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

http://dx.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

V originále

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://dx.doi.org/10.1137/120884237.

@article{17954,
   author = {Plešinger, Martin and Hnětynková, Iveta and Strakoš, Zdeněk},
   article_number = {3},
   doi = {http://dx.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://dx.doi.org/10.1137/120884237.