The submatrix constraint problem of matrix equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"><mml:mrow><mml:mi mathvariant="italic">AXB</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="italic">CYD</mml:mi><mml:mo>=</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:math>

Li, Jiao‐fen; Hu, Xi-Yan; Zhang, Lei

doi:10.1016/j.amc.2009.08.051

Cited by 13 publications

(6 citation statements)

References 21 publications

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“…Similar method was constructed to solve matrix equations (4) with generalized bisymmetric in [28]. In particular, Li et al [29] proposed an elegant algorithm for solving the generalized Sylvester (Lyapunov) matrix equation + = with bisymmetric and symmetric , the two unknown matrices include the given central principal submatrix and leading principal submatrix, respectively. This method shunned the difficulties in numerical instability and computational complexity, and solved the problem, completely.…”

Section: Problem 2 Givenmentioning

confidence: 99%

An Iterative Method for the Least-Squares Problems of a General Matrix Equation Subjects to Submatrix Constraints

Dai

Liang

Shen

2013

Journal of Applied Mathematics

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An iterative algorithm is proposed for solving the least-squares problem of a general matrix equation∑i=1t‍MiZiNi=F, whereZi(i=1,2,…,t) are to be determined centro-symmetric matrices with given central principal submatrices. For any initial iterative matrices, we show that the least-squares solution can be derived by this method within finite iteration steps in the absence of roundoff errors. Meanwhile, the unique optimal approximation solution pair for given matricesZ~ican also be obtained by the least-norm least-squares solution of matrix equation∑i=1t‍MiZ-iNi=F-, in whichZ-i=Zi-Z~i, F-=F-∑i=1t‍MiZ~iNi. The given numerical examples illustrate the efficiency of this algorithm.

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Section: Problem 2 Givenmentioning

confidence: 99%

An Iterative Method for the Least-Squares Problems of a General Matrix Equation Subjects to Submatrix Constraints

Dai

Liang

Shen

2013

Journal of Applied Mathematics

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“…The optimal approximation is the matrix that satisfies the centrosymmetric and the minimum residual restriction, and is closed to the given matrix X * in Frobenius norm (may be spectral norm or others). About the optimal approximation problem, we refer the reader to references 8, 7, 9, 21, 11, 27, 28.…”

Section: The Solution Of Problem IImentioning

confidence: 99%

Numerical solutions of AXB = C for centrosymmetric matrix X under a specified submatrix constraint

Li¹,

Hu²,

Zhang³

2011

Numer. Linear Algebra Appl.

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We say that X = [x i j ] n i, j=1 is centrosymmetric if x i j = x n− j+1,n−i+1 , 1 i, j n. In this paper, we present an efficient algorithm for minimizing AX B −C where · is the Frobenius norm, A ∈ R m×n , B ∈ R n×s , C ∈ R m×s and X ∈ R n×n is centrosymmetric with a specified central submatrix [x i j ] p i, j n− p . Our algorithm produces a suitable X such that AX B = C in finitely many steps, if such an X exists. We show that the algorithm is stable in any case, and we give results of numerical experiments that support this claim.Problem II Let X * ∈ R n×n be given and X * c (q) = X q (be given in Problem I) and let S E denote the solution set of Problem I. Find X ∈ S E such thatOur results are natured extension of results obtain in [8,7,9]. That work was motivated by results of serval investigators including [16][17][18][19][20]. These references describe application in which such problems arise. We are particularly influenced by Zhao et al. [21], who considered the case AX = B for symmetric centrosymmetric matrix X under central principal submatrix constraint. In these papers, inevitably, Moore-Penrose generalized inverses and some complicated matrix decompositions such as canonical correlation decomposition (CCD) [22] and general singular value decomposition (GSVD) [23] are involved. Because of the obvious difficulties in numerical instability and computational complexity, those constructional solutions narrow down their applications. Indeed, it is impractical to find a solution by those formulas if the matrix size is large. In the present paper, we extend and develop the above research, however, in a totally different way.In this paper we are only concerned with iteration method, and the main idea is based onComparing the two side of J n X J n = X , we obtain X 33 = J (n−q)/2 X 11 J (n−q)/2 , X 22 = J q X 22 J q , X 23 = J q X 21 J (n−q)/2 , X 31 = J (n−q)/2 X 13 J (n−q)/2 , and X 32 = J (n−q)/2 X 12 J q . Substituting X 13 , X 23 , X 33 , and X 32 into (3) and noticing that X 22 = J q X 22 J q , we have (2).

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“…In recent years, there has been interest in the submatrix constraint problem of centrosymmetric and bisymmetric matrices 34,35 . Li et al proposed the extended form of the CG for least squares (CGLS) algorithm for minimizing

| | A X B + C Y D - E | |,

where X is symmetric centrosymmetric with a specified central submatrix and Y is symmetric with a specified central submatrix 36 . In Peng, 37 the generalized CGLS algorithm was proposed to minimize

| | A_{1} X_{1} B_{1} + A_{2} X_{2} B_{2} + \dots + A_{l} X_{l} B_{l} - C | |,

over reflexive matrices with a specified central principal submatrix.…”

Section: Introductionmentioning

confidence: 99%

“…where X is symmetric centrosymmetric with a specified central submatrix and Y is symmetric with a specified central submatrix. 36 In Peng, 37 the generalized CGLS algorithm was proposed to minimize…”

Section: Introductionmentioning

confidence: 99%

Least‐squares partially bisymmetric solutions of coupled Sylvester matrix equations accompanied by a prescribed submatrix constraint

Hajarian

Chronopoulos

2020

Math Methods in App Sciences

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The coupled Sylvester matrix equations (CSMEs) A1XB1+C1YD1=E1,A2XB2+C2YD2=E2, appear frequently in various fields of mathematics and engineering such as in control systems and signal processing. In this investigation, we establish and analyze the generalized conjugate directions (GCDs) method for solving the CSMEs over partially bisymmetric matrices X and Y with a prescribed submatrix constraint. We show that the GCD method with the arbitrary initial bisymmetric matrices can compute the least‐squares partially bisymmetric solutions with a prescribed submatrix constraint within a finite number of iterations in the absence of round‐off errors. Numerical examples illustrate the efficiency and simplicity of the GCD method and confirm the theoretical results.

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The submatrix constraint problem of matrix equation AXB+CYD=E

Cited by 13 publications

References 21 publications

An Iterative Method for the Least-Squares Problems of a General Matrix Equation Subjects to Submatrix Constraints

An Iterative Method for the Least-Squares Problems of a General Matrix Equation Subjects to Submatrix Constraints

Numerical solutions of AXB = C for centrosymmetric matrix X under a specified submatrix constraint

Least‐squares partially bisymmetric solutions of coupled Sylvester matrix equations accompanied by a prescribed submatrix constraint

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