3 but well proceed less formally at the rst j 1 step

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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 round-o error accumulation. The modi ed Gram-Schmidt procedure, as illustrated in Figure 9.4.3, is much 62 Solution Techniques for Elliptic Problems more stable and less sensitive to round-o error di culties. Mathematically, the steps in the two Arnoldi procedures are identical however, the modi ed Gram-Schmidt version avoids cancellations of nearly equal quantities. In cases of severe round-o 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 Gram-Schmidt 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.

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