HSS Iteration Research Articles

Efficient parameterized HSS iteration methods for complex symmetric linear systems

Based on the new HSS (NHSS) iteration method proposed by Pour and Goughery (2015) and the efficient PSHSS iteration method by Zeng and Ma (2016), we introduce an efficient parameterized HSS (PNHSS) and a parameterized single-step HSS (PS*HSS) iteration methods for solving a class of complex symmetric linear systems. Convergence properties of the PNHSS and the PS*HSS iteration methods are studied, which show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess under a loose restriction on the parameter ω. Furthermore, we derive an upper bound for the spectral radius of the PNHSS iteration matrix, and the quasi-optimal parameters α∗ and ω∗ which minimize the above upper bound are also considered. Both theoretical and numerical results show that the PNHSS and the PS*HSS iteration methods outperform the NHSS and the SHSS iteration methods. Little difference about the computational efficiency from the point of view of the CPU times between the PS*HSS, the PNHSS and the PSHSS iteration methods is justified by using the experimental optimal parameters. However, sometimes the PS*HSS and the PNHSS iteration methods are more efficient than the PSHSS iteration method when the experimental optimal parameters are not used.

Regularized HSS iteration methods for saddle-point linear systems

We propose a class of regularized Hermitian and skew-Hermitian splitting methods for the solution of large, sparse linear systems in saddle-point form. These methods can be used as stationary iterative solvers or as preconditioners for Krylov subspace methods. We establish unconditional convergence of the stationary iterations and we examine the spectral properties of the corresponding preconditioned matrix. Inexact variants are also considered. Numerical results on saddle-point linear systems arising from the discretization of a Stokes problem and of a distributed control problem show that good performance can be achieved when using inexact variants of the proposed preconditioners.

A Modified HSS Iteration Method for Solving the Complex Linear Matrix Equation AXB = C

在这，为解决复杂线性矩阵方程 AXB = C 的切开的纸，修改 Hermitian 和 skew-Hermitian (MHSS ) 重复方法被介绍了。作为理论分析表演， MHSS 重复方法将在某些条件下面收敛。在这个方法的每次重复与真实对称的积极明确的系数矩阵要求四个线性矩阵方程的答案，尽管原来的系数矩阵是建筑群和 non-Hermitian。另外，新重复方法的最佳的参数被建议。数字结果证明那个 MHSS 重复方法有效、柔韧。[从作者抽象]

On choices of iteration parameter in HSS method

The HSS iteration method is effective to solve non-Hermitian positive definite linear systems, but the choice of its optimal parameter is a difficult and challenging problem in theoretical analysis and practical computations. In this paper, we obtain an accurate estimate to the optimal parameter of the HSS iteration method by adopting a reasonable and simple optimization principle. Numerical experiments show that this principle is feasible to produce an accurate estimate to the optimal parameter of the HSS iteration 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.

A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems

By utilizing the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration technique, we establish a non-alternating PHSS (NPHSS) iteration method for solving large sparse non-Hermitian positive definite linear systems. The convergence analysis demonstrates that the iterative series produced by the NPHSS method converge to the unique solution of the linear system when the parameters satisfy some moderate conditions. We also give a possible optimal upper bound for the iterative spectral radius. Moreover, to reduce the computational cost, we establish an inexact variant of the NPHSS (INPHSS) iteration method whose convergence property is studied. Both theoretical and numerical results validate that the NPHSS method outperforms the PHSS method when the Hermitian part of the coefficient matrix is dominant.

The variable HSS iteration based on the bacterial foraging optimisation algorithm

In the Hermitian and skew-Hermitian splitting HSS iteration method, the determination of the optimal parameter is a tough task when solving a large sparse non-Hermitian positive definite linear systems. In this paper, we present the variable HSS iteration with the non-fixed positive constant. For obtaining the approximate optimal parameters of this method, we solve the minimising the optimisation model based on the bacterial foraging optimisation BFO algorithm. Numerical experiments have shown that the new strategy is feasible and effective than the HSS iteration method.

On the convergence of a new splitting iterative method for non-Hermitian positive definite linear systems

In this paper we present a new splitting method for solving a linear systems with non-Hermitian positive definite coefficient matrix. This splitting overcomes the computation complexity of HSS. The spectral radius and some norm properties of the iteration matrix are discussed. With the results obtained, we study the reasonable choices of the parameter and introduce a preconditioner. Moreover, an accelerated algorithm is proposed. Finally, the numerical examples show the new method is much more efficient than the HSS (or the NSS and the PSS) iteration method.

A practical formula for computing optimal parameters in the HSS iteration methods

In the HSS iteration methods proposed by Bai, Golub and Ng Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM. J...

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.

A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems

We present a generalization of the preconditioned modified HSS (PMHSS) iteration method for solving two classes of complex symmetric indefinite linear systems and show that under suitable conditions the new iteration method is unconditionally convergent. Each iteration of the new method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. Numerical experiments are given to show the effectiveness of the new method.

A practical formula for computing optimal parameters in the HSS iteration methods

In the HSS iteration methods proposed by Bai, Golub and Ng [Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM. J. Matrix Anal. Appl. 24 (2003) 603–626], the determination of the optimal parameter is a tough task when solving a non-Hermitian positive definite linear system. In this paper, a new and simple strategy for obtaining the optimal parameter is proposed, which computes the optimal parameter by solving a cubic polynomial equation. The coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew-symmetric parts of the coefficient matrix of the real linear system. Numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the HSS iteration methods as it leads to fast convergence of the method.

A new splitting and preconditioner for iteratively solving non-Hermitian positive definite systems

A new iteration method for solving a linear system with coefficient matrix being non-Hermitian positive definite is presented in this note. We study the spectral radius and contraction properties of the iteration matrix and then analyze the best possible choice of the parameter. With the results obtained, we show that the new method is convergent for a non-Hermitian positive definite linear system and propose a preconditioner to improve the condition number of the system. The numerical examples show that the new method is much more efficient than the HSS (or PSS) iteration method.

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.

On preconditioned MHSS iteration methods for complex symmetric linear systems

We propose a preconditioned variant of the modified HSS ( MHSS ) iteration method for solving a class of complex symmetric systems of linear equations. Under suitable conditions, we prove the conve...

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.

On the HSS iteration methods for positive definite Toeplitz linear systems

We study the HSS iteration method for large sparse non-Hermitian positive definite Toeplitz linear systems, which first appears in Bai, Golub and Ng’s paper published in 2003 [Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003) 603–626], and HSS stands for the Hermitian and skew-Hermitian splitting of the coefficient matrix A . In this note we use the HSS iteration method based on a special case of the HSS splitting, where the symmetric part H = 1 2 ( A + A T ) is a centrosymmetric matrix and the skew-symmetric part S = 1 2 ( A − A T ) is a skew-centrosymmetric matrix for a given Toeplitz matrix. Hence, fast methods are available for computing the two half-steps involved in the HSS and IHSS iteration methods. Some numerical results illustrate their effectiveness.

Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems

By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian splitting (PSS), and then establish a class of PSS methods similar to the Hermitian (or normal) and skew-Hermitian splitting (HSS or NSS) method for iteratively solving the positive-definite systems of linear equations. Theoretical analysis shows that the PSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrix and independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved. When we specialize the PSS to block triangular (or triangular) and skew-Hermitian splitting (BTSS or TSS), the PSS method naturally leads to a BTSS or TSS iteration method, which may be more practical and efficient than the HSS and NSS iteration methods. Applications of the BTSS method to the linear systems of block two-by-two structures are discussed in detail. Numerical experiments further show the effectiveness of our new methods.

Explicit Solutions of Linear Matrix Equations

Explicit Solutions of Linear Matrix Equations