12 we will have to use iterative methods in order to

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Unformatted text preview: e methods in order to take full advantage of the sparsity in A. Nevertheless, some additional time and space savings are possible. For example, a symmetric, positive de nite matrix A may be factored as A = LDLT where 2 6 6 L=6 6 6 4 (9.1.8a) 3 2 1 d1 7 l21 1 7 6 d2 7 l31 l32 D=6 7 6 ... 4 ... ... . . . 7 5 dN lN 1 lN 2 1 Computing the product in (9.1.8a) using (9.1.8b) yields i;1 X2 di = aii ; dk lik 3 7 7: 7 5 k=1 i;1 1 (a ; X d l l ) lji = d ji k jk ik i k=1 j = i+1 i+2 ::: N (9.1.9a) i = 1 2 : : : N: (9.1.9b) The solution phase follows by substituting (9.1.8a) into (9.1.1) to get Ax = LDLT x = b: (9.1.8b) 6 Solution Techniques for Elliptic Problems Letting LT x = y Dy = z Lz = b: (9.1.10) The solution is obtained after forward, diagonal, and backward substitution steps, which have the scalar form i;1 X zi = bi ; lik zk i = 1 2 ::: N (9.1.11a) k=1 yi = zi=di xi = yi ; N X k=i+1 lkixk i = 1 2 ::: N i = N N ; 1 : : : 1: (9.1.11b) (9.1.11c) Banded versions of the factorization and solution steps can also be developed. The block tridiagonal algorithm 3] exploits the fact that A has a tridiagonal structure with entries that are matrices (cf. (8.1.4)). This too has (approximately) the same number of operations as banded Gaussian elimination (9.1.6, 9.1.7). A di erent ordering of the equations and unknowns, however, can signi cantly reduce ll-in of the band and, hence, the order of operations. Nested dissection, developed by George 1, 2], is known to be optimal in certain situations. The dissection process is illustrated for a 4 4 mesh in Figure 9.1.3. Alternate unknowns are eliminated rst to create the coarser mesh of \macro elements" shown at the right of Figure 9.1.3. Midside nodes of these macro elements are eliminated next to leave, in this case, a single unknown at the center of the domain (bottom). Although we will not describe how to do the dissection for a more general mesh and problem, one can visualize the process and essential idea. The structure of the linear systems obtained by using t...
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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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