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Article: Toward solution of matrix equation X=Af(X)B+C

TitleToward solution of matrix equation X=Af(X)B+C
Authors
KeywordsClosed form solutions
Conjugated and transpose
Iteration
Matrix equations
Numerical solution
Issue Date2011
PublisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laa
Citation
Linear Algebra and Its Applications, 2011, v. 435 n. 6, p. 1370-1398 How to Cite?
Abstract
This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation X=Af(X) B+C with f(X)=X T, f(X)=X and f(X)=X H, where X is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of W=AWB+C where the dimensions of the coefficient matrices A,B and C are the same as those of the original equation. Closed-form solutions of equation X=Af(X) B+C can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation X=Af(X) B+C. Necessary and sufficient conditions are established to guarantee the convergence of the iterations. © 2011 Elsevier Inc. All rights reserved.
Persistent Identifierhttp://hdl.handle.net/10722/157122
ISSN
2013 Impact Factor: 0.983
2013 SCImago Journal Rankings: 1.102
ISI Accession Number ID
Funding AgencyGrant Number
National Natural Science Foundation of China60904007
61074111
China Postdoctoral Science Foundation20100480059
Foundation for Innovative Research Group of the National Natural Science Foundation of China601021002
Harbin Institute of TechnologyHITQNJS.2009.054
Heilongjiang Postdoctoral Foundation of ChinaLRB10-194
HKU CRCG201007176243
Funding Information:

This work is supported in part by the National Natural Science Foundation of China under Grant numbers 60904007 and 61074111, the China Postdoctoral Science Foundation under Grant number 20100480059, the Foundation for Innovative Research Group of the National Natural Science Foundation of China under Grant 601021002, the Development Program for Outstanding Young Teachers at Harbin Institute of Technology under Grant number HITQNJS.2009.054, the Heilongjiang Postdoctoral Foundation of China under Grant No. LRB10-194, and by HKU CRCG 201007176243.

References

 

Author Affiliations
  1. The University of Hong Kong
  2. Harbin Institute of Technology
DC FieldValueLanguage
dc.contributor.authorZhou, Ben_US
dc.contributor.authorLam, Jen_US
dc.contributor.authorDuan, GRen_US
dc.date.accessioned2012-08-08T08:45:25Z-
dc.date.available2012-08-08T08:45:25Z-
dc.date.issued2011en_US
dc.identifier.citationLinear Algebra and Its Applications, 2011, v. 435 n. 6, p. 1370-1398en_US
dc.identifier.issn0024-3795en_US
dc.identifier.urihttp://hdl.handle.net/10722/157122-
dc.description.abstractThis paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation X=Af(X) B+C with f(X)=X T, f(X)=X and f(X)=X H, where X is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of W=AWB+C where the dimensions of the coefficient matrices A,B and C are the same as those of the original equation. Closed-form solutions of equation X=Af(X) B+C can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation X=Af(X) B+C. Necessary and sufficient conditions are established to guarantee the convergence of the iterations. © 2011 Elsevier Inc. All rights reserved.en_US
dc.languageengen_US
dc.publisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laaen_US
dc.relation.ispartofLinear Algebra and Its Applicationsen_US
dc.subjectClosed form solutionsen_US
dc.subjectConjugated and transposeen_US
dc.subjectIterationen_US
dc.subjectMatrix equationsen_US
dc.subjectNumerical solutionen_US
dc.titleToward solution of matrix equation X=Af(X)B+Cen_US
dc.typeArticleen_US
dc.identifier.emailZhou, B: binzhoulee@163.comen_US
dc.identifier.emailLam, J: james.lam@hku.hk-
dc.identifier.emailDuan, GR: g.r.duan@hit.edu.cn-
dc.identifier.authorityLam, J=rp00133en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1016/j.laa.2011.03.003en_US
dc.identifier.scopuseid_2-s2.0-79958852535en_US
dc.identifier.hkuros208790-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-79958852535&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume435en_US
dc.identifier.issue6en_US
dc.identifier.spage1370en_US
dc.identifier.epage1398en_US
dc.identifier.isiWOS:000292439100018-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridDuan, GR=8937477100en_US
dc.identifier.scopusauthoridLam, J=7201973414en_US
dc.identifier.scopusauthoridZhou, B=7401906664en_US
dc.identifier.citeulike9235554-

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