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Band Generalization of the Golub-Kahan Bidiagonalization, Generalized Jacobi Matrices, and the Core Problem
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SYSNO ASEP 0446564 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Band Generalization of the Golub-Kahan Bidiagonalization, Generalized Jacobi Matrices, and the Core Problem Author(s) Hnětynková, Iveta (UIVT-O) SAI, RID, ORCID
Plešinger, M. (CZ)
Strakoš, Z. (CZ)Source Title SIAM Journal on Matrix Analysis and Applications. - : SIAM Society for Industrial and Applied Mathematics - ISSN 0895-4798
Roč. 36, č. 2 (2015), s. 417-434Number of pages 18 s. Language eng - English Country US - United States Keywords total least squares problem ; multiple right-hand sides ; core problem ; Golub-Kahan bidiagonalization ; generalized Jacobi matrices Subject RIV BA - General Mathematics R&D Projects GA13-06684S GA ČR - Czech Science Foundation (CSF) UT WOS 000357407800004 EID SCOPUS 84936749626 DOI 10.1137/140968914 Annotation The concept of the core problem in total least squares (TLS) problems with single right-hand side introduced in [C. C. Paige and Z. Strakoš, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 861-875] separates necessary and sufficient information for solving the problem from redundancies and irrelevant information contained in the data. It is based on orthogonal transformations such that the resulting problem decomposes into two independent parts. One of the parts has nonzero right-hand side and minimal dimensions and it always has the unique TLS solution. The other part has trivial (zero) right-hand side and maximal dimensions. Assuming exact arithmetic, the core problem can be obtained by the Golub-Kahan bidiagonalization. Extension of the core concept to the multiple right-hand sides case $AX/approx B$ in [I. Hnětynková, M. Plešinger, and Z. Strakoš, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 917--931], which is highly nontrivial, is based on application of the singular value decomposition. In this paper we prove that the band generalization of the Golub-Kahan bidiagonalization proposed in this context by Björck also yields the core problem. We introduce generalized Jacobi matrices and investigate their properties. They prove useful in further analysis of the core problem concept. This paper assumes exact arithmetic. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2016
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