Hermitian And skew-Hermitian Splitting Iteration Method Research Articles

On the convergence of the minimum residual HSS iteration method

Recently, by applying the minimum residual technique to the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, a minimum residual HSS (MRHSS) iteration method was proposed for solving non-Hermitian positive definite linear systems. Although the MRHSS iteration method is very efficient, it is conditionally convergent. In this work, we further study the convergence of the MRHSS iteration method, and show that it can unconditionally convergent if its parameters are determined by minimizing a new norm of the residual. Numerical results verify that the MRHSS method discussed in this work is also very efficient.

Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems

By applying the minimum residual technique to the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, we introduce a non-stationary iteration method named minimum residual HSS (MRHSS) iteration method to solve non-Hermitian positive definite linear systems. The convergence property of the MRHSS iteration method together with the property of the iteration parameters are carefully studied. Numerical results verify the effectiveness and robustness of the MRHSS iteration method.

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.

The generalized HSS method with a flexible shift-parameter for non-Hermitian positive definite linear systems

Based on the Hermitian and skew-Hermitian splitting (HSS), we come up with a generalized HSS iteration method with a flexible shift-parameter for solving the non-Hermitian positive definite system of linear equations. This iteration method utilizes the optimization technique to obtain the optimal value of the flexible shift-parameter at iteration process. Both theory and experiment have shown that the new strategy is efficient.

Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for a class of linear matrix equations

Recently, Wang et al. (2013) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method to approximate the solution of linear matrix equation. To further improve the efficiency of the HSS method, we in this paper establish a parameterized preconditioned HSS (PPHSS) iteration method by introducing two Hermitian positive definite preconditioners and two more different iteration parameters for the HSS method. The convergence properties of the PPHSS method used for solving linear matrix equation AXB=C are strictly analyzed and the quasi-optimal values of the iteration parameters for the PPHSS method are also derived. Numerical results demonstrate the feasibility and the efficiency of the PPHSS iteration method.

Two-parameter generalized Hermitian and skew-Hermitian splitting iteration method

It is known that the Hermitian and skew-Hermitian splitting (HSS) iteration method is an efficient solver for non-Hermitian positive-definite linear system of equations. Benzi [A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 360–374] proposed a generalized HSS (GHSS) iteration method. In this paper, we present a two-parameter version of the GHSS (TGHSS) method and investigate its convergence properties. To show the effectiveness of the proposed method the TGHSS iteration method is applied to image restoration and convection–diffusion problems and the results are compared with those of the HSS and GHSS methods.

A single-step HSS method for non-Hermitian positive definite linear systems

In this paper, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method, a single-step HSS (SHSS) iteration method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that, under a loose restriction on the iteration parameter, the SHSS method is convergent to the unique solution of the linear system. Furthermore, we derive an upper bound for the spectral radius of the SHSS iteration matrix, and the quasi-optimal parameter is obtained to minimize the above upper bound. Numerical experiments are reported to the efficiency of the SHSS method; numerical comparisons show that the proposed SHSS method is superior to the HSS method under certain conditions.

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

Based on the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, an efficient Uzawa–HSS iteration method has been proposed to solve the nonsingular saddle-point problems. In this paper, we discuss the feasibility of the Uzawa–HSS method used for solving singular saddle-point problems. The semi-convergence properties of the Uzawa–HSS iteration method are carefully analyzed, which show that the iterative sequence generated by the Uzawa–HSS method converges to a solution of the singular saddle-point problem if the iteration parameters satisfy suitable restrictions. Numerical results verify the robustness and efficiency of the Uzawa–HSS method.

The Picard–HSS iteration method for absolute value equations

Recently Bai and Yang in (Appl Numer Math 59:2923–2936, 2009) proposed the Picard–Hermitian and skew-Hermitian splitting (HSS) iteration method to solve the system of nonlinear equations \(Ax=\varphi (x)\), where \(A\in \mathbb {C}^{n \times n}\) is a non-Hermitian positive definite matrix and \(\varphi :\mathbb {D}\subset \mathbb {C}^n \rightarrow \mathbb {C}^n\) is continuously differentiable function defined on the open complex domain \(\mathbb {D}\) in the \(n\)-dimensional complex linear space \(\mathbb {C}^n\). In this paper, we focus our attention to the absolute value equation (AVE) \(Ax=\varphi (x)\) where \(\varphi (x)=|x|+b\), where \(b\in \mathbb {C}^n\). Since the function \(\varphi \) in AVE is not continuously differentiable function the convergence analysis of the Picard–HSS iteration method for this problem needs to be investigated. We give sufficient conditions for the convergence of the Picard–HSS iteration method for AVE. Some numerical experiments are given to show the effectiveness of the method and to compare with two available methods.

On $${\varvec{m}}$$ m -step Hermitian and skew-Hermitian splitting preconditioning methods

Preconditioners are often applied in various Krylov subspace iteration methods to improve their computing efficiency. In this paper, we consider solving linear systems with a non-Hermitian positive definite coefficient matrix using preconditioned Krylov subspace iteration methods such as the generalized minimal residual (GMRES) method. An $$m$$ -step polynomial preconditioner is designed based on the Hermitian and skew-Hermitian splitting (HSS) iteration method proposed by Bai et al. (SIAM J Matrix Anal Appl 24:603–626, 2003). The proposed preconditioned system is solved by fully utilizing the HSS iteration method. Theoretical and experimental results show that the proposed $$m$$ -step preconditioner is efficient in accelerating GMRES for solving a non-Hermitian positive definite linear system.

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.

On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation [formula omitted

In this paper, a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving the linear matrix equation AXB=C is proposed. This method is formed by extending the corresponding HSS iteration method for solving Ax=b. The analysis shows that the HSS iteration method will converge under certain assumptions. Moreover, the optimal parameter of this iteration method is presented. Numerical results show that this new method is more efficient and robust than existing methods.

On Hermitian and Skew-Hermitian Splitting Ietration Methods for the Continuous Sylvester Equations

We present a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems

For the singular, non-Hermitian, and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the semi-convergence of the Hermitian and skew-Hermitian splitting (HSS) iteration methods. We then investigate the semi-convergence factor and estimate its upper bound for the HSS iteration method. If the semi-convergence condition is satisfied, it is shown that the semi-convergence rate is the same as that of the HSS iteration method applied to a linear system with the coefficient matrix equal to the compression of the original matrix on the range space of its Hermitian part, that is, the matrix obtained from the original matrix by restricting the domain and projecting the range space to the range space of the Hermitian part. In particular, an upper bound is obtained in terms of the largest and the smallest nonzero eigenvalues of the Hermitian part of the coefficient matrix. In addition, applications of the HSS iteration method as a preconditioner for Krylov subspace methods such as GMRES are investigated in detail, and several examples are used to illustrate the theoretical results and examine the numerical effectiveness of the HSS iteration method served either as a preconditioner for GMRES or as a solver.

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.