The double-step scale splitting method for solving complex Sylvester matrix equation

Dehghan, Mehdi; Shirilord, Akbar

doi:10.1007/s40314-019-0921-6

Cited by 22 publications

(5 citation statements)

References 49 publications

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“…Moreover, the error estimates p t=1 A t X(k)B t -C F compared to the previous iteration and the first iteration are provided by (14) and (15), respectively. In particular, the relative error at each iteration gets smaller than the previous (nonzero) error, as in (16).…”

Section: Theorem 42 If (10)mentioning

confidence: 99%

“…In practical applications, we solve the linear matrix equations of large dimensions by effective iterative methods. There are several ideas to formulate an iterative procedure, namely, one can use matrix sign function [5], block recursion [6,7], Krylov subspace [8,9], Hermitian and skew-Hermitian splitting [10,11], and other related research works; see, e.g., [12][13][14][15]. In the recent decade, the ideas of gradients, hierarchical identification and minimization of associated norm-error functions have encouraged and brought about many researches; see, e.g., [16][17][18][19][20][21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

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Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

Kittisopaporn

Chansangiam

2020

Adv Differ Equ

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In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form p t=1 A t XB t = C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson's equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

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Section: Theorem 42 If (10)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

Kittisopaporn

Chansangiam

2020

Adv Differ Equ

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“…28 In Hajarian, 29 the CGs squared (CGS) and bi-CG stabilized (Bi-CGSTAB) methods were developed for solving the Sylvester-transpose and periodic Sylvester matrix equations. In Dehghan and Shirilord, 30,31 double-step scale splitting (DSS) approaches were used to solve complex Sylvester matrix equation. Dehghan and Shirilord extended the modified Hermitian and skew-Hermitian splitting (MHSS) method for solving large sparse Sylvester equation with non-Hermitian and complex symmetric positive definite/semidefinite matrices.…”

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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“…In many problems in scientific computing we encounter with matrix equations. Matrix equations are one of the most interesting and intensively studied classes of mathematical problems and play vital roles in applications, and many researchers have studied matrix equations and their applications, see [6,7,8,14,16,17,21,22] and their references. Nowadays, the continuous Sylvester equation is possibly the most famous and the most broadly employed linear matrix equation, and is given as…”

Section: Introductionmentioning

confidence: 99%

Minimal residual Hermitian and skew-Hermitian splitting iteration method for the continuous Sylvester equation

Bahramizadeh,

Nazari,

Zak

et al. 2020

Preprint

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By applying the minimal residual technique to the Hermitian and skew-Hermitian (HSS) iteration scheme, we introduce a non-stationary iteration method named minimal residual Hermitian and skew-Hermitian (MRHSS) iteration method to solve the continuous Sylvester equation. Numerical results verify the effectiveness and robustness of the MRHSS iteration method versus the HSS method for the continuous Sylvester equation. Moreover, by numerical computation, we show that the MRHSS splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the continuous Sylvester equation.

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The double-step scale splitting method for solving complex Sylvester matrix equation

Cited by 22 publications

References 49 publications

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

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

Minimal residual Hermitian and skew-Hermitian splitting iteration method for the continuous Sylvester equation

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