( I − T ) − 1 c = ( I − T ) − 1 c = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 23 / 38

  44. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (2/5) The previous lemma implies that   ∞ k →∞ T k x ( 0 ) + k →∞ x ( k ) � T j  c lim = lim  j = 0 0 + ( I − T ) − 1 c = ( I − T ) − 1 c = Hence, the sequence { x ( k ) } converges to the vector x ≡ ( I − T ) − 1 c and x = T x + c . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 23 / 38

  45. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (3/5) R n , we have To prove the converse, we will show that for any z ∈ I lim k →∞ T k z = 0 . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 24 / 38

  46. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (3/5) R n , we have To prove the converse, we will show that for any z ∈ I lim k →∞ T k z = 0 . Again, by the theorem on convergent matrices, this is equivalent to ρ ( T ) < 1. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 24 / 38

  47. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (3/5) R n , we have To prove the converse, we will show that for any z ∈ I lim k →∞ T k z = 0 . Again, by the theorem on convergent matrices, this is equivalent to ρ ( T ) < 1. Let z be an arbitrary vector, and x be the unique solution to x = T x + c . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 24 / 38

  48. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (3/5) R n , we have To prove the converse, we will show that for any z ∈ I lim k →∞ T k z = 0 . Again, by the theorem on convergent matrices, this is equivalent to ρ ( T ) < 1. Let z be an arbitrary vector, and x be the unique solution to x = T x + c . Define x ( 0 ) = x − z , and, for k ≥ 1, x ( k ) = T x ( k − 1 ) + c . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 24 / 38

  49. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (3/5) R n , we have To prove the converse, we will show that for any z ∈ I lim k →∞ T k z = 0 . Again, by the theorem on convergent matrices, this is equivalent to ρ ( T ) < 1. Let z be an arbitrary vector, and x be the unique solution to x = T x + c . Define x ( 0 ) = x − z , and, for k ≥ 1, x ( k ) = T x ( k − 1 ) + c . Then { x ( k ) } converges to x . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 24 / 38

  50. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (4/5) Also, x − x ( k ) = ( T x + c ) − � T x ( k − 1 ) + c � � x − x ( k − 1 ) � = T Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 25 / 38

  51. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (4/5) Also, x − x ( k ) = ( T x + c ) − � T x ( k − 1 ) + c � � x − x ( k − 1 ) � = T so � x − x ( k − 1 ) � x − x ( k ) = T Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 25 / 38

  52. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (4/5) Also, x − x ( k ) = ( T x + c ) − � T x ( k − 1 ) + c � � x − x ( k − 1 ) � = T so � x − x ( k − 1 ) � x − x ( k ) = T T 2 � x − x ( k − 2 ) � = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 25 / 38

  53. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (4/5) Also, x − x ( k ) = ( T x + c ) − � T x ( k − 1 ) + c � � x − x ( k − 1 ) � = T so � x − x ( k − 1 ) � x − x ( k ) = T T 2 � x − x ( k − 2 ) � = . . = . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 25 / 38

  54. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (4/5) Also, x − x ( k ) = ( T x + c ) − � T x ( k − 1 ) + c � � x − x ( k − 1 ) � = T so � x − x ( k − 1 ) � x − x ( k ) = T T 2 � x − x ( k − 2 ) � = . . = . T k � x − x ( 0 ) � = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 25 / 38

  55. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (4/5) Also, x − x ( k ) = ( T x + c ) − � T x ( k − 1 ) + c � � x − x ( k − 1 ) � = T so � x − x ( k − 1 ) � x − x ( k ) = T T 2 � x − x ( k − 2 ) � = . . = . T k � x − x ( 0 ) � = T k z = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 25 / 38

  56. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (5/5) Hence k →∞ T k � x − x ( 0 ) � k →∞ T k z lim = lim Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 26 / 38

  57. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (5/5) Hence k →∞ T k � x − x ( 0 ) � k →∞ T k z lim = lim � x − x ( k ) � = lim k →∞ Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 26 / 38

  58. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (5/5) Hence k →∞ T k � x − x ( 0 ) � k →∞ T k z lim = lim � x − x ( k ) � = lim k →∞ = 0 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 26 / 38

  59. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Proof (5/5) Hence k →∞ T k � x − x ( 0 ) � k →∞ T k z lim = lim � x − x ( k ) � = lim k →∞ = 0 R n was arbitrary, so by the theorem on convergent But z ∈ I matrices, T is convergent and ρ ( T ) < 1. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 26 / 38

  60. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Corollary � T � < 1 for any natural matrix norm and c is a given vector, then the sequence { x ( k ) } ∞ k = 0 defined by x ( k ) = T x ( k − 1 ) + c converges, for any x ( 0 ) ∈ I R n , to a vector x ∈ I R n , with x = T x + c , and the following error bounds hold: Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 27 / 38

  61. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Corollary � T � < 1 for any natural matrix norm and c is a given vector, then the sequence { x ( k ) } ∞ k = 0 defined by x ( k ) = T x ( k − 1 ) + c converges, for any x ( 0 ) ∈ I R n , to a vector x ∈ I R n , with x = T x + c , and the following error bounds hold: (i) � x − x ( k ) � ≤ � T � k � x ( 0 ) − x � Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 27 / 38

  62. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Corollary � T � < 1 for any natural matrix norm and c is a given vector, then the sequence { x ( k ) } ∞ k = 0 defined by x ( k ) = T x ( k − 1 ) + c converges, for any x ( 0 ) ∈ I R n , to a vector x ∈ I R n , with x = T x + c , and the following error bounds hold: (i) � x − x ( k ) � ≤ � T � k � x ( 0 ) − x � � T � k 1 −� T � � x ( 1 ) − x ( 0 ) � (ii) � x − x ( k ) � ≤ Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 27 / 38

  63. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence Results for General Iteration Methods Corollary � T � < 1 for any natural matrix norm and c is a given vector, then the sequence { x ( k ) } ∞ k = 0 defined by x ( k ) = T x ( k − 1 ) + c converges, for any x ( 0 ) ∈ I R n , to a vector x ∈ I R n , with x = T x + c , and the following error bounds hold: (i) � x − x ( k ) � ≤ � T � k � x ( 0 ) − x � � T � k 1 −� T � � x ( 1 ) − x ( 0 ) � (ii) � x − x ( k ) � ≤ Corollary to The proof of the following corollary is similar to that for the the Fixed-Point Theorem for a single nonlinear equation. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 27 / 38

  64. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Outline The Gauss-Seidel Method 1 The Gauss-Seidel Algorithm 2 Convergence Results for General Iteration Methods 3 Application to the Jacobi & Gauss-Seidel Methods 4 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 28 / 38

  65. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Using the Matrix Formulations We have seen that the Jacobi and Gauss-Seidel iterative techniques can be written T j x ( k − 1 ) + c j x ( k ) = and T g x ( k − 1 ) + c g x ( k ) = Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 29 / 38

  66. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Using the Matrix Formulations We have seen that the Jacobi and Gauss-Seidel iterative techniques can be written T j x ( k − 1 ) + c j x ( k ) = and T g x ( k − 1 ) + c g x ( k ) = using the matrices T j = D − 1 ( L + U ) T g = ( D − L ) − 1 U and respectively. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 29 / 38

  67. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Using the Matrix Formulations We have seen that the Jacobi and Gauss-Seidel iterative techniques can be written T j x ( k − 1 ) + c j x ( k ) = and T g x ( k − 1 ) + c g x ( k ) = using the matrices T j = D − 1 ( L + U ) T g = ( D − L ) − 1 U and respectively. If ρ ( T j ) or ρ ( T g ) is less than 1, then the corresponding sequence { x ( k ) } ∞ k = 0 will converge to the solution x of A x = b . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 29 / 38

  68. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Example For example, the Jacobi method has x ( k ) = D − 1 ( L + U ) x ( k − 1 ) + D − 1 b , Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 30 / 38

  69. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Example For example, the Jacobi method has x ( k ) = D − 1 ( L + U ) x ( k − 1 ) + D − 1 b , and, if { x ( k ) } ∞ k = 0 converges to x , Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 30 / 38

  70. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Example For example, the Jacobi method has x ( k ) = D − 1 ( L + U ) x ( k − 1 ) + D − 1 b , and, if { x ( k ) } ∞ k = 0 converges to x , then x = D − 1 ( L + U ) x + D − 1 b Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 30 / 38

  71. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Example For example, the Jacobi method has x ( k ) = D − 1 ( L + U ) x ( k − 1 ) + D − 1 b , and, if { x ( k ) } ∞ k = 0 converges to x , then x = D − 1 ( L + U ) x + D − 1 b This implies that D x = ( L + U ) x + b and ( D − L − U ) x = b Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 30 / 38

  72. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Example For example, the Jacobi method has x ( k ) = D − 1 ( L + U ) x ( k − 1 ) + D − 1 b , and, if { x ( k ) } ∞ k = 0 converges to x , then x = D − 1 ( L + U ) x + D − 1 b This implies that D x = ( L + U ) x + b and ( D − L − U ) x = b Since D − L − U = A , the solution x satisfies A x = b . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 30 / 38

  73. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods The following are easily verified sufficiency conditions for convergence of the Jacobi and Gauss-Seidel methods. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 31 / 38

  74. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods The following are easily verified sufficiency conditions for convergence of the Jacobi and Gauss-Seidel methods. Theorem If A is strictly diagonally dominant, then for any choice of x ( 0 ) , both the Jacobi and Gauss-Seidel methods give sequences { x ( k ) } ∞ k = 0 that converge to the unique solution of A x = b . Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 31 / 38

  75. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Is Gauss-Seidel better than Jacobi? Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 32 / 38

  76. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Is Gauss-Seidel better than Jacobi? No general results exist to tell which of the two techniques, Jacobi or Gauss-Seidel, will be most successful for an arbitrary linear system. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 32 / 38

  77. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Is Gauss-Seidel better than Jacobi? No general results exist to tell which of the two techniques, Jacobi or Gauss-Seidel, will be most successful for an arbitrary linear system. In special cases, however, the answer is known, as is demonstrated in the following theorem. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 32 / 38

  78. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods (Stein-Rosenberg) Theorem If a ij ≤ 0, for each i � = j and a ii > 0, for each i = 1 , 2 , . . . , n , then one and only one of the following statements holds: (i) 0 ≤ ρ ( T g ) < ρ ( T j ) < 1 (ii) 1 < ρ ( T j ) < ρ ( T g ) (iii) ρ ( T j ) = ρ ( T g ) = 0 (iv) ρ ( T j ) = ρ ( T g ) = 1 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 33 / 38

  79. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods (Stein-Rosenberg) Theorem If a ij ≤ 0, for each i � = j and a ii > 0, for each i = 1 , 2 , . . . , n , then one and only one of the following statements holds: (i) 0 ≤ ρ ( T g ) < ρ ( T j ) < 1 (ii) 1 < ρ ( T j ) < ρ ( T g ) (iii) ρ ( T j ) = ρ ( T g ) = 0 (iv) ρ ( T j ) = ρ ( T g ) = 1 For the proof of this result, see pp. 120–127. of Young, D. M., Iterative solution of large linear systems, Academic Press, New York, 1971, 570 pp. Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 33 / 38

  80. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation Convergence of the Jacobi & Gauss-Seidel Methods Two Comments on the Thoerem For the special case described in the theorem, we see from part (i), namely 0 ≤ ρ ( T g ) < ρ ( T j ) < 1 Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 34 / 38

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