Hermitian And skew-Hermitian Splitting Iteration Research Articles

Developing HSS iteration schemes for solving the quadratic matrix equation AX2+BX+C=0$AX^{2}+BX+C=0$

AbstractThe quadratic matrix equation (QME) occurs in the branches such as the quadratic eigenvalue problems and quasi‐birth‐death processes. Also, the numerical solution of QMEs is an essential step in many computational methods for linear‐quadratic and robust control, filtering, controller order reduction, inner‐outer factorization, spectral factorization, and other applications. In this study, schemes are presented to solve the QME based on the Hermitian and skew‐Hermitian splitting (HSS). It is shown that the proposed schemes converge to the solutions of the QME. Finally, some examples are solved to discover the application of the schemes in comparison with Newton's method.

Hermitian and skew-Hermitian splitting methods for solving a tensor equation

The present paper deals with the numerical solution of non-Hermitian positive definite tensor equation under the Einstein product. Firstly, we extend the Hermitian and skew-Hermitian splitting (HSS) method to solve the tensor equation. Then we propose a new Hermitian splitting (NHS) method under some certain conditions, which is expected to converge faster than the HSS iteration. We also present the optimal parameters of both the HSS and NHS methods. Moreover, we apply the Smith technique to give two modified methods which can greatly accelerate the convergence rate. The performed numerical examples illustrate that the proposed methods are feasible and efficient.

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.

On partially inexact HSS iteration methods for the complex symmetric linear systems in space fractional CNLS equations

The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal-plus-Toeplitz matrix. The Hermitian and skew-Hermitian splitting (HSS) method and the partially inexact HSS (PIHSS) method are employed to solve the discretized linear system. In the inner iteration processes of the HSS method, we only need to solve the linear sub-systems associated with the Hermitian part inexactly by the conjugate gradient (CG) method, resulting in PIHSS iteration method. Theoretical analyses show that both HSS and PIHSS methods are unconditionally convergent. Numerical examples are given to demonstrate the effectiveness of the HSS iteration and the PIHSS iteration.

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.

Skew-Hermitian based iterations for nine-point approximations of convection–diffusion problems

We study skew-Hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high-order compact (HOC) approximations of two-dimensional convection–diffusion problems. Such discretisations lead to nine-point system matrices with block-tridiagonal structures. The nonsymmetric discretisation matrix for a constant-coefficient grid-aligned flow problem is shown to be positive definite and thus Hermitian and skew-Hermitian splitting (HSS) methods can be considered for the solution of the corresponding linear system. For the solution of this linear system using the HSS iteration, we derive an analytical expression for the optimal value for the upper bound of the contraction factor. Comparison of the HSS iteration with triangular and skew-Hermitian splitting (TSS) methods and their corresponding block variants (BTSS) are carried out for various test problems. Cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given. Our results show that the combination of high-order compact discretisations and skew-Hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two-dimensional convection–diffusion equations.

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.

Generalized Preconditioned Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive-Definite Linear Systems

In this paper, a generalized preconditioned Hermitian and skew-Hermitian splitting (GPHSS) iteration method for a non-Hermitian positive-definite matrix is studied, which covers standard Hermitian and skew-Hermitian splitting (HSS) iteration and also many existing variants. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From practical point of view, we have analyzed and implemented inexact generalized preconditioned Hermitian and skew-Hermitian splitting (IGPHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments from three-dimensional convection-diffusion equation show that the GPHSS and IGPHSS iterations are efficient and competitive with standard HSS iteration and AHSS iteration.

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 Newton-HSS Methods for Systems of Nonliear Equations with Positive-Definite Jacobian Matrices

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implementations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step. Mathematics subject classification: 65F10, 65W05.