QRT: A Qr-based tridiagonalization algorithm for nonsymmetric matrices

Sidje, R. B. and Burrage, K. (2005) QRT: A Qr-based tridiagonalization algorithm for nonsymmetric matrices. Siam Journal On Matrix Analysis And Applications, 26 3: 878-900. doi:10.1137/040612476

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Author Sidje, R. B.
Burrage, K.
Title QRT: A Qr-based tridiagonalization algorithm for nonsymmetric matrices
Journal name Siam Journal On Matrix Analysis And Applications   Check publisher's open access policy
ISSN 0895-4798
Publication date 2005
Sub-type Article (original research)
DOI 10.1137/040612476
Open Access Status File (Publisher version)
Volume 26
Issue 3
Start page 878
End page 900
Total pages 23
Editor Henk A van der Vorst
Place of publication Philadelphia, PA, United States
Publisher Society for Industrial and Applied Mathematics
Collection year 2005
Language eng
Abstract The stable similarity reduction of a nonsymmetric square matrix to tridiagonal form has been a long-standing problem in numerical linear algebra. The biorthogonal Lanczos process is in principle a candidate method for this task, but in practice it is confined to sparse matrices and is restarted periodically because roundoff errors affect its three-term recurrence scheme and degrade the biorthogonality after a few steps. This adds to its vulnerability to serious breakdowns or near-breakdowns, the handling of which involves recovery strategies such as the look-ahead technique, which needs a careful implementation to produce a block-tridiagonal form with unpredictable block sizes. Other candidate methods, geared generally towards full matrices, rely on elementary similarity transformations that are prone to numerical instabilities. Such concomitant difficulties have hampered finding a satisfactory solution to the problem for either sparse or full matrices. This study focuses primarily on full matrices. After outlining earlier tridiagonalization algorithms from within a general framework, we present a new elimination technique combining orthogonal similarity transformations that are stable. We also discuss heuristics to circumvent breakdowns. Applications of this study include eigenvalue calculation and the approximation of matrix functions.
Keyword Matrix Reduction
Nonsymmetric Tridiagonalization
Mathematics, Applied
General Matrix
Efficient Computation
Q-Index Code C1

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Created: Wed, 15 Aug 2007, 06:10:43 EST