THE CORE PROBLEM WITHIN A LINEAR APPROXIMATION PROBLEM AX ~ B WITH MULTIPLE RIGHT-HAND SIDES

• ID: 2588, RIV: 10159151
• ISSN: 0895-4798, ISBN: not specified
• source: SIAM Journal on Matrix Analysis and Applications
• keywords: total least squares problem; multiple right-hand sides; core problem; linear approximation problem; error-in-variables modeling; orthogonal regression; singular value decomposition
• authors: Iveta Hnětynková, Martin Plesinger, Zdeněk Strakoš
• authors from KNM: Strakoš Zdeněk, Hnětynková Iveta

Abstract

This paper focuses on total least squares (TLS) problems AX ~ B with multiple right-hand sides. Existence and uniqueness of a TLS solution for such problems was analyzed in the paper [I. Hnetynkova et al., SIAM J. Matrix Anal. Appl., 32, 2011, pp. 748-770]. For TLS problems with single right-hand sides the paper [C. C. Paige and Z. Strakos, SIAM J. Matrix Anal. Appl., 27, 2006, pp. 861-875] showed how necessary and sufficient information for solving Ax ~ b can be revealed from the original data through the so-called core problem concept. In this paper we present a theoretical study extending this concept to problems with multiple right-hand sides. The data reduction we present here is based on the singular value decomposition of the system matrix A. We show minimality of the reduced problem; in this sense the situation is analogous to the single right-hand side case. Some other properties of the core problem, however, cannot be extended to the case of multiple right-hand sides.