Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

Deng, Yuanbei; Bai, Zhong‐Zhi; Gao, Yong-Hua

doi:10.1002/nla.496

Cited by 78 publications

(32 citation statements)

References 25 publications

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“…Starke and Niethammer reported an iterative method for solutions of CT Sylvester equations by using the SOR (successive overrelaxation) technique [28], Mukaidani, Xu, and Mizukami discussed an iterative algorithm for generalized algebraic Lyapunov equations [24], and El Guennouni et al [13] used Krylov methods to solve Sylvester equations, and Bai constructed a class of unconditional convergent Hermitian and skew-Hermitian splitting (HSS) iteration methods for solving the CT Sylvester equations [1]. See also [3,4,7,8,14,15,22,23,25,27]. Recently, Ding and Chen [10] developed a gradient-based iterative method for solving coupled Sylvester matrix equations.…”

Section: Introductionmentioning

confidence: 98%

“…(1.1) is quite general and includes many matrix equations as special cases. For example, continuous-time (CT) Sylvester equation AX þ XB ¼ C [1]; discrete-time (DT) Sylvester equation AXB T þ X ¼ C [16][17][18]; generalized Sylvester matrix equation AXB T þ CXD T ¼ F [19,20], which are often encountered in stability analysis of control systems and robust control; see also [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

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A note on the iterative solutions of general coupled matrix equation

Zhang

2011

Applied Mathematics and Computation

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

A note on the iterative solutions of general coupled matrix equation

Zhang

2011

Applied Mathematics and Computation

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“…More results on the matrix equations can be found in [39][40][41][42][43][44]. Now we consider the sparsity requirements on the matrices K and M. For different structures, the sparsity requirements differ from each other, so it is difficult to give the general form of elements in S 3 .…”

Section: Corollary 21mentioning

confidence: 99%

The matrix pencil nearness problem in structural dynamic model updating

Yuan

Dai

2015

J Eng Math

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This paper considers the problem of finding the least change adjustment to a given matrix pencil. The desired matrix properties, including satisfaction of the characteristic equation, symmetry, positive semidefiniteness, and sparsity, are imposed as side constraints to form the optimal matrix pencil approximation problem. This problem is related to the frequently encountered engineering problem of a structural modification to the dynamic behavior of a structure. Conditions ensuring the feasible region of the matrix pencil nearness problem are analyzed using a matrix decomposition technique. An unconstrained minimization formulation is presented in terms of the Lagrangian multiplier method and solved by a subgradient algorithm. Numerical results are included to illustrate the performance and application of the proposed method.

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“…The idea of transforming the coefficient matrix into a Schur or Hessenberg form to compute (1) have been presented in [16,17]. When the linear matrix (1) is inconsistent, a finite iterative method to solving its Hermitian minimum norm solutions has been presented in [18]. An efficient iterative method based on Hermitian and skew Hermitian splitting has been proposed in [19].…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

The Relaxed Gradient Based Iterative Algorithm for the Symmetric (Skew Symmetric) Solution of the Sylvester Equation AX + XB = C

Zhang

Sheng

2017

Mathematical Problems in Engineering

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In this paper, we present two different relaxed gradient based iterative (RGI) algorithms for solving the symmetric and skew symmetric solution of the Sylvester matrix equation + = . By using these two iterative methods, it is proved that the iterative solution converges to its true symmetric (skew symmetric) solution under some appropriate assumptions when any initial symmetric (skew symmetric) matrix 0 is taken. Finally, two numerical examples are given to illustrate that the introduced iterative algorithms are more efficient.

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Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

Cited by 78 publications

References 25 publications

A note on the iterative solutions of general coupled matrix equation

A note on the iterative solutions of general coupled matrix equation

The matrix pencil nearness problem in structural dynamic model updating

The Relaxed Gradient Based Iterative Algorithm for the Symmetric (Skew Symmetric) Solution of the Sylvester Equation AX + XB = C

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