Article: Toward solution of matrix equation X=Af(X)B+C
| Title | Toward solution of matrix equation X=Af(X)B+C | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Authors | Zhou, B2 Lam, J1 Duan, GR2 | ||||||||||||||
| Keywords | Closed form solutions Conjugated and transpose Iteration Matrix equations Numerical solution | ||||||||||||||
| Issue Date | 2011 | ||||||||||||||
| Publisher | Elsevier 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?] DOI: http://dx.doi.org/10.1016/j.laa.2011.03.003 | ||||||||||||||
| 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. | ||||||||||||||
| ISSN | 0024-3795 2011 Impact Factor: 0.974 2011 SCImago Journal Rankings: 0.047 | ||||||||||||||
| DOI | http://dx.doi.org/10.1016/j.laa.2011.03.003 | ||||||||||||||
| ISI Accession Number ID | WOS:000292439100018
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 | References in Scopus | ||||||||||||||
| Grants | Iterative Solutions to Linear Matrix Equations Arising from Control Theory |
| dc.contributor.author | Zhou, B | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| dc.contributor.author | Lam, J | ||||||||||||||
| dc.contributor.author | Duan, GR | ||||||||||||||
| dc.date.accessioned | 2012-08-08T08:45:25Z | ||||||||||||||
| dc.date.available | 2012-08-08T08:45:25Z | ||||||||||||||
| dc.date.issued | 2011 | ||||||||||||||
| dc.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. © 2011 Elsevier Inc. All rights reserved. | ||||||||||||||
| dc.description.grant | Iterative Solutions to Linear Matrix Equations Arising from Control Theory | ||||||||||||||
| dc.description.grantcode | 103955 | ||||||||||||||
| dc.description.nature | Link_to_subscribed_fulltext | ||||||||||||||
| dc.identifier.citation | Linear 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.citeulike | 9235554 | ||||||||||||||
| dc.identifier.doi | http://dx.doi.org/10.1016/j.laa.2011.03.003 | ||||||||||||||
| dc.identifier.epage | 1398 | ||||||||||||||
| dc.identifier.hkuros | 208790 | ||||||||||||||
| dc.identifier.isi | WOS:000292439100018
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.issn | 0024-3795 2011 Impact Factor: 0.974 2011 SCImago Journal Rankings: 0.047 | ||||||||||||||
| dc.identifier.issue | 6 | ||||||||||||||
| dc.identifier.scopus | eid_2-s2.0-79958852535 | ||||||||||||||
| dc.identifier.spage | 1370 | ||||||||||||||
| dc.identifier.uri | http://hdl.handle.net/10722/157122 | ||||||||||||||
| dc.identifier.volume | 435 | ||||||||||||||
| dc.language | eng | ||||||||||||||
| dc.publisher | Elsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laa | ||||||||||||||
| dc.publisher.place | United States | ||||||||||||||
| dc.relation.ispartof | Linear Algebra and Its Applications | ||||||||||||||
| dc.relation.references | References in Scopus | ||||||||||||||
| dc.subject | Closed form solutions | ||||||||||||||
| dc.subject | Conjugated and transpose | ||||||||||||||
| dc.subject | Iteration | ||||||||||||||
| dc.subject | Matrix equations | ||||||||||||||
| dc.subject | Numerical solution | ||||||||||||||
| dc.title | Toward solution of matrix equation X=Af(X)B+C | ||||||||||||||
| dc.type | Article |
Author Affiliations
- The University of Hong Kong
- Harbin Institute of Technology