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Unformatted text preview: norm was less than 10;7. The value of ! = 2=(1 + h) was
used with the SOR method. Results are shown in Table 9.3.1. The two methods are
comparable. Apparent convergence is at a linear rate in h, i.e., doubling h approximately 9.3. Conjugate Gradient Methods 39 doubles the number of SOR and conjugate gradient iterations to convergence. Since an
SOR iteration is less costly than a conjugate gradient step, we may infer that the SOR
procedure is the faster.
1=h SOR CG
10 31 27
20 64 54
40 122 107
Table 9.3.1: Number of SOR and conjugate gradient iterations to convergence for Example 9.3.2.
The following theorem con rms the ndings of the previous example. Theorem 9.3.3. Let A be a symmetric and positive de nite matrix, then iterates of the
conjugate gradient method satisfy kx( ) ; xk A where k kA and 2 p ;1
2 p 2 + 1 kx(0) ; xkA
2 (9.3.14) were de ned in (9.3.4a, 9.3.4c). Proof. cf. 4], Section 6.11. Examining (9.3.14) and (9.3.4c), we see that convergence is fastest when the eigenvalues of A are clustered together, i.e., when 2 (A) 1.
Example 9.3.3. The factor
R = p 2+1
2 determines the convergence rate of the conjugate gradient method. Since the condition
number 2 depends on the eigenvalues of A, it would seem that we have to examine the
Aq = q:
Using (9.2.8a), let us write this relation as
(D ; L ; U)q = q 40 Solution Techniques for Elliptic Problems where D, L, and U were de ned by (9.2.8b-9.2.8c). Multiplying by D;1 and using
(9.2.10b), we have
MJ q = (I ; D;1 )q
where MJ is the Jacobi iteration matrix. For the Laplacian operator, D = I and we have
= 1 ; , where is an eigenvalue of MJ . Still con ning our attention to the Laplacian
operator, we may use (9.2.17d) to evaluate and, hence, obtain n
= 1 ; = 4 x sin2 mJ + 4 y sin2 2K
2 m = 1 2 ::: J ;1 n = 1 2 : : : K ; 1: For simplicity, let us focus on a square grid (J = K ) where
= sin2 mJ + sin2 nJ
2 m n = 1 2 : : : J ; 1: The smallest eigenvalue occurs with m = n = 1 and the largest occurs with m = n =
J ; 1 thus,
2 (J ; 1) :
min = 2 sin
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14