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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), GaussSeidel 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, GaussSeidel 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 GaussSeidel method. The
process begins by using the GaussSeidel 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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 Spring '14
 JosephE.Flaherty

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