Least-squares solutions of generalized inverse eigenvalue problem over Hermitian–Hamiltonian matrices with a submatrix constraint

In this paper, a gradient-based iterative algorithm is proposed for finding the least-squares solutions of the following constrained generalized inverse eigenvalue problem: given X ∈ C n × m , Λ = diag ( λ 1 , λ 2 , … , λ m ) ∈ C m × m , find A ∗ , B ∗ ∈ C n × n , such that ‖ A X - B X Λ ‖ is minimi...

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Vydáno v:	Computational & applied mathematics Ročník 37; číslo 1; s. 593 - 603
Hlavní autoři:	Cai, Jing, Chen, Jianlong
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Cham Springer International Publishing 01.03.2018 Springer Nature B.V
Témata:	Algorithms Applications of Mathematics Applied physics Computational mathematics Computational Mathematics and Numerical Analysis Eigenvalues Generalized inverse Iterative algorithms Iterative methods Least squares Mathematical Applications in Computer Science Mathematical Applications in the Physical Sciences Mathematics Mathematics and Statistics Matrix Roundoff error Hermitian–Hamiltonian matrix 65J22 Optimal approximation Submatrix constraint Generalized inverse eigenvalue problem 15A29
ISSN:	0101-8205, 2238-3603, 1807-0302
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Abstract	In this paper, a gradient-based iterative algorithm is proposed for finding the least-squares solutions of the following constrained generalized inverse eigenvalue problem: given X ∈ C n × m , Λ = diag ( λ 1 , λ 2 , … , λ m ) ∈ C m × m , find A ∗ , B ∗ ∈ C n × n , such that ‖ A X - B X Λ ‖ is minimized, where A ∗ , B ∗ are Hermitian–Hamiltonian except for a special submatrix. For any initial constrained matrices, a solution pair ( A ∗ , B ∗ ) can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least-norm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained. A numerical example is given to show the efficiency of the proposed algorithm.
AbstractList	In this paper, a gradient-based iterative algorithm is proposed for finding the least-squares solutions of the following constrained generalized inverse eigenvalue problem: given X∈Cn×m, Λ=diag(λ1,λ2,…,λm)∈Cm×m, find A∗,B∗∈Cn×n, such that ‖AX-BXΛ‖ is minimized, where A∗,B∗ are Hermitian–Hamiltonian except for a special submatrix. For any initial constrained matrices, a solution pair (A∗,B∗) can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least-norm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained. A numerical example is given to show the efficiency of the proposed algorithm. In this paper, a gradient-based iterative algorithm is proposed for finding the least-squares solutions of the following constrained generalized inverse eigenvalue problem: given X ∈ C n × m , Λ = diag ( λ 1 , λ 2 , … , λ m ) ∈ C m × m , find A ∗ , B ∗ ∈ C n × n , such that ‖ A X - B X Λ ‖ is minimized, where A ∗ , B ∗ are Hermitian–Hamiltonian except for a special submatrix. For any initial constrained matrices, a solution pair ( A ∗ , B ∗ ) can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least-norm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained. A numerical example is given to show the efficiency of the proposed algorithm.
Author	Chen, Jianlong Cai, Jing
Author_xml	– sequence: 1 givenname: Jing surname: Cai fullname: Cai, Jing email: caijing@zjhu.edu.cn organization: School of Science, Huzhou University, Department of Mathematics, Southeast University – sequence: 2 givenname: Jianlong surname: Chen fullname: Chen, Jianlong organization: Department of Mathematics, Southeast University
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Cites_doi	10.1016/j.amc.2007.05.035 10.1080/03081087.2014.922969 10.1007/BFb0039443 10.1007/s11741-004-0055-x 10.1016/j.cam.2008.05.015 10.1016/0024-3795(88)90223-6 10.1093/imamci/2.4.335 10.1137/140972494
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References	AntoniouALuWSPractical optimization: algorithm and engineering applications2007New YorkSpringer1128.90001 DaiHBaiZZWeiYOn the solvability condition and numerical algorithm for the parameterized generalized inverse eigenvalue problemSIAM J Matrix Anal Appl201536707726335576910.1137/1409724941317.65101 MoghaddamMRMirzaeiHGhanbariKOn the generalized inverse eigenvalue problem of constructing symmetric pentadiagonal matrices from three mixed eigendataLinear Multilinear Algebra201563611541166329196210.1080/03081087.2014.9229691315.15009 PritchardAJSalamonDThe linear quadratic control problem for retarded systems with delays in control and observationIMA J Math Control Inf1985233536210.1093/imamci/2.4.3350646.34078 JamshidiMAn overview on the solutions of the algebra matrix riccati equation and related problemsLarge Scale Syst Theory Appl198011671926179860453.93025 GhanbariKMingarelliAGeneralized inverse eigenvalue problem for symmetric matricesInt J Appl Math20004219920919343341172.15300 YuanYXDaiHA generalized inverse eigenvalue problem in structural dynamic model updatingJ Comput Appl Math200922614249250187810.1016/j.cam.2008.05.0151175.65049 MoRHLiWThe inverse eigenvalue problem of hermitian and generalized skew-Hamiltonian matrices with a submatrix constraint and its approximationActa Mathematica Scientia201131A369170128169871240.65129 YuanYXGeneralized inverse eigenvalue problems for symmetric arrow-head matricesInt J Comput Math Sci2010462682712765319 LiuZYTanYXTianZLGeneralized inverse eigenvalue problem for centrohermitian matricesJ Shanghai Univ200484448454210971610.1007/s11741-004-0055-x WeiPZhangZZXieDXGeneralized inverse eigenvalue problem for Hermitian generalized Hamiltonian matricesChin J Eng Math201027582082628100481240.65131 ZhouKMDoyleJGloverKRobust and optimal control1995Upper Saddle RiverPrentice Hall0999.49500 Mehrmann VL (1991) The autonomous linear quadratic control problem: theory and numerical solution. J Shanghai Univ. Springer, Heidelberg YuanYXOn the two class of best approximation problemsMath Numer Sinica2001234294361881817 GhanbariKA survey on inverse and generalized inverse eigenvalue problems of jacobi matricesAppl Math Comput20081952355363238121610.1016/j.amc.2007.05.0351156.65036 HighamNJComputing a nearest symmetric positive semidefinite matrixLinear Algebra Appl19881310311894399710.1016/0024-3795(88)90223-60649.65026 JiangZLuQOn optimal approximation of a matrix under a spectral restrictionMath Numer Sine1986847528640300592.65023 GaoYQWeiPZhangZZXieDXGeneralized inverse eigenvalue problem for reflexive and anti-reflexive matrices.Numer Math J Chin Univ201234321422230892411289.65088 YQ Gao (363_CR3) 2012; 34 MR Moghaddam (363_CR12) 2015; 63 RH Mo (363_CR11) 2011; 31A YX Yuan (363_CR16) 2010; 4 363_CR10 H Dai (363_CR2) 2015; 36 K Ghanbari (363_CR4) 2008; 195 M Jamshidi (363_CR7) 1980; 1 ZY Liu (363_CR9) 2004; 8 K Ghanbari (363_CR5) 2000; 4 NJ Higham (363_CR6) 1988; 13 P Wei (363_CR14) 2010; 27 YX Yuan (363_CR17) 2009; 226 AJ Pritchard (363_CR13) 1985; 2 YX Yuan (363_CR15) 2001; 23 KM Zhou (363_CR18) 1995 A Antoniou (363_CR1) 2007 Z Jiang (363_CR8) 1986; 8
References_xml	– reference: GhanbariKMingarelliAGeneralized inverse eigenvalue problem for symmetric matricesInt J Appl Math20004219920919343341172.15300 – reference: MoRHLiWThe inverse eigenvalue problem of hermitian and generalized skew-Hamiltonian matrices with a submatrix constraint and its approximationActa Mathematica Scientia201131A369170128169871240.65129 – reference: MoghaddamMRMirzaeiHGhanbariKOn the generalized inverse eigenvalue problem of constructing symmetric pentadiagonal matrices from three mixed eigendataLinear Multilinear Algebra201563611541166329196210.1080/03081087.2014.9229691315.15009 – reference: JamshidiMAn overview on the solutions of the algebra matrix riccati equation and related problemsLarge Scale Syst Theory Appl198011671926179860453.93025 – reference: WeiPZhangZZXieDXGeneralized inverse eigenvalue problem for Hermitian generalized Hamiltonian matricesChin J Eng Math201027582082628100481240.65131 – reference: GhanbariKA survey on inverse and generalized inverse eigenvalue problems of jacobi matricesAppl Math Comput20081952355363238121610.1016/j.amc.2007.05.0351156.65036 – reference: GaoYQWeiPZhangZZXieDXGeneralized inverse eigenvalue problem for reflexive and anti-reflexive matrices.Numer Math J Chin Univ201234321422230892411289.65088 – reference: ZhouKMDoyleJGloverKRobust and optimal control1995Upper Saddle RiverPrentice Hall0999.49500 – reference: YuanYXDaiHA generalized inverse eigenvalue problem in structural dynamic model updatingJ Comput Appl Math200922614249250187810.1016/j.cam.2008.05.0151175.65049 – reference: AntoniouALuWSPractical optimization: algorithm and engineering applications2007New YorkSpringer1128.90001 – reference: Mehrmann VL (1991) The autonomous linear quadratic control problem: theory and numerical solution. J Shanghai Univ. Springer, Heidelberg – reference: LiuZYTanYXTianZLGeneralized inverse eigenvalue problem for centrohermitian matricesJ Shanghai Univ200484448454210971610.1007/s11741-004-0055-x – reference: YuanYXOn the two class of best approximation problemsMath Numer Sinica2001234294361881817 – reference: DaiHBaiZZWeiYOn the solvability condition and numerical algorithm for the parameterized generalized inverse eigenvalue problemSIAM J Matrix Anal Appl201536707726335576910.1137/1409724941317.65101 – reference: YuanYXGeneralized inverse eigenvalue problems for symmetric arrow-head matricesInt J Comput Math Sci2010462682712765319 – reference: JiangZLuQOn optimal approximation of a matrix under a spectral restrictionMath Numer Sine1986847528640300592.65023 – reference: HighamNJComputing a nearest symmetric positive semidefinite matrixLinear Algebra Appl19881310311894399710.1016/0024-3795(88)90223-60649.65026 – reference: PritchardAJSalamonDThe linear quadratic control problem for retarded systems with delays in control and observationIMA J Math Control Inf1985233536210.1093/imamci/2.4.3350646.34078 – volume-title: Robust and optimal control year: 1995 ident: 363_CR18 – volume: 195 start-page: 355 issue: 2 year: 2008 ident: 363_CR4 publication-title: Appl Math Comput doi: 10.1016/j.amc.2007.05.035 – volume: 1 start-page: 167 year: 1980 ident: 363_CR7 publication-title: Large Scale Syst Theory Appl – volume: 63 start-page: 1154 issue: 6 year: 2015 ident: 363_CR12 publication-title: Linear Multilinear Algebra doi: 10.1080/03081087.2014.922969 – ident: 363_CR10 doi: 10.1007/BFb0039443 – volume: 34 start-page: 214 issue: 3 year: 2012 ident: 363_CR3 publication-title: Numer Math J Chin Univ – volume: 8 start-page: 47 year: 1986 ident: 363_CR8 publication-title: Math Numer Sine – volume: 8 start-page: 448 issue: 4 year: 2004 ident: 363_CR9 publication-title: J Shanghai Univ doi: 10.1007/s11741-004-0055-x – volume: 23 start-page: 429 year: 2001 ident: 363_CR15 publication-title: Math Numer Sinica – volume: 226 start-page: 42 issue: 1 year: 2009 ident: 363_CR17 publication-title: J Comput Appl Math doi: 10.1016/j.cam.2008.05.015 – volume: 27 start-page: 820 issue: 5 year: 2010 ident: 363_CR14 publication-title: Chin J Eng Math – volume: 4 start-page: 268 issue: 6 year: 2010 ident: 363_CR16 publication-title: Int J Comput Math Sci – volume: 13 start-page: 103 year: 1988 ident: 363_CR6 publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(88)90223-6 – volume: 2 start-page: 335 year: 1985 ident: 363_CR13 publication-title: IMA J Math Control Inf doi: 10.1093/imamci/2.4.335 – volume: 36 start-page: 707 year: 2015 ident: 363_CR2 publication-title: SIAM J Matrix Anal Appl doi: 10.1137/140972494 – volume: 4 start-page: 199 issue: 2 year: 2000 ident: 363_CR5 publication-title: Int J Appl Math – volume-title: Practical optimization: algorithm and engineering applications year: 2007 ident: 363_CR1 – volume: 31A start-page: 691 issue: 3 year: 2011 ident: 363_CR11 publication-title: Acta Mathematica Scientia
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SubjectTerms	Algorithms Applications of Mathematics Applied physics Computational mathematics Computational Mathematics and Numerical Analysis Eigenvalues Generalized inverse Iterative algorithms Iterative methods Least squares Mathematical Applications in Computer Science Mathematical Applications in the Physical Sciences Mathematics Mathematics and Statistics Matrix Roundoff error
Title	Least-squares solutions of generalized inverse eigenvalue problem over Hermitian–Hamiltonian matrices with a submatrix constraint
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