File Download
  • No File Attached
 
Links for fulltext
(May Require Subscription)
 
Supplementary

Article: Toward solution of matrix equation X=Af(X)B+C
  • Basic View
  • Metadata View
  • XML View
TitleToward solution of matrix equation X=Af(X)B+C
 
AuthorsZhou, B2
Lam, J1
Duan, GR2
 
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
 
CitationLinear Algebra and Its Applications, 2011, v. 435 n. 6, p. 1370-1398 [How to Cite?]
DOI: http://dx.doi.org/10.1016/j.laa.2011.03.003
 
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.
 
ISSN0024-3795
2012 Impact Factor: 0.968
2012 SCImago Journal Rankings: 0.868
 
DOIhttp://dx.doi.org/10.1016/j.laa.2011.03.003
 
ISI Accession Number IDWOS:000292439100018
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.

 
ReferencesReferences in Scopus
 
DC FieldValue
dc.contributor.authorZhou, B
 
dc.contributor.authorLam, J
 
dc.contributor.authorDuan, GR
 
dc.date.accessioned2012-08-08T08:45:25Z
 
dc.date.available2012-08-08T08:45:25Z
 
dc.date.issued2011
 
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.
 
dc.description.natureLink_to_subscribed_fulltext
 
dc.identifier.citationLinear Algebra and Its Applications, 2011, v. 435 n. 6, p. 1370-1398 [How to Cite?]
DOI: http://dx.doi.org/10.1016/j.laa.2011.03.003
 
dc.identifier.citeulike9235554
 
dc.identifier.doihttp://dx.doi.org/10.1016/j.laa.2011.03.003
 
dc.identifier.epage1398
 
dc.identifier.hkuros208790
 
dc.identifier.isiWOS:000292439100018
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.

 
dc.identifier.issn0024-3795
2012 Impact Factor: 0.968
2012 SCImago Journal Rankings: 0.868
 
dc.identifier.issue6
 
dc.identifier.scopuseid_2-s2.0-79958852535
 
dc.identifier.spage1370
 
dc.identifier.urihttp://hdl.handle.net/10722/157122
 
dc.identifier.volume435
 
dc.languageeng
 
dc.publisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laa
 
dc.publisher.placeUnited States
 
dc.relation.ispartofLinear Algebra and Its Applications
 
dc.relation.referencesReferences in Scopus
 
dc.subjectClosed form solutions
 
dc.subjectConjugated and transpose
 
dc.subjectIteration
 
dc.subjectMatrix equations
 
dc.subjectNumerical solution
 
dc.titleToward solution of matrix equation X=Af(X)B+C
 
dc.typeArticle
 
<?xml encoding="utf-8" version="1.0"?>
<item><contributor.author>Zhou, B</contributor.author>
<contributor.author>Lam, J</contributor.author>
<contributor.author>Duan, GR</contributor.author>
<date.accessioned>2012-08-08T08:45:25Z</date.accessioned>
<date.available>2012-08-08T08:45:25Z</date.available>
<date.issued>2011</date.issued>
<identifier.citation>Linear Algebra and Its Applications, 2011, v. 435 n. 6, p. 1370-1398</identifier.citation>
<identifier.issn>0024-3795</identifier.issn>
<identifier.uri>http://hdl.handle.net/10722/157122</identifier.uri>
<description.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. &#169; 2011 Elsevier Inc. All rights reserved.</description.abstract>
<language>eng</language>
<publisher>Elsevier Inc. The Journal&apos;s web site is located at http://www.elsevier.com/locate/laa</publisher>
<relation.ispartof>Linear Algebra and Its Applications</relation.ispartof>
<subject>Closed form solutions</subject>
<subject>Conjugated and transpose</subject>
<subject>Iteration</subject>
<subject>Matrix equations</subject>
<subject>Numerical solution</subject>
<title>Toward solution of matrix equation X=Af(X)B+C</title>
<type>Article</type>
<description.nature>Link_to_subscribed_fulltext</description.nature>
<identifier.doi>10.1016/j.laa.2011.03.003</identifier.doi>
<identifier.scopus>eid_2-s2.0-79958852535</identifier.scopus>
<identifier.hkuros>208790</identifier.hkuros>
<relation.references>http://www.scopus.com/mlt/select.url?eid=2-s2.0-79958852535&amp;selection=ref&amp;src=s&amp;origin=recordpage</relation.references>
<identifier.volume>435</identifier.volume>
<identifier.issue>6</identifier.issue>
<identifier.spage>1370</identifier.spage>
<identifier.epage>1398</identifier.epage>
<identifier.isi>WOS:000292439100018</identifier.isi>
<publisher.place>United States</publisher.place>
<identifier.citeulike>9235554</identifier.citeulike>
</item>
Author Affiliations
  1. The University of Hong Kong
  2. Harbin Institute of Technology