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 llin 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.
 Spring '14
 JosephE.Flaherty

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