Unformatted text preview: 9.1.7b) The algorithm (9.1.6,9.1.7) ignores embedded zeros within the band. Accounting for
these is typically not necessary with a direct solution method since, as will be discussed,
they become nonzero during the elimination process.
The reduction in time and space complexity is signi cant when p N . Approximate
operation counts for the factorization and forward and backward substitution phases of
the full and banded procedures are given in Table 9.1.1. In order to provide some meaning
to these estimates, consider the solution of a Dirichlet problem for Laplace's equation
on a square. With x = y the resulting algebraic system (8.1.4) has the form shown
in Figure 9.1.2. Using (8.1.2), reveals that the diagonal elements of A are unity and all
o -diagonal terms are ;1=4. Each block of A is (J ; 1) (J ; 1) and there are J ; 1
blocks. Thus, N = (J ; 1)2 and p = J ; 1. Using this data with the estimates shown in
Table 9.1.1 gives the approximate operation counts reported on the right of Table 9.1.1.
Even modest values of J indicate the impracticality of ignoring the sparsity present in
3 =3 Np(p + 1)
Factor J 6=3
Solve N 2 N (2p + 1)
Solve J 4
Table 9.1.1: Approximate operation counts for solving full and banded N N linear
systems having a bandwidth of 2p + 1 by Gaussian elimination (left). Approximate
operation counts when solving a Dirichlet problem for Laplace's equation on a J J
square mesh (right) which has N = (J ; 1)2 and p = J ; 1.
The factorization of A creates nonzero entries within the band. Hence, for the Laplacian operator, the storage needed for L and U using banded Gaussian elimination is
approximately 2J 3 , while the nonzero entries of A only require 5J 2 memory locations. 9.1. Direct Solution Methods 5 J
∆x ∆x 1 k=0
j=0 1 J Figure 9.1.2: Uniform square mesh (left) and structure of the corresponding matrix A
for the solution of Laplace's equation using centered nite di erences (8.1.2).
We will have to use iterativ...
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