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Unformatted text preview: Then, letting v = Vc
we write (9.4.3) as w = Wd (9.4.7) (r(0) ; AVc Wd) = 0: The requirement that this result hold for all w 2 W is equivalent to it holding for all
possible choices of d. For this to occur, we must have
(r(0) ; AVc W) = 0: 58 Solution Techniques for Elliptic Problems When the inner product corresponds to (9.4.2c), we have WT (r(0) ; AVc) = 0:
Thus, c satis es WT AVc = WT r(0) :
Assuming that the M M matrix WT AV is invertible, we nd
c = (WT AV);1 WT r(0) (9.4.8a) ~
x = x(0) + V(WT AV);1WT r(0) : (9.4.8b) and, using (9.4.2a) and (9.3.5c),
Let's look at two methods where the subspaces V and W are one dimensional.
Example 9.4.1. Let V = W = spanfv1 g where v1 is an N -vector and choose v1 = r(0) .
Then, using (9.4.8)
(0) )T (0)
c = (r T r (0)
(r(0) ) Ar
x = x(0) + cr(0) :
We recognize this as being one step of the steepest descent algorithm (Figure 9.3.1).
Example 9.4.2. In minimum residual (MR) iteration, we select V = spanfv1g = r(0)
and W = spanfw1g = Ar(0) . Using (9.4.8)
(0) )T T (0)
c = (r T A r (0)
(r(0) ) AT Ar
x = x(0) + cr(0) :
The matrix A need not be symmetric but only be positive de nite for MR to converge
( 4], Section 5.3).
Many iterative techniques for symmetric and non-symmetric matrices utilize Krylov
subspaces for the trial space V . A Krylov subspace of dimension M is KM (A r(0) ) = spanfr(0) Ar(0) A2r(0) : : : AM ;1r(0) g: (9.4.9) 9.4. Krylov Subspace Methods 59 It seems clear that vectors v 2 KM have the form v = p(A)r(0) where p(A) is a polynomial
of degree not exceeding M ; 1 in A. Methods using Krylov subspaces as trial spaces
are called Krylov subspace methods or simply Krylov methods. Di erent approaches are
distinguished by their choice of the test space W , with the two most popular choices
being KM and AKM . The generalized minimal residual (GMRES) method, in particular,
uses the latter choice. Thus, if we de ne the Krylov matrix KM = r(0) Ar(0) A2r(0) : : : AM r(0) ]
we could select VM = KM WM = AKM choose the M th solution iterate according to (9.4.2a) and (9.3.5c) as x(M ) = x(0) + VM c(M ) (9.4.10) and determine the M th-G...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14