Continuous Sylvester Equations Research Articles

Retracted: The MGHSS for Solving Continuous Sylvester Equation AX + XB = C

Retracted: The MGHSS for Solving Continuous Sylvester Equation AX + XB = C

On the Generalized Richardson Iteration of the Continuous Sylvester Equation

On the Generalized Richardson Iteration of the Continuous Sylvester Equation

On circulant and skew-circulant splitting algorithms for (continuous) Sylvester equations

We present a circulant and skew-circulant splitting (CSCS) iterative method for solving large sparse continuous Sylvester equations AX+XB=C, where the coefficient matrices A and B are Toeplitz matrices. A theoretical study shows that if the circulant and skew-circulant splitting factors of A and B are positive semi-definite (not necessarily Hermitian), and at least one factor is positive definite, then the CSCS method converges to the unique solution of the Sylvester equation. In addition, our analysis gives an upper bound for the convergence factor of the CSCS iteration which depends only on the eigenvalues of the circulant and skew-circulant splitting matrices. A computational comparison with alternative methods reveals the efficiency and reliability of the proposed method.

On estimating the separation associated with the periodic continuous Sylvester equation

On estimating the separation associated with the periodic continuous Sylvester equation

Shift-Splitting Iteration Method and Its Variants for Solving Continuous Sylvester Equations

Shift-Splitting Iteration Method and Its Variants for Solving Continuous Sylvester Equations

The MGHSS for Solving Continuous Sylvester Equation A X + X B = C

This paper proposes the modified generalization of the HSS (MGHSS) to solve a large and sparse continuous Sylvester equation, improving the efficiency and robustness. The analysis shows that the MGHSS converges to the unique solution of AX + XB = C unconditionally. We also propose an inexact variant of the MGHSS (IMGHSS) and prove its convergence under certain conditions. Numerical experiments verify the efficiency of the proposed methods.

On inexact alternating direction implicit iteration for continuous Sylvester equations

AbstractIn this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX + XB = C, where the coefficient matrices A and B are assumed to be positive semi‐definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations.

On preconditioned normal and skew-Hermitian splitting iteration method for continuous Sylvester equations AX + XB = C*

In this paper, we present a preconditioned normal and skew-Hermitian splitting (PNSS) iteration method for continuous Sylvester equations AX + XB = C with positive definite/semi-definite matrices. Theoretical analysis shows that the PNSS methods will converge unconditionally to the exact solution of the continuous Sylvester equations. An inexact variant of the PNSS iteration method(IPNSS) and the analysis of its convergence property in detail have been established. Numerical experiments further show that this new method is more efficient and robust than the existing ones.

Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations

Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants.

A Parallel Two-Stage Iteration Method for Solving Continuous Sylvester Equations

In this paper we propose a parallel two-stage iteration algorithm for solving large-scale continuous Sylvester equations. By splitting the coefficient matrices, the original linear system is transformed into a symmetric linear system which is then solved by using the SYMMLQ algorithm. In order to improve the relative parallel efficiency, an adjusting strategy is explored during the iteration calculation of the SYMMLQ algorithm to decrease the degree of the reduce-operator from two to one communications at each step. Moreover, the convergence of the iteration scheme is discussed, and finally numerical results are reported showing that the proposed method is an efficient and robust algorithm for this class of continuous Sylvester equations on a parallel machine.

Preconditioned Positive-Definite and Skew-Hermitian Splitting Iteration Methods for Continuous Sylvester Equations AX + XB = C

AbstractIn this paper, we present a preconditioned positive-definite and skew-Hermitian splitting (PPSS) iteration method for continuous Sylvester equations AX + XB = C with positive definite/semi-definite matrices. The analysis shows that the PPSS iteration method will converge under certain assumptions. An inexact variant of the PPSS iteration method (IPPSS) has been presented and the analysis of its convergence property in detail has been discussed. Numerical results show that this new method is more efficient and robust than the existing ones.

General analytical forms for the solution of the Sylvester and Lyapunov equations for continuous and discrete dynamic systems

An approach to forming analytical solutions of the discrete and continuous Sylvester and Lyapunov linear algebraic matrix equations is described. The approach is based on reducing the square matrix to the Jordan normal form. Examples, algorithms, and implementations in Matlab are presented.

On PMHSS Iteration Methods for Continuous Sylvester Equations

On PMHSS Iteration Methods for Continuous Sylvester Equations

A new generalization of the Hermitian and skew-Hermitian splitting method for solving the continuous Sylvester equation

The two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method is applied to solve the continuous Sylvester equation [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. The matrix-vector form of the equation is considered to present convergence properties of the TGHSS method. Some numerical examples are given to show the effectiveness and accuracy of the method. Comparison with other methods is also included.

An iterative method for solving the continuous sylvester equation by emphasizing on the skew-hermitian parts of the coefficient matrices

ABSTRACTWe present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (CGNR)-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matrices and (which are the skew-Hermitian part of A and B, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods.

A generalization of the Hermitian and skew-Hermitian splitting iteration method for solving Sylvester equations

This paper is concerned with a generalization of the Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations. The analysis shows that the GHSS iteration method will converge under certain assumptions. Numerical results show that this new method is more efficient and robust than the existing ones.

On modified HSS iteration methods for continuous Sylvester equations

We introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semidefinite matrices. It is found that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration in this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. Numerical results show that the MHSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equations

This paper is concerned with some generalizations of the Hermitian and skew-Hermitian splitting (HSS) iteration for solving continuous Sylvester equations. The main contents we will introduce are the normal and skew-Hermitian splitting (NSS) iteration methods for the continuous Sylvester equations. It is shown that the new schemes can outperform the standard HSS method in some situations. Theoretical analysis shows that the NSS methods converge unconditionally to the exact solution of the continuous Sylvester equations. Moreover, we derive the upper bound of the contraction factor of the NSS iterations. Numerical experiments further show the effectiveness of our new methods.

A Generalized HSS Iteration Method for Continuous Sylvester Equations

Based on the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish a generalized HSS (GHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The GHSS method is essentially a four-parameter iteration which not only covers the standard HSS iteration but also enables us to optimize the iterative process. An exact parameter region of convergence for the method is strictly proved and a minimum value for the upper bound of the iterative spectrum is derived. Moreover, to reduce the computational cost, we establish an inexact variant of the GHSS (IGHSS) iteration method whose convergence property is discussed. Numerical experiments illustrate the efficiency and robustness of the GHSS iteration method and its inexact variant.

Nested splitting CG-like iterative method for solving the continuous Sylvester equation and preconditioning

We present a nested splitting conjugate gradient iteration method for solving large sparse continuous Sylvester equation, in which both coefficient matrices are (non-Hermitian) positive semi-definite, and at least one of them is positive definite. This method is actually inner/outer iterations, which employs the Sylvester conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and Hermitian positive definite splitting of the coefficient matrices. Convergence conditions of this method are studied and numerical experiments show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods.