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Unformatted text preview: r( ) = (p( ) )TAp( ) = 0 6= : (9.3.13) Proof. This has essentially been proven by the prior developments. An algorithm for the conjugate gradient method is presented in Figure 9.3.3. Some
comments on the algorithm follow:
1. Equation (9.3.9b) was used to modify the expression (9.3.7) for . 2. The procedure requires storage for the nonzero elements of A and for x( ) , p( ) ,
and r( ) . An additional vector is needed to store the product Ap( ) . Thus, storage
costs remain modest relative to the direct methods of Section 9.1.
3. The procedure requires a matrix multiplication (Ap( ) ) and computation of two
inner products ((r( ) )T r( ) and (p( ) )T Ap( ) ) per step.
4. Unlike SOR methods, there are no acceleration parameters to determine. 38 Solution Techniques for Elliptic Problems procedure conjugate gradient
p(0) = r(0) = b ; Ax(0)
=0 while not converged do
= (r( ))T r( ) =(p( ) )T Ap( )
( +1) = x( ) + p( )
r( +1) = r( ) ; Ap( )
= (r( +1) )T r( +1) =(r( ))T r( )
p( +1) = r( +1) + p( )
= +1 end while Figure 9.3.3: Conjugate gradient algorithm.
The conjugate gradient method is both a direct and an iterative method as indicated
by the following theorem. Theorem 9.3.2. Let A be a positive de nite, symmetric, N N matrix. Then the
conjugate gradient method converges to the exact solution in no more than N steps. Proof. By Theorem 9.3.1, the residuals r( ) , = 0 1 : : : N ; 1, are mutually orthogonal.
Since the space is N -dimensional, the residual r(N ) must be zero hence, the method
converges in N steps. While convergence is achieved in N steps, the hope is to produce acceptable approximations of x in far fewer than N steps when N is large. Practically, convergence may
not be achieved in N steps when round-o errors are present.
Example 9.3.2 ( 5], Section 14.2). Consider the solution of Laplace's equation on a
square region with x = y = h. Let the Dirichlet boundary conditions be prescribed
so that the exact solution is
u(x y) = ex sin y:
Solutions were calculated using SOR and conjugate gradient iterations until the change
in the solution in the L2...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14