The growth-factor bound for the Bunch-Kaufman factorization is tight

Alex Druinsky; Sivan Toledo

doi:10.1137/100801548

The growth-factor bound for the Bunch-Kaufman factorization is tight

Alex Druinsky, Sivan Toledo

School of Computer Science

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the growth-factor bound in the Bunch-Kaufman factorization method is essentially tight. The method factors a symmetric matrix A into A = P^TLDL^TP, where P is a permutation matrix, L is lower triangular, and D is block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The method uses one of several partial pivoting rules that ensure bounded in the elements of the reduced matrix and the factor D (growth in L is not bounded).We show that the exponential bound is essentially tight, thereby solving a question that has been open since 1977.

Original language	English
Pages (from-to)	928-937
Number of pages	10
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	32
Issue number	3
DOIs	https://doi.org/10.1137/100801548
State	Published - 2011

Keywords

Bunch-Kaufman factorization
Growth factor
Numerical stability
Symmetric indefinite matrices

All Science Journal Classification (ASJC) codes

Analysis

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10.1137/100801548

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abstract = "We show that the growth-factor bound in the Bunch-Kaufman factorization method is essentially tight. The method factors a symmetric matrix A into A = PTLDLTP, where P is a permutation matrix, L is lower triangular, and D is block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The method uses one of several partial pivoting rules that ensure bounded in the elements of the reduced matrix and the factor D (growth in L is not bounded).We show that the exponential bound is essentially tight, thereby solving a question that has been open since 1977.",

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AB - We show that the growth-factor bound in the Bunch-Kaufman factorization method is essentially tight. The method factors a symmetric matrix A into A = PTLDLTP, where P is a permutation matrix, L is lower triangular, and D is block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The method uses one of several partial pivoting rules that ensure bounded in the elements of the reduced matrix and the factor D (growth in L is not bounded).We show that the exponential bound is essentially tight, thereby solving a question that has been open since 1977.

KW - Bunch-Kaufman factorization

KW - Growth factor

KW - Numerical stability

KW - Symmetric indefinite matrices

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DO - 10.1137/100801548

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JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 3

ER -

The growth-factor bound for the Bunch-Kaufman factorization is tight

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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