THE GENERALIZED LOCAL HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR THE NON-HERMITIAN GENERALIZED SADDLE POINT PROBLEMS

For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 （2012） 8816-8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting （GLHSS） iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.     

非Hermitian正定线性方程组的外推的HSS迭代方法

为了高效地求解大型稀疏非Hermitian正定线性方程组，在白中治、Golub和Ng提出的Hermitian和反Hermitian分裂（HSS）迭代法的基础上，通过引入新的参数并结合迭代法的松弛技术，对HSS迭代方法进行加速，提出了一种新的外推的HSS迭代方法（EHSS），并研究了该方法的收敛性.数值例子表明：通过参数值的选择，新方法比HSS方法具有更快的收敛速度和更少的迭代次数，选择了合适的参数值后，可以提高HSS方法的收敛效率.     

ON HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR 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/semi-definite 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.     

连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速

对于求解大型稀疏连续Sylvester方程,Bai提出了非常有效的Hermitian和反Hermitian分裂(HSS)迭代法.为了进一步提高求解这类方程的效率,本文建立一种广义正定和反Hermitian分裂(GPSS)迭代法,并且提出不精确GPSS(IGPSS)迭代法从而可以降低计算成本.对GPSS迭代法及其不精确变...     

一类弱非线性方程组的Picard-MHSS迭代方法

修正的Hermite/反Hermite分裂(MHSS)迭代方法是一类求解大型稀疏复对称线性代数方程组的无条件收敛的迭代算法.基于非线性代数方程组的特殊结构和性质,我们选取Picard迭代为外迭代方法,MHSS迭代作为内迭代方法,构造了求解大型稀疏弱非线性代数方程组的Picard-MHSS和非线性MHSS-like方法.这两类方法的优点是不需要在每次迭代时均精确计算和存储Jacobi矩阵,仅需要在迭代过程中求解两个常系数实对称正定子线性方程组.除此之外,在一定条件下,给出了两类方法的局部收敛性定理.数值结果证明了这两类方法是可行、有效和稳健的.     

二阶锥线性互补问题的广义模系矩阵分裂迭代算法

通过将二阶锥线性互补问题转化为等价的不动点方程,介绍了一种广义模系矩阵分裂迭代算法,并研究了该算法的收敛性.进一步,数值结果表明广义模系矩阵分裂迭代算法能够有效地求解二阶锥线性互补问题.     

关于非Hermitian正定线性代数方程组的超松弛HSS方法

本文针对求解大型稀疏非Hermitian正定线性方程组的HSS迭代方法,利用迭代法的松弛技术进行加速,提出了一种具有三个参数的超松弛HSS方法(SAHSS)和不精确的SAHSS方法(ISAHSS),它采用CG和一些Krylov子空间方法作为其内部过程,并研究了SAHSS和ISAHSS方法的收敛性.数值例子验证了新方法的有效性.     

GENERALIZED CONJUGATE A-ORTHOGONAL RESIDUAL SQUARED METHOD FOR COMPLEX NON-HERMITIAN LINEAR SYSTEMS

Recently numerous numerical experiments on realistic calculation have shown that the conjugate A-orthogonal residual squared （CORS） method is often competitive with other popular methods. However, the CORS method, like the CGS method, shows irreg- ular convergence, especially appears large intermediate residual norm, which may lead to worse approximate solutions and slower convergence rate. In this paper, we present a new product-type method for solving complex non-Hermitian linear systems based on the bicon- jugate A-orthogonal residual （BiCOR） method, where one of the polynomials is a BiCOR polynomial, and the other is a BiCOR polynomial with the same degree corresponding to different initial residual. Numerical examples are given to illustrate the effectiveness of the proposed method.     

几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.     

一类H矩阵线性互补问题的预处理二步模基矩阵分裂迭代方法

本文我们利用预处理技术推广了求解线性互补问题的二步模基矩阵分裂迭代法,并针对H-矩阵类给出了新方法的收敛性分析,得到的理论结果推广了已有的一些方法.     

ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR 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 implemen- tations 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.     

关于鞍点问题的广义预处理HSS-SOR交替分裂迭代方法

本文研究了鞍点问题的迭代法. 在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂（PHSS）迭代法的基础上,通过结合GSOR迭代格式,利用两个参数加速,提出了一种广义预处理HSS-SOR交替分裂迭代法,并研究了该方法的收敛性.数值结果表明本文所给方法是有效的.     

解强刚性块线代数方程组的两类L-收敛迭代法

对解强刚性块线代数方程组X=(A(?)J)X φ,本文提出了L-收敛的最佳单参数迭代法(L-OOPI)和L-收敛的多参数迭代直接法(L-MPID),并给出了数值例子．数例表明,对于强刚性块线代数方程组,该二迭代法是有效的．     

A NOTE ON COMPARISON THEOREM OF TWO-STAGE ITERATIVE METHOD FOR HERMITIAN POSITIVE DEFINITE LINEAR SYSTEMS

We discuss two-stage iterative methods for the solution of linear system Ax = b, and give a new proof of the comparison theorems of two-stage iterative method for an Hermitian positive definite matrix. Meanwhile, we put forward two new versions of well known comparison theorem and apply them to some examples.     

鞍点问题的广义位移分裂预条件子

对于大型稀疏非Hermitian正定线性方程组,Bai等人提出了一种位移分裂预条件子(J.Comput.Math.,24(2006)539-552).本文将这种思想用到鞍点问题上并提出了一种广义位移分裂(Generalized Shift Splitting,GSS)预条件子,同时证明了该预条件子所对应分裂迭代法的无条件收敛性.最后用数值算例验证了新预条件子的有效性.     

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.     

FAST PARALLELIZABLE METHODS FOR COMPUTING INVARIANT SUBSPACES OF HERMITIAN MATRICES

We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big O notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.     

Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems

By further generalizing the modified skew-Hermitian triangular splitting iteration methods studied in [L. Wang, Z.-Z. Bai, Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math. 44 (2004) 363-386], in this paper, we present a new iteration scheme, called the product-type skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this method. Moreover, when it is applied to precondition the Krylov subspace methods, the preconditioning property of the product-type skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that the product-type skew-Hermitian triangular splitting iteration method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

求解带Toeplitz矩阵的线性互补问题的一类预处理模系矩阵分裂迭代法

针对系数矩阵为对称正定Toeplitz矩阵的线性互补问题,本文提出了一类预处理模系矩阵分裂迭代方法.先通过变量替换将线性互补问题转化为一类非线性方程组,然后选取Strang或T.Chan循环矩阵作为预优矩阵,利用共轭梯度法进行求解.我们分析了该方法的收敛性.数值实验表明,该方法是高效可行的.     

A FLEXIBLE PRECONDITIONED ARNOLDI METHOD FOR SHIFTED LINEAR SYSTEMS

We are interested in the numerical solution of the large nonsymmetric shifted linear system, （A ＋ αI）x -= b, for many different values of the shift a in a wide range. We apply the Saad＇s flexible preconditioning technique to the solution of the shifted systems. Such flexible preconditioning with a few parameters could probably cover all the shifted systems with the shift in a wide range. Numerical experiments report the effectiveness of our approach on some problems.