The Core Problem within a Linear Approximation Problem
$AXapprox B$ with Multiple Right-Hand Sides

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The Core Problem within a Linear Approximation Problem $AXapprox B$ with Multiple Right-Hand Sides

Přehled o publikaci

J 2013

The Core Problem within a Linear Approximation Problem $AXapprox B$ with Multiple Right-Hand Sides

PLEŠINGER, Martin; Iveta HNĚTYNKOVÁ and Zdeněk STRAKOŠ

Basic information

Original name

The Core Problem within a Linear Approximation Problem $AXapprox B$ with Multiple Right-Hand Sides

Authors

PLEŠINGER, Martin (203 Czech Republic, belonging to the institution); Iveta HNĚTYNKOVÁ (203 Czech Republic) and Zdeněk STRAKOŠ (203 Czech Republic)

Edition

SIAM Journal on Matrix Analysis and Appliccations, 2013, 0895-4798

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

General mathematics

Country of publisher

United States of America

Confidentiality degree

is not subject to a state or trade secret

References:

URL

RIV identification code

RIV/46747885:24510/13:#0000992

Organization

Faculty of Science, Humanities and Education – Technical University of Liberec – Repository

DOI

https://doi.org/10.1137/120884237

UT WoS

000325092700004

Keywords in English

total least squares problem; multiple right-hand sides; core problem; linear approximation problem; error-in-variables modeling; orthogonal regression; singular value decomposition

Changed: 24/3/2015 20:24, Martin Plešinger

Abstract

In the original language

This paper focuses on total least squares (TLS) problems $AXapprox B$ 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 $Axapprox b$ 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.

Cite

PLEŠINGER, Martin; Iveta HNĚTYNKOVÁ and Zdeněk STRAKOŠ. The Core Problem within a Linear Approximation Problem $AXapprox B$ with Multiple Right-Hand Sides. SIAM Journal on Matrix Analysis and Appliccations. 2013, vol. 34, No 3, p. 917-931. ISSN 0895-4798. Available from: https://doi.org/10.1137/120884237.

@article{17954,
   author = {Plešinger, Martin and Hnětynková, Iveta and Strakoš, Zdeněk},
   article_number = {3},
   doi = {https://doi.org/10.1137/120884237},
   keywords = {total least squares problem; multiple right-hand sides; core problem; linear approximation problem; error-in-variables modeling; orthogonal regression; singular value decomposition},
   language = {eng},
   issn = {0895-4798},
   journal = {SIAM Journal on Matrix Analysis and Appliccations},
   title = {The Core Problem within a Linear Approximation Problem $AXapprox B$ with Multiple Right-Hand Sides},
   url = {http://epubs.siam.org/doi/abs/10.1137/120884237},
   volume = {34},
   year = {2013}
}

TY  - JOUR
ID  - 17954
AU  - Plešinger, Martin - Hnětynková, Iveta - Strakoš, Zdeněk
PY  - 2013
TI  - The Core Problem within a Linear Approximation Problem $AXapprox B$ with Multiple Right-Hand Sides
JF  - SIAM Journal on Matrix Analysis and Appliccations
VL  - 34
IS  - 3
SP  - 917-931
EP  - 917-931
SN  - 0895-4798
KW  - total least squares problem
KW  - multiple right-hand sides
KW  - core problem
KW  - linear approximation problem
KW  - error-in-variables modeling
KW  - orthogonal regression
KW  - singular value decomposition
UR  - http://epubs.siam.org/doi/abs/10.1137/120884237
N2  - This paper focuses on total least squares (TLS) problems $AXapprox B$ 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 $Axapprox b$ 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.
ER  -

PLEŠINGER, Martin; Iveta HNĚTYNKOVÁ and Zdeněk STRAKOŠ. The Core Problem within a Linear Approximation Problem \${}AXapprox B\${} with Multiple Right-Hand Sides. \textit{SIAM Journal on Matrix Analysis and Appliccations}. 2013, vol.~34, No~3, p.~917-931. ISSN~0895-4798. Available from: https://doi.org/10.1137/120884237.