15b the procedure fails if uii 0 i 1 2 n pivoting

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Unformatted text preview: ails if uii = 0, i = 1 2 : : : N . Pivoting may be necessary if this should occur. The above procedures ignore sparsity in A which, as we'll show, is not practical. As a rst step towards this end, let us consider the banded structure of A. De nition 9.1.1. A matrix A is called a band matrix of bandwidth p + q + 1 if aij = 0 for j > i + p and i > j + q . 9.1. Direct Solution Methods 3 p+1 q+1 A= 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 p+q+1 q+1 L= 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 p+1 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 U= Figure 9.1.1: Structure of a band matrix A of bandwidth p + q + 1 (left) and of its lower and upper triangular factors L (right) and U (bottom), respectively. Elements not in the shaded regions are zero. When using Gaussian elimination with a band matrix, it is easily shown that the factors L and U, respectively, have the structures of the lower and upper portions of A (Figure 9.1.1) thus, lij = 0 for i > j + q and j > i and uij = 0 for i > j and j > i + p. For a band matrix with q = p, the factorization (9.1.3) and forward and backward substitution (9.1.5) phases of Gaussian elimination become i;1 X uij = aij ; lik ukj j = i i +1 ::: i+p (9.1.6a) k=max(1 j ;p) i;1 1 (a ; X l u ) lji = u ji jk ki ii k=max(1 j ;p) j = i +1 i+2 ::: i+p i = 1 2 : : : N: (9.1.6b) 4 Solution Techniques for Elliptic Problems yi = bi ; i;1 X k=max(1 i;p) lik yk X 1 (y ; min(N i+p) u y ) xi = u i ik k ii k=i+1 i = 1 2 ::: N (9.1.7a) i = N N ; 1 : : : 1: (...
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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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