The Growth-Factor Bound for the Bunch-Kaufman Factorization Is Tight | Zendy

Alex Druinsky | Zendy; Sivan Toledo | Zendy

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The Growth-Factor Bound for the Bunch-Kaufman Factorization Is Tight

Author(s) -

Alex Druinsky,

Sivan Toledo

Publication year - 2011

Publication title -

siam journal on matrix analysis and applications

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.268

H-Index - 101

eISSN - 1095-7162

pISSN - 0895-4798

DOI - 10.1137/100801548

Subject(s) - mathematics , triangular matrix , factorization , bounded function , diagonal , block matrix , matrix (chemical analysis) , combinatorics , upper and lower bounds , permutation (music) , exponential function , permutation matrix , diagonal matrix , matrix decomposition , pure mathematics , mathematical analysis , algorithm , geometry , physics , eigenvalues and eigenvectors , materials science , quantum mechanics , composite material , circulant matrix , acoustics , invertible matrix

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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