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Unformatted text preview: = 2 sin
2J Hence, using (9.3.4c)
When J 2
= sin (J 2; 1) =2J :
1 we may approximate this as p = sin(J ; 1) =2J
Thus, 2J : 1 ; =2J 1 ; :
1 + =2J
Convergence is, therefore, at the same rate as the SOR method (Example 9.2.9).
9.3.2 R Preconditioned Conjugate Gradient Iterations From Theorem 9.3.3, we see that the performance of the conjugate gradient method
improves when the eigenvalues of A are clustered about a point. This suggests the
possibility of preconditioning A by a positive de nite matrix M and solving M;1 Ax = M;1b: (9.3.15) 9.3. Conjugate Gradient Methods 41 If the eigenvalues of M;1 A were clustered, the conjugate gradient procedure may converge at a faster rate. The preconditioner M should be chosen to minimize the solution
time. There are, however, competing priorities. Thus, for example, the optimal choice of
M as far as clustering eigenvalues is concerned is M = A. This choice requires a direct
solution of the original system and, thus, has an extreme cost. The optimal choice of
M as far as computational e ort is concerned is M = I. This is the conjugate gradient
algorithm and, thus, no improvement has been provided. The search for the best preconditioning is still an active area of research with optimality dependent on many factors
including sparsity and intended computer architecture.
The preconditioning shown in (9.3.15) is called a left preconditioning. A right preconditioning is AM;1 w = b x = M;1 w: (9.3.16a) A symmetric preconditioning is C;1AC;T w = C;1 b x = C;T w (9.3.16b) where C;T denotes the transpose of C;1. The preconditioning matrix C need not be
symmetric since C;1 AC;T is symmetric and positive de nite when A is. The matrix C
may be a Cholesky factor of M, i.e., M = CCT : (9.3.16c) In this case, C would be lower triangular and it could be obtained from the symmetric
factorization of M given by (9.1.8, 9.1.9) or (9.1.12, 9.3.11).
The preconditioned conjugate gradient (PCG) algorithm with the symmetric preconditioning may be implemented by applying the conjugate gradient proced...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14