An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1, A2XB2=C2

Peng, Zhuohua; Hu, Xi-Yan; Zhang, Lei

doi:10.1016/j.amc.2006.01.071

Cited by 33 publications

(18 citation statements)

References 10 publications

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“…Recently, some finite iterative algorithms have also been developed to solve some coupled matrix equations. In [9], an algorithm was constructed to solve the reflexive with respect to the generalized reflection matrix P solution for matrix equation (AXB, GXH) = (C, D). By this algorithm, a solution can be obtained within finite iteration steps for any initial reflexive matrix.…”

Section: Introductionmentioning

confidence: 99%

Iterative solutions to coupled Sylvester-conjugate matrix equations

Wu¹,

Feng²,

Duan³

et al. 2010

Computers & Mathematics with Applications

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a b s t r a c tThis paper is concerned with iterative solutions to the coupled Sylvester-conjugate matrix equation with a unique solution. By applying a hierarchical identification principle, an iterative algorithm is established to solve this class of complex matrix equations. With a real representation of a complex matrix as a tool, a sufficient condition is given to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices. In addition, a sufficient convergence condition that is easier to compute is also given by the original coefficient matrices. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.

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Section: Introductionmentioning

confidence: 99%

Iterative solutions to coupled Sylvester-conjugate matrix equations

Wu¹,

Feng²,

Duan³

et al. 2010

Computers & Mathematics with Applications

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“…Least squares solutions to linear matrix equations or coupled linear matrix equations, have also been well studied in the literature in the past few years. Conjugated-gradient based iterative methods are developed in [26,27] respectively for solving the least squares symmetric solution of the linear matrix equation AXB = C and the least-squares reflexive solution of the matrix equations…”

Section: Introductionmentioning

confidence: 99%

Weighted least squares solutions to general coupled Sylvester matrix equations

Zhou

Duan

et al. 2009

Journal of Computational and Applied Mathematics

129

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MSC: 15A09 15A12 15A24Keywords: Weighted least squares solutions Weighted generalized inverses Coupled Sylvester matrix equations Gradient based iterative algorithms Maximal convergence rate a b s t r a c t This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.

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“…He also indicated that if A ∈ C m×n is a generalized reflexive matrix and rank(A) = n, then its least-squares problem can be reduced to two independent least-squares problems for matrices of smaller dimensions. This dimension-reduced idea was applied to solve the singular value decomposition [2], the inverse eigenvalue problem [3], and the least-squares problem [4].…”

Section: Introductionmentioning

confidence: 99%

Procrustes problems for (P,Q,η)-reflexive matrices

Jia

Wang

Wei

2010

Journal of Computational and Applied Mathematics

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a b s t r a c tThis paper characterizes (P, Q , η)-reflexive matrices, showing that a (P, Q , η)-reflexive matrix can be represented in terms of k matrices A a,b ∈ C ma×n b , where a + b = η(mod k), m a and n b are dimensions of the τ a -and τ b -eigenspaces of P and Q , respectively. A general solution of Procrustes problems of (P, Q , η)-reflexive matrices is presented in terms of lower-order matrices A 1,η−1 , . . . , A η−1,1 , A η,k , A η+1,k−1 , . . . , A k,η under the condition that P and Q are unitary.Crown

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An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1, A2XB2=C2

Cited by 33 publications

References 10 publications

Iterative solutions to coupled Sylvester-conjugate matrix equations

Iterative solutions to coupled Sylvester-conjugate matrix equations

Weighted least squares solutions to general coupled Sylvester matrix equations

Procrustes problems for (P,Q,η)-reflexive matrices

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