218a or using 9215b in component form qjk xqj1

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Unformatted text preview: rs more di cult to analyze than the eigenvalue problem (9.2.17b) for the Jacobi method however, there is a transformation that simpli es things considerably. Let qjk = (j+k)=2rjk and substitute this relationship into (9.2.18b) to obtain (j +k+2)=2 r ; (j +k+1)=2 ( r (j +k+1)=2 ( r jk x j ;1 k + y rj k;1) = x j +1 k + y rj k+1): Dividing by the common factor yields 1=2 r jk = x (rj;1 k + rj+1 k ) + y (rj k;1 + rj k+1): This is the same eigenvalue problem as (9.2.17b) for Jacobi's method with replaced by 1=2 thus, using (9.2.17d) n = 2 = 1 ; 4 x sin2 mJ ; 4 y sin2 2K ]2 m = 1 2 ::: J ;1 2 n = 1 2 : : : K ; 1: (9.2.18c) In particular, (MGS ) = 2 (MJ ): (9.2.18d) Thus, according to (9.2.7b), Gauss-Seidel iterations converge twice as fast as Jacobi iterations. In the special case of Laplace's equation on a square mesh with large J we obtain the asymptotic approximation 2 (9.2.19) (MGS ) 1 ; J 2 : The results of Example 9.2.5 generalize as indicated by the following theorem. Theorem 9.2.4. Suppose that A satis es aij =aii < 0, i 6= j , then one and only one of the following conditions can occur: 1. (MJ ) = (MGS ) = 0, 2. 0 < (MGS ) < (MJ ) < 1, 18 Solution Techniques for Elliptic Problems 3. (MJ ) = (MGS ) = 1, or 4. 1 < (MJ ) < (MGS ). Proof. cf. 6], p. 70. In the important Case 2, Gauss-Seidel iterations always converge faster than Jacobi iterations. 9.2.2 Successive Over Relaxation \Relaxation" is a procedure that can accelerate the convergence rate of virtually any iteration. At present, it suits our purposes to apply it to the Gauss-Seidel method. The process begins by using the Gauss-Seidel method (9.2.14) to compute a \provisional" iterate i;1 n X aij ( +1) X aij ( ) bi ( +1) xi = ; a xj ; ^ xj + a (9.2.20a) ii j =1 ii j =i+1 aii and concludes with the nal iterate x(i +1) = !x(i +1) + (1 ; !)x(i ) ^ i = 1 2 : : : N: (9.2.20b) The acceleration parameter ! is to be chosen so that the iteration (9.2.20) converges as fast as possible. In particular, (9.2.20) is called Gaus...
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