Unformatted text preview: ry e cient. It ignores sparsity and
does too much searching for levels. These conditions can be remedied. For example, the
search for levels does not depend on the values of the elements of A and can be separated
from the elimination process. It can, thus, be done in advance and be used for several
matrices having the same structure. However, because the algorithm is based on levels of
ll rather than numerical information, some large elements of A may be dropped because
of their locations rather than their sizes. This may result in a poor preconditioning and,
hence, require more iterations than necessary for convergence.
A threshold elimination procedure (ILUT) would identify small elements of A and set
them to zero during the factorization. At its simplest, this can be done by modifying the
algorithm of Figure 9.3.10 as follows: 54 Solution Techniques for Elliptic Problems
1. eliminate the de nition and updates of levij ,
2. replace the tests on levik
respectively, p and levij p by tests on aik 6= 0 and aij 6= 0, 3. insert a rule for dropping aik following its calculation (aik = aik =akk ), and
4. replace the statements within the second j loop by rules for dropping aij .
Rules for dropping elements are described by Saad 4], Section 10.4. Brie y, he suggests
1. dropping elements aik (Item 3 above) and aij (Item 4 above) that are less than a
tolerance relative to the size (e.g., the L2 norm) of row i, and
2. only retaining the p largest elements aij (Item 4 above) of row i without dropping
the diagonal element.
The goal of the second dropping rule is to control the number of elements per row
(or column) of L. The rst rule avoids retaining unnecessarily small elements in the
factorization. Problems 1. Consider the method of steepest descent and, using (9.3.1) and (9.3.2b), show that
E (y) = 2 (y ; x)T A(y ; x) ; 2 xT Ax:
2. Show that the SSOR procedure (9.3.25) has the form of (9.3.21) with M and N
given by (9.3.26).
3. Indicate the algorithm for solving (9.3.27c - 9.3.27e) when D and L correspond to
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