Modified Hermitian And skew-Hermitian Splitting Research Articles

Novel minimum residual MHSS iteration method for solving complex symmetric linear systems

For solving complex symmetric systems of linear equations, based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme, a minimum residual MHSS (MRMHSS) iteration method was proposed by applying the minimum residual technique to the MHSS iteration process. We have known that the MRMHSS iteration method is very efficient, while the convergence condition is difficult to verify in practice. In this work, we improve the MRMHSS iteration method by determining the involved iteration parameters using a new norm and prove that the so obtained novel variant can be unconditionally convergent. Numerical results are reported to demonstrate the applicability and effectiveness of the proposed method.

Minimum residual modified HSS iteration method for a class of complex symmetric linear systems

For solving a class of complex symmetric system of linear equations, we apply the minimum residual technique to the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme and propose an iteration method referred to as minimum residual MHSS (MRMHSS) iteration method. Compared with the classical MHSS method, the MRMHSS method has two more iteration parameters, which can be automatically and easily computed. Then, some properties of the MRMHSS iteration method are carefully studied. Finally, we use four examples to test the performance of the MRMHSS iteration method by comparing its numerical results with three other iteration methods.

A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation

In this study, based on the MHSS (Modified Hermitian and skew-Hermitian splitting) method, we will present a generalized MHSS approach for solving large sparse Sylvester equation with non-Hermitian and complex symmetric positive definite/semi-definite matrices. The new method (GMHSS) is a four-parameter iteration procedure where the iterative sequence is unconditionally convergent to the unique solution of the Sylvester equation. Then to improve the GMHSS method, the inexact version of the GMHSS iterative method (IGMHSS) will be described and will be analyzed. Also, by using a new idea, we try to minimize the upper bound of the spectral radius of iteration matrix. Two test problems are given to illustrate the efficiency of the new approach.

An MHSS-like iteration method for two-by-two linear systems with application to FDE optimization problems

In this paper, we exploit the numerical solvers for a fractional differential equations (FDE) optimization problem with constraints given by fractional elliptic state equations. The discretized linear system can be rewritten as a two-by-two linear system. By making use of the strategy of modified Hermitian and skew-Hermitian splitting (MHSS) iteration method proposed by Bai, we propose an MHSS-like iteration method. Comparing with the MHSS iteration method, the MHSS-like method only requires one to solve the two-by-two linear systems by a sparse Cholesky factorization and a fast Fourier transform. Hence, the MHSS-like method has less workload than the MHSS method. The convergence properties and the quasi-optimal parameters are analyzed in detail. Numerical examples are used to testify the efficiency of the new method.

An efficient two-step iterative method for solving a class of complex symmetric linear systems

In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems.

Complex-extrapolated MHSS iteration method for singular complex symmetric linear systems

In this paper, by extrapolating the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method with a complex relaxation parameter, a complex-extrapolated MHSS (CMHSS) iteration method is present for solving a class of complex singular symmetric of linear equations. We study the semi-convergence properties of the CMHSS iteration method and the extent of the optimal iterative parameters. Furthermore, the convergence conditions also hold for solving nonsingular complex systems. Numerical experiments are given to verify the effectiveness of the CMHSS iteration method for solving both singular and nonsingular complex symmetric systems.

On the modified Hermitian and skew-Hermitian splitting iteration methods for a class of weakly absolute value equations

In this paper, based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method, the nonlinear MHSS-like iteration method is presented to solve a class of the weakly absolute value equations (AVE). By using a smoothing approximate function, the convergence properties of the nonlinear MHSS-like iteration method are presented. Numerical experiments are reported to illustrate the feasibility, robustness, and effectiveness of the proposed method.

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 semi-convergence of modified HSS method for a class of complex singular linear systems

In this paper, we discuss the semi-convergence of the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving a broad class of complex singular linear systems. Some semi-convergence theories of the MHSS iteration method are established and are weaker than those appeared in previously published works.

Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

A GENERALIZED PRECONDITIONED MHSS METHOD FOR A CLASS OF COMPLEX SYMMETRIC LINEAR SYSTEMS

Based on the MHSS (Modified Hermitian and skew-Hermitian splitting) and preconditioned MHSS methods, we will present a generalized preconditioned MHSS method for solving a class of complex symmetric linear systems. The new method (GPMHSS) is essentially a two-parameter iteration method where the iterative sequence is unconditionally convergent to the unique solution of the linear system. A parameter region of the convergence for our method is provided. An efficient preconditioner is presented for the actual implementation of the new method. Some numerical results are given to show its effectiveness.

The semi-convergence properties of MHSS method for a class of complex nonsymmetric singular linear systems

We use the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method to solve a class of complex nonsymmetric singular linear systems. The semi-convergence properties of the MHSS method are studied by analyzing the spectrum of the iteration matrix. Moreover, after investigating the semi-convergence factor and estimating its upper bound for the MHSS iteration method, an optimal iteration parameter that minimizes the upper bound of the semi-convergence factor is obtained. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the MHSS method served both as a preconditioner for GMRES method and as a solver.

On Structured Variants of Modified HSS Iteration Methods for Complex Toeplitz Linear Systems

The Modified Hermitian and skew-Hermitian splitting (MHSS) iteration method was presented and studied by Bai, Benzi and Chen (Computing, 87(2010), 93-111) for solving a class of complex symmetric linear systems. In this paper, using the properties of Toeplitz matrix, we propose a class of structured MHSS iteration methods for solving the complex Toeplitz linear system. Theoretical analysis shows that the structured MHSS iteration method is unconditionally convergent to the exact solution. When the MHSS iteration method is used directly to complex symmetric Toeplitz linear systems, the computational costs can be considerately reduced by use of Toeplitz structure. Finally, numerical experiments show that the structured MHSS iteration method and the structured MHSS preconditioner are efficient for solving the complex Toeplitz linear system.

On semi-convergence of modified HSS iteration methods

We discuss semi-convergence of the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving a broad class of complex symmetric singular linear systems. The semi-convergence theory of the MHSS iteration method is established. In addition, numerical examples show the effectiveness of the MHSS iteration method when it is used as a solver or as a preconditioner (for the restarted GMRES method).

Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems

We consider the numerical solution of a class of non-Hermitian positive-definite linear systems by the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method. We show that the MHSS iteration method converges unconditionally even when the real and the imaginary parts of the coefficient matrix are nonsymmetric and positive semidefinite and, at least, one of them is positive definite. At each step the MHSS iteration method requires to solve two linear sub-systems with real nonsymmetric positive definite coefficient matrices. We propose to use inner iteration methods to compute approximate solutions of these linear sub-systems. We illustrate the performance of the MHSS method and its inexact variant by two numerical examples.

Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices

Modified Hermitian and skew-Hermitian splitting (MHSS) method is an unconditionally convergent iterative method for solving large sparse complex symmetric systems of linear equations. By making use of the MHSS iteration as the inner solver for the inexact Newton method, we establish a class of inexact Newton-MHSS methods for solving large sparse systems of nonlinear equations with complex symmetric Jacobian matrices at the solution points. The local and semi-local convergence properties are analyzed under some proper assumptions. Moreover, by introducing a backtracking linear search technique, a kind of global convergence inexact Newton-MHSS methods are also presented and analyzed. Numerical results are given to examine the feasibility and effectiveness of the inexact Newton-MHSS methods.

Modified HSS iteration methods for a class of complex symmetric linear systems

In this paper, we introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. We consider acceleration of the MHSS iteration by Krylov subspace methods. Numerical experiments on a few model problems are used to illustrate the performance of the new method.