关于PageRank的广义二级分裂迭代方法

本文研究计算PageRank的迭代法,在Gleich等人提出的内/外迭代方法的基础上,提出了具有三个参数的广义二级分裂迭代法,该方法包含了内/外迭代法和幂迭代法,并研究了该方法的收敛性.基于该方法的收缩因子的计算公式,讨论了迭代参数可能的选择,通过参数的选择能有效提高内/外迭代法的收敛效率.     

关于Leontief产出方程的二级分裂迭代方法

研究Leontief投入产出模型中计算产出向量的迭代方法,基于Leontief产出方程,在矩阵规模很大,直接计算逆矩阵很困难的条件下,通过引入参数并运用二级分裂迭代思想和松弛技术,提出了Leontief产出方程的二级分裂迭代方法,给出了该方法的收敛理论.利用给出的收敛因子的计算方法,讨论了参数的优化选择,数值实例验证了此方法的有效性,表明优化参数能有效提高迭代方法的收敛效率.     

对称正定Toeplitz方程组的多级迭代求解

二级迭代法亦称内外迭代法. 多级迭代法由多个二级迭代嵌套而成.这些方法特别适合于并行计算,同时可以理解为古典迭代法的延伸或共轭梯度法的预处理子.本文讨论了对称正定Toeplitz线性方程组多级迭代法. 首先,基于Toeplitz矩阵的结构, 我们给出了多级块Jacobi分裂,然后证明了每一级分裂均为P-正则分裂, 并证明了当每一级内迭代次数均为偶数时,迭代法的收敛性. 最后通过数值实例验证了此方法的有效性.     

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.     

一种求解鞍点问题的广义对称超松弛迭代法

本文研究了鞍点问题的迭代算法.利用新的待定参数加速迭代格式并结合SSOR分裂的方法,获得了有两个参数的广义对称超松弛迭代法及其收敛性条件.数值例子表明选择适当的参数值可以提高算法的收敛效率,推广和改进了SOR-like迭代法.     

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.     

关于Stokes和线性Navier-Stokes方程的广义维数分裂迭代方法

本文研究了鞍点问题的迭代法.在Benzi等人提出的维数分裂(DS)迭代方法的基础上,提出了具有三个参数的广义维数分裂(GDS)迭代法,该方法包含了DS迭代法,理论分析表明该方法是无条件收敛的.通过对有限差分法和有限元法离散的Stokes问题及有限元法离散的Oseen问题的数值结果表明,本文所给方法是有效的.     

基于分数阶扩散方程的离散线性代数方程组迭代方法研究

本文构建一类双参数拟Toeplitz分裂（TQTS）迭代方法求解变系数非定常空间分数阶扩散方程.TQTS迭代法是基于QTS迭代法引入双参技术建立而成,通过选取适当的参数使迭代矩阵谱半径变得更小,从而有效提升收敛的速度.然后对TQTS迭代法进行收敛性分析,获得相应的收敛区域,并对迭代法中涉及的参数进行讨论,获得使迭代矩阵谱半径上界达到最小的最优参数的表达式.最后通过数值仿真实验验证TQTS迭代法的有效性,实验结果表明TQTS迭代法改进效果十分突出,在迭代时间和步数上均有明显的减小.     

关于鞍点问题的预处理HSS-SOR交替分裂迭代方法

为了高效地求解大型稀疏鞍点问题,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SOR-like迭代格式对原有迭代算法进行加速,提出了一种预处理HSS-SOR交替分裂迭代方法,并研究了该算法的收敛性.数值例子表明:通过参数值的选择,新算法比SOR-like和PHSS算法都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,可以提高算法的收敛效率.     

一类复对称线性方程组的单步HSS迭代法

基于修正的埃尔米特和反埃尔米特分裂(MHSS)及预处理的MHSS(PMHSS)迭代法,提出了关于一类复对称线性方程组的单步MHSS(SMHSS)和单步PMHSS(SPMHSS)迭代法,进一步利用优化技巧给出了位移参数的动态选择格式,得出相应的带有灵活位移的SMHSS方法及SPMHSS迭代法.理论分析表明,迭代参数α在较弱的约束条件下,SMHSS迭代法收敛于复对称线性方程组的唯一解.同时,得到了SMHSS迭代矩阵的谱半径的上界,并且求得使上述上界最小的最优参数α~*.进一步给出了SPMHSS方法的收敛性分析.MHSS法和SMHSS迭代法之间的数值比较表明,在某些情况下,SMHSS迭代法比MHSS迭代法更优.     

Efficiently implementable iterative methods for linear elliptic variational inequalities with constraints on the gradient of solution

We construct and investigate a new iterative solution method for a finite-dimensional constrained saddle point problem. The results are applied to prove the convergence of different iterativemethods formesh approximations of variational inequalities with constraints on the gradient of solution. In particular, we prove the convergence of two-stage iterative methods. The main advantage of the proposed methods is the simplicity of their implementation. The numerical testing demonstrates high convergence rate of the methods.     

Comparison results on preconditioned SOR-type iterative method for Z-matrices linear systems

In this paper, we present some comparison theorems on preconditioned iterative method for solving Z-matrices linear systems, Comparison results show that the rate of convergence of the Gauss–Seidel-type method is faster than the rate of convergence of the SOR-type iterative method.     

New preconditioned AOR iterative method for Z-matrices

In this paper, we present a new preconditioned AOR-type iterative method for solving the linear system Ax=b, where A is a Z-matrix, and prove its convergence. Then we give some comparison theorems to show that the rate of convergence of the preconditioned AOR-type iterative method is faster than the rate of convergence of the AOR-type iterative method. Finally, we give two numerical examples to illustrate our results.     

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

The finite difference or the finite element discretizations of many differential or integral equations often result in a class of systems of weakly nonlinear equations. In this paper, by reasonably applying both the multisplitting and the two-stage iteration techniques, and in accordance with the special properties of this system of weakly nonlinear equations, we first propose a general multisplitting two-stage iteration method through the two-stage multiple splittings of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a multisplitting two-stage AOR method, which particularly uses the AOR-like iteration as inner iteration and is substantially a relaxed variant of the afore-presented method. These two methods have a forceful parallel computing function and are much more suitable to the high-speed multiprocessor systems. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only directionally differentiable. When the system matrix is either an H-matrix or a monotone matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new multisplitting two-stage iteration methods, and investigate the influence of the multiple splittings as well as the relaxation parameters upon the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for parallel solving of the system of weakly nonlinear equations. This revised version was published online in August 2006 with corrections to the Cover Date.     

矩阵分裂序列与线性二级迭代法

本文讨论线性非定常二级迭代法的收敛性．对于一般的基于矩阵分裂序列的迭代法,针对分裂序列本身找到了一种新的且相对较弱的收敛性条件,并因此得到了由非定常二级迭代法推广而来的广义二级迭代法的收敛结果．从而,用一种新的方法证明了非定常二级迭代法的收敛性．     

The alternating-direction iterative method for saddle point problems

In the paper, a new alternating-direction iterative method is proposed based on matrix splittings for solving saddle point problems. The convergence analysis for the new method is given. When the better values of parameters are employed, the proposed method has faster convergence rate and less time cost than the Uzawa algorithm with the optimal parameter and the Hermitian and skew-Hermitian splitting iterative method. Numerical examples further show the effectiveness of the method.