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Unformatted text preview: However, v1 v1 = 1 and h11 = v1 Av1, so v1 is orthogonal to v2 . It is also clear that v2 is
a linear combination of v1 and Av1 and, hence, an element of K2(A v1). The extension
of the induction argument to higher values of j is similar.
From the for loop within Arnoldi's algorithm, we may infer hj+1 j vj+1 = Avj ; j
X or Avj = j +1
X
i=1 i=1 hij vi hij vi Let VM be the N M matrix whose
(M + 1) M Hessenberg matrix
2
h11
6 h21
6
HM = 6
6
6
4 j = 1 2 ::: M j = 1 2 : : : M: (9.4.12a) (9.4.12b) columns are v1 v2 : : : vM and let HM be the h12
h22
h32 ... h1M
h2M
h3M
... hM +1 M 3
7
7
7:
7
7
5 (9.4.13) Then (9.4.12) can be written in matrix form as AVM = VM +1HM : (9.4.14a) Additionally, let HM be the matrix obtained by deleting the last row of HM . Then,
either by using (9.4.12a) or (9.4.14a), we have AVM = VM HM + wM eT
M (9.4.14b) where, from Arnoldi's algorithm, wM = hM +1 M vM +1 and eM is the M th column of the
T
identity matrix. Finally, multiplying (9.4.14b) by VM and using the orthogonality of its
columns, we obtain
T
VM AVM = HM : (9.4.14c) The Arnoldi algorithm as stated in Figure 9.4.2 is subject to roundo error accumulation. The modi ed GramSchmidt procedure, as illustrated in Figure 9.4.3, is much 62 Solution Techniques for Elliptic Problems more stable and less sensitive to roundo error di culties. Mathematically, the steps in
the two Arnoldi procedures are identical however, the modi ed GramSchmidt version
avoids cancellations of nearly equal quantities. In cases of severe roundo error accumulation, Householder transformations can be used to construct the orthogonal basis. We
will not do this here, but interested readers may consult Saad 4], Section 6.3. procedure marnoldi
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 wj
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 9.4.3: Arnoldi modi ed GramSchmidt orthogonal basis construction for KM .
We are now in a position to improve the stability o...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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