210a which has the form of 922 with bj d1b mj

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Unformatted text preview: ij b x(j ) + a i ii j =1 j 6=i aii i = 1 2 : : : N: (9.2.11) Example 9.2.2. The Jacobi method for the Poisson equation (8.1.2a) is ( Ujk +1) = () () () () x (Uj ;1 k + Uj +1 k ) + y (Uj k;1 + Uj k+1) + xy fjk j = 1 2 ::: J ; 1 k = 1 2 : : : K ; 1: (9.2.12) Updates to the solution at (j k) are computed as a weighted average of solutions at its four neighboring points. Contrary to solutions obtained by direct methods, parallel computational techniques are easily used with Jacobi's method since the solution state at iteration + 1 is explicit. It's easy to show that Jacobi iteration converges when A satis es some rather restrictive properties. De nition 9.2.2. A matrix A is strictly diagonally dominant if N X j =1 j 6=i jaij j < jaiij i = 1 2 : : : N: (9.2.13) 9.2. Basic Iterative Solution Methods 13 Theorem 9.2.3. The Jacobi iteration converges in the maximum norm when A is strictly diagonally dominant. Proof. If A is strictly diagonally dominant, we may use (9.2.8) and (9.2.10) to obtain kMJ k1 = 1max iN N X jaij j j =1 j 6=i jaii j < 1: Convergence of Jacobi's method is too slow for practical serial computation, although it may be used for parallel computation. Gauss-Seidel iteration uses the latest solution information as soon as it becomes available. Thus, when computing x(i +1) according to (9.2.11), we could use the latest iterates x(j +1) , j = 1 2 : : : i ; 1, on the right to obtain x(i +1) i;1 X aij N b ( +1) X aij ( ) xj ; a xj + a i =; ii j =i ii j =1 aii i = 1 2 : : : N: (9.2.14) In the matrix form of (9.2.9), this is equivalent to (D ; L)x( +1) = Ux( ) + b (9.2.15a) which has the form of (9.2.2) with MGS = (D ; L);1 U ^ bGS = (D ; L);1b: (9.2.15b) Example 9.2.3. The Gauss-Seidel iteration for the Poisson equation (8.1.2a) is ( Ujk +1) = ( +1) () ( +1) () x (Uj ;1 k + Uj +1 k ) + y (Uj k;1 + Uj k+1) + xy fjk j = 1 2 ::: J ; 1 k = 1 2 : : : K ; 1: (9.2.16) The solution process depends on the order in which the equations are written. As described above and as shown in Figure 9.2.1, row ordering has been assumed....
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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