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# The Core Problem within a Linear Approximation Problem \$AX/approx B\$ with Multiple Right-Hand Sides

1. 1. 0426569 - UIVT-O 2014 RIV US eng J - Článek v odborném periodiku
Hnětynková, Iveta - Plešinger, Martin - Strakoš, Z.
The Core Problem within a Linear Approximation Problem \$AX/approx B\$ with Multiple Right-Hand Sides.
SIAM Journal on Matrix Analysis and Applications. Roč. 34, č. 3 (2013), s. 917-931. ISSN 0895-4798
Grant CEP: GA ČR GA13-06684S
Grant ostatní:GA ČR(CZ) GA201/09/0917; GA MŠk(CZ) EE2.3.09.0155; GA MŠk(CZ) EE2.3.30.0065
Program:GA
Institucionální podpora: RVO:67985807
Klíčová slova: total least squares problem * multiple right-hand sides * core problem * linear approximation problem * error-in-variables modeling * orthogonal regression * singular value decomposition
Kód oboru RIV: BA - Obecná matematika
Impakt faktor: 1.806, rok: 2013

This paper focuses on total least squares (TLS) problems \$AX/approx B\$ with multiple right-hand sides. Existence and uniqueness of a TLS solution for such problems was analyzed in the paper [I. Hnětynková 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. Strakoš, SIAM J. Matrix Anal. Appl., 27, 2006, pp. 861--875] showed how necessary and sufficient information for solving \$Ax/approx 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.