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Unformatted text preview: MRES iterate from (9.4.8a) as c(M ) = (KT AT AKM );1 KT AT r(0) :
M
We see that:
1. The GMRES procedure is the extension of the MR procedure to higherdimensional
spaces.
2. The dimension of the Krylov space increases by one after each iteration.
3. As may be expected from our earlier work, x(M ) minimizes kr(M )k2 = kb ; Ax(M ) k2
Unfortunately, the procedure just outlined is unstable (sensitive to roundo error
accumulation). It will be necessary to select a more well conditioned basis for KM and
we'll do this by constructing a set of mutually orthogonal vectors. This orthogonal
projection onto KM is usually done by Arnoldi's method and a pseudocode procedure for
its construction appears in Figure 9.4.2. The procedure begins with an initial vector v1 60 Solution Techniques for Elliptic Problems which is the initial residual r(0) normalized to have a unit Euclidean length. Successive
vectors vj are chosen to be orthogonal to the previous ones and to have unit lengths.
Indeed, the algorithm of Figure 9.4.2 is just classical GramSchmidt orthogonalization
modi ed by the presence of A to produce an orthonormal basis for the Krylov subspace KM (A v1) = spanfv1 Av1 A2v1 : : : AM ;1v1 g: (9.4.11) The algorithm will halt prematurely if any unnormalized vector wj has a zero Euclidean
length. The Boolean variable quit is set to true should this happen. Proving these
assertions may be done with an induction argument ( 4], Section 6.3), but we'll proceed
less formally. At the rst (j = 1) step of the algorithm, we obtain procedure arnoldi
quit = (kr(0) k2 = 0)
if not quit then
v1 = r(0) =kr(0) k2
end if
j=1 while (j M ) and (not quit) do
wj = Avj
for i = 1 to j do
hij = viT Avj
wj = wj ; hij vi
end for
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 Figure 9.4.2: Arnoldi GramSchmidt orthogonal basis construction for KM . h21 v2 = w1 = Av1 ; h11 v1 :
Taking an inner product with v1
T
T
T
h21 v1 v2 = v1 Av1 ; h11 v1 v1: 9.4. Krylov Subspace Methods 61 T
T...
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 Spring '14
 JosephE.Flaherty

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