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Unformatted text preview: f the GMRES procedure. Again,
choose x(M ) according to (9.4.10) and calculate the residual b ; Ax(M ) = b ; A(x(0) ; VM c(M )) = r(0) ; AVM c(M ) :
Using (9.4.14) b ; Ax(M ) = v1 ; VM +1HM c(M ) = VM +1( e1 ; HM c(M ) ) (9.4.15a) where e1 is the rst column of the identity matrix and (Figure 9.4.3)
= kr(0) k2: (9.4.15b) 9.4. Krylov Subspace Methods 63 Since the GMRES approximation minimizes kb ; Ax(M ) k2, our task is to nd c(M ) as
the minimizer of kVM +1( e1 ; HM c(M ) )k2. Actually, since VM is orthogonal, it su ces
to minimize k e1 ; HM c(M ) k2. The approximate solution is then given by (9.4.10).
The minimizer c(M ) is computationally inexpensive. It requires the solution of an
(M + 1) M least-squares problem where M is typically small relative to N . An algorithm for the GMRES procedure appears in Figure 9.4.4. The procedure is basically
the Arnoldi-modi ed Gram-Schmidt algorithm with a least-squares procedure. Some
additional comments follow.
HM = 0
r(0) = b ; Ax(0)
= kr(0) k2
quit = ( = 0)
if not quit then
v1 = r(0) = end if
j=1 while (j M ) and (not quit) do
wj = Avj
for i = 1 to j do
hij = viT wj
wj = wj ; hij vi
hj+1 j = kwj k2
quit = (hj+1 j = 0)
if not quit then
vj+1 = wj =hj+1 j end if j =j+1 end while
m=j;1 Determine c(M ) as the minimizer of k e1 ; HM c(M ) k2
x(M ) = x(0) + VM c(M )
Figure 9.4.4: Generalized minimum residual algorithm.
1. As stated, the GMRES procedure does not calculate the solution at each step.
Thus, it is di cult to know when to stop. It would be better to calculate x(j) 64 Solution Techniques for Elliptic Problems
during the procedure and to check for convergence by, e.g., monitoring the size of
2. If the procedure terminates before completing M steps (quit = true), then x(j) is
the esact solution. This is the only way that the GMRES procedure can terminate
prematurely. Although shown, the least-squares solution need not be calculated in
3. The GMRES procedure becomes impractical because of the growth of memory and...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14