JACOBI S ITERATION METHOD

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1 ITERATION METHODS These are methods which compute a sequence of progressively accurate iterates to approximate the solution of Ax = b. We need such methods for solving many large linear systems. Sometimes the matrix is too large to be stored in the computer memory, making a direct method too difficult to use. More importantly, the operations cost of n for Gaussian elimination is too large for most large systems. With iteration methods, the cost can often be reduced to something of cost O ( n ) or less. Even when a special form for A can be used to reduce the cost of elimination, iteration will often be faster. There are other, more subtle, reasons, which we do not discuss here.

2 JACOBI S ITERATION METHOD We begin with an example. Consider the linear system 9x + x + x = b x + 0x + x = b x + 4x + x = b In equation #k, solve for x k : x = 9 [b x x x = 0 [b x x x = [b x 4x

3 [ Let x (0) = x (0), x (0), x (0) T be an initial guess to the solution x. Then define x (k+) = [ b x (k) x (k) 9 x (k+) = 0 x (k+) = [ b x (k) x (k) [ b x (k) 4x (k) for k = 0,,,.... This is called the Jacobi iteration method or the method of simultaneous replacements.

4 NUMERICAL EXAMPLE Let b = [0, 9, 0 T. The solution is x = [,, T. To measure the error, we use Error = x x (k) = max x i x (k) i i

5 k x (k) x (k) x (k) Error Ratio E E E E E E E E E E E E E 0.447

6 GAUSS-SEIDEL ITERATION METHOD Again consider the linear system and solve for x k in equation #k: 9x + x + x = b x + 0x + x = b x + 4x + x = b x = 9 [b x x x = 0 [b x x x = [b x 4x

7 Now immediately use every new iterate: x (k+) = [ b x (k) x (k) 9 x (k+) = 0 x (k+) = [ b x (k+) x (k) [ b x (k+) 4x (k+) for k = 0,,,.... This is called the Gauss-Seidel iteration method or the method of successive replacements.

8 NUMERICAL EXAMPLE Let b = [0, 9, 0 T. The solution is x = [,, T. To measure the error, we use Error = x x (k) = max x i x (k) i i k x (k) x (k) x (k) Error Ratio E E E E E E E The values of Ratio do not approach a limiting value with larger values of the iteration index k.

9 A GENERAL SCHEMA Rewrite Ax = b as Nx = b + Px () with A = N P a splitting of A. Choose N to be nonsingular. Usually we want Nz = f to be easily solvable for arbitray f. The iteration method is Nx (k+) = b + Px (k), k = 0,,,..., () EXAMPLE. Let N be the diagonal of A, and let P = N A. The iteration method is the Jacobi method: for k = 0,,.... a i,i x (k+) i = b i n j= j i a i,j x (k) j, i n

10 EXAMPLE. Let N be the lower triangular part of A, including its diagonal, and let P = N A. The iteration method is the Gauss-Seidel method: for k = 0,,..., i j= a i,j x (k+) j = b i n j=i+ a i,j x (k) j, i n EXAMPLE. Another method could be defined by letting N be the tridiagonal matrix formed from the diagonal, super-diagonal, and sub-diagonal of A, with P = N A: a, a, 0 0. a, a, a..,. N = an,n a n,n a n,n 0 0 a n,n a n,n Solving Nx (k+) = b + Px (k) uses the algorithm for tridiagonal systems from 6.4.

11 CONVERGENCE When does the iteration method () converge? Subtract () from (), obtaining ( N x x (k+)) = P (x x (k)) x x (k+) = N P (x x (k)) e (k+) = Me (k), M = N P with e (k) x x (k) Return now to the matrix and vector norms of 6.5. Then e (k+) e M (k), k 0 () Thus the error e (k) converges to zero if M <, with e (k) M k e (0), k 0

12 EXAMPLE For the earlier example with the Jacobi method, x (k+) = [ b x (k) x (k) 9 x (k+) = 0 x (k+) = M = [ b x (k) x (k) [ b x (k) 4x (k) M = 7. = This is consistent with the earlier table of values, although the actual convergence rate was better than predicted by ().

13 EXAMPLE For the earlier example with the Gauss-Seidel method, M = M = 0. x (k+) = [ b x (k) x (k) 9 x (k+) = 0 x (k+) = [ b x (k+) x (k) [ b x (k+) 4x (k+) = This too is consistent with the earlier numerical results. 99

14 DIAGONALLY DOMINANT MATRICES Matrices A for which a i,i > n a i,j, i =,..., n j= j i are called diagonally dominant.

15 JACOBI ITERATION For the Jacobi iteration method, M = 0 a, a, a,n a, a, a, 0 a,n a,..... a n, a n,n 0 a n,n a n,n With diagonally dominant matrices A, M = max i n n j= j i a i,j a i,i < (4) Thus the Jacobi iteration method for solving Ax = b is convergent.

16 GAUSS-SEIDEL ITERATION Assuming A is diagonally dominant, we can show that the Gauss-Seidel iteration will also converge. However, constructing M = N P is not reasonable for this method and an alternative approach is needed. Return to the error equation Ne (k+) = Pe (k) and write it in component form for the Gauss-Seidel method: i j= a i,j e (k+) j e (k+) i = i = j= n j=i+ a i,j e (k+) j a i,i a i,j e (k) j, i n n j=i+ a i,j a i,i e (k) j (5)

17 Introduce i α i = j= a i,j a i,i, β i = n j=i+ with α = β n = 0. Taking bounds in (5), a i,j a i,i, i n e (k+) i α i e (k+) + β i e (k), i =,..., n (6) Let l be an index for which e (k+) l = max Then using i = l in (6), Define η = max i i n e(k+) i = e (k+) e (k+) α l e (k+) + β l e (k) e (k+) β i α i. Then β l α l e (k) e (k+) η e (k)

18 For A diagonally dominant, it can be shown that η M (7) where M is for the definition of M for the Jacobi method, given earlier in (4) as n M = max a i,j i n = max (α i + β i ) < i n j= j i a i,i Consequently, for A diagonally dominant, the Gauss-Seidel method also converges and it does so more rapidly than the Jacobi method in most cases. Showing (7) follows by showing β i α i (α i + β i ) 0, i n For our earlier example with A of order, we have µ = 0.75 This is not as good as computing M directly for the Gauss-Seidel method, but it does show that the rate of convergence is better than for the Jacobi method.

19 CONVERGENCE: AN ADDENDUM Since M < is satisfied if N satisfies M = N P N P, N P < Using P = N A, this can be rewritten as A N < N We also want to choose N so that systems Nz = f is easily solvable. GENERAL CONVERGENCE THEOREM: Nx (k+) = b + Px (k), k = 0,,,..., will converge, for all right sides b and all initial guesses x (0), if and only if all eigenvalues λ of M = N P satisfy λ <. This is the basis of deriving other splittings A = N P that lead to convergent iteration methods.

20 RESIDUAL CORRECTION METHODS Let N be an invertible approximation of the matrix A; let x (0) x for the solution of Ax = b. Define r (0) = b Ax (0) Since Ax = b for the true solution x, r (0) = Ax Ax (0) = A(x x (0) ) = Ae (0) with e (0) = x x (0). Let ê (0) be the solution of and then define Repeat this process inductively. Nê (0) = r (0) x () = x (0) + ê (0)

21 RESIDUAL CORRECTION For k = 0,,..., define r (k) = b Ax (k) Nê (k) = r (k) x (k+) = x (k) + ê (k) This is the general residual correction method. To see how this fits into our earlier framework, proceed as follows: x (k+) = x (k) + ê (k) = x (k) + N r (k) = x (k) + N (b Ax (k) ) Thus, Nx (k+) = Nx (k) + b Ax (k) = b + (N A)x (k) = b + Px (k) Sometimes the residual correction scheme is a preferable way of approaching the development of an iterative method.

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