17a i n n 1 1 917b the algorithm 916917

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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 A. Full Banded Full Banded 3 =3 Np(p + 1) Factor N Factor J 6=3 J4 Solve N 2 N (2p + 1) Solve J 4 2J 3 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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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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