Stability of the Bruhat decomposition
Date
1997
Authors
Odeh, Omar Hasan
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Abstract
For any nonsingular n x n matrix A there exists a decomposition Ab= VIIU, where II is a unique permutation matrix and V,U are upper triangular matrices. When ITTVII is lower triangular and U has unit diagonal, such a decomposition is called the left Bruhat decomposition of A. The Bruhat decomposition, known from the theory of linear algebraic groups, was discussed by Kolotilina and Y eremin in the context of solving large sparse linear systems. We give an algorithm for computing the left Bruhat decomposition and consider its application to solving non-sparse linear systems.
The numerical stability of an algorithm for computing a matrix factorization depends on the growth of elements in the derived matrices. For example, for Gaussian elimination with partial pivoting (GEPP), the growth of elements in the derived matrices is measured using the growth factor YoP = maxla;~I/ maxia;,I, where A(i) = [a;n is the ith l,J,I; J,k derived matrix. GEPP is well-known to be unstable ( YoP is exponential inn) for classes of matrices given by Wilkinson and recently (from a practical application) by Foster. We show that the left Bruhat decomposition gives at most linear growth inn for these classes, demonstrating its stability.
In order to obtain a practical, general-purpose algorithm for solving non-sparse linear systems, we specify a partial pivoting strategy for the Bruhat decomposition (BDPP). Simple relationships are determined between the derived matrices, permutation matrices and the matrices of multipliers associated with BDPP (applied to matrix A) and those of GEPP (applied to AT p, where p is the permutation matrix that reverses the order of the columns). All real matrices that give the maximum exponential growth factor of 2•-1 with GEPP were characterized by Higham and Higham. We show that BDPP gives a growth factor of at most 2 for these matrices. Thus BDPP is stable for matrices that give the maximum exponential growth factor with GEPP, as well as for the matrix of Foster. For linear systems in which GEPP is unstable, application of BDPP is a practical alternative.
We introduce a bipartite graph model for the Bruhat decomposition of a pattern, and we develop an algorithm for computing the patterns of the factors in the left Brubat decomposition. This algorithm is shown to model the corresponding numerical algorithm.