NDLI: Bruhat Decomposition and Numerical Stability

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Olesky, D. D. Odeh, O. H. van den Driessche, P.
Copyright Year	1998
Abstract	For a real nonsingular n-by-n matrix A, there exists a decomposition $A=V\Pi U$, where $\Pi$ is a permutation matrix and V,U are upper triangular matrices. When $\Pi^TV\Pi$ is lower triangular and U is normalized, such a decomposition is called the left Bruhat decomposition of A. An algorithm for computing the left Bruhat decomposition is given. For classes of matrices introduced by Wilkinson and recently (from a practical application) by Foster that have an exponential growth factor when Gaussian elimination with partial pivoting (GEPP) is applied, left Bruhat decomposition has at most linear growth. A partial pivoting strategy for Bruhat decomposition is also developed, and an explicit equivalence between GEPP and Bruhat decomposition with partial pivoting (BDPP) is derived. This equivalence implies that the growth factor for GEPP on A equals the growth factor for BDPP on $\rho A^T$, where $\rho$ is the permutation matrix that reverses the rows of $A^T$. BDPP is shown to give a growth factor of at most 2 when applied to any matrix for which GEPP gives the maximal growth factor of 2 n - 1.
Starting Page	89
Ending Page	98
Page Count	10
File Format	PDF
ISSN	08954798
DOI	10.1137/S0895479896303314
e-ISSN	10957162
Journal	SIAM Journal on Matrix Analysis and Applications (SJMAEL)
Issue Number	1
Volume Number	19
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-31
Access Restriction	Subscribed
Subject Keyword	growth factor Gaussian elimination Factorization of matrices Direct methods for linear systems and matrix inversion Bruhat decomposition partial pivoting numerical stability
Content Type	Text
Resource Type	Article
Subject	Analysis

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