关于线性互补问题的模系矩阵分裂迭代方法

模系矩阵分裂迭代方法是求解大型稀疏线性互补问题的有效方法之一.本文的目标是归纳总结模系矩阵分裂迭代方法的最新发展和已有成果,主要内容包括相应的多分裂迭代方法, 二级多分裂迭代方法和两步多分裂迭代方法, 以及这些方法的收敛理论.     

H-矩阵非线性互补问题基于模的矩阵分裂迭代法改进的收敛性定理北大核心CSCD

该文在较弱的条件下,证明了解一类H-矩阵非线性互补问题基于模的矩阵分裂迭代法和相应的加速迭代法的收敛性定理.这意味着对于分裂A=M-N有更多的选择,使得基于模的矩阵分裂迭代法得以收敛.改进的收敛性定理扩展了基于模的矩阵分裂迭代法的应用范围.     

解线性互补问题的预处理加速模Gauss-Seidel迭代方法

本文提出了解线性互补问题的预处理加速模系Gauss-Seidel迭代方法,当线性互补问题的系统矩阵是M-矩阵时证明了方法的收敛性,并给出了该预处理方法关于原方法的一个比较定理.数值实验显示该预处理迭代方法明显加速了原方法的收敛.     

求解一类非线性互补问题的松弛two-sweep模系矩阵分裂迭代法

本文构造了求解一类非线性互补问题的松弛two-sweep模系矩阵分裂迭代法. 理论分析建立了新方法在系数矩阵为正定矩阵或H₊矩阵时的收敛性质．数值实验结果表明新方法是行之有效的, 并且在最优参数下松弛two-sweep模系矩阵分裂迭代法在迭代步数和时间上均优于传统的模系矩阵分裂迭代法和two-sweep模系矩阵分裂迭代法.     

求解增广线性系统的局部多分裂迭代法

为了在高性能计算机上求解增广线性系统,基于并行多分裂的两种技巧,本文提出一种局部多分裂迭代格式,给出当增广线性系统的矩阵为M-矩阵和H-矩阵时新方法的收敛性理论.并讨论预条件矩阵的特征值情形.     

矩阵多分裂迭代法的收敛条件

本文给出了求解非奇异线性方程组的矩阵多分裂并行迭代法的一些新的收敛结果.当系数矩阵单调和多分裂序列为弱正则分裂时,得到了几个与已有的收敛准则等价的条件,并且证明了异步迭代法在较弱条件下的收敛性.对于同步迭代,给出了与异步迭代不同且较为宽松的收敛条件.     

M—矩阵分裂的迭代矩阵

1 迭代矩阵谱半径的代数重数 设A=M—N是M-矩阵的正则分裂。一般地,mult_0(A)与mult_1(M~(-1)N)不一定相等.我们研究在弱正则分裂下使mult_0(A)=mult_1(M~(-1)N)的条件. 引理 1.1 设A∈R~(nn)是有“性质C”的M-矩阵,rank(A)=n—1.则mult_0(A)=1. 证明 显然. 引理 1.2 设A=M—N是奇异不可约M-矩阵的弱正则分裂,则     

M-矩阵平方根的一类迭代法

矩阵平方根在数学的许多应用中起着重要的作用.本文研究M-矩阵平方根的计算问题,提出一种计算正则M-矩阵平方根的迭代方法.首先将这个问题转化为M-矩阵代数Riccati方程,进而提出一种有效的方法来求解这个特殊的MARE.理论分析表明,该方法在一定条件下是收敛的.数值实验表明该方法是可行的,且优于二项式迭代法.     

求解一类隐式互补问题的加速模系矩阵分裂迭代方法

本文提出求解一类隐式互补问题的加速模系矩阵分裂迭代法.通过将隐式互补问题重新表述为一个等价的不动点方程,建立一类新的基于模系的两步矩阵分裂方法,并在一定条件下证明了方法的收敛性.数值实验表明,该方法在迭代步数上优于传统的模系矩阵分裂迭代方法.     

奇异M─矩阵的图相容弱正则分裂

设Ａ是奇异Ｍ－矩阵，Ａ＝Ｍ－Ｎ是Ａ的图相容弱正则分裂．本文研究迭代矩阵Ｍ－１Ｎ的谱性质，得到与迭代矩阵的指数有关的一个定理：ｉｎｄ０（Ａ）＝ｉｎｄ１（Ｍ－１Ｎ）．它推广了Ｈ．Ｓｃｈｎｅｉｄｅｒ和作者的结果．     

Two-Stage Multisplitting Iteration Methods Using Modulus-Based Matrix Splitting as Inner Iteration for Linear Complementarity Problems

The matrix multisplitting iteration method is an effective tool for solving large sparse linear complementarity problems. However, at each iteration step we have to solve a sequence of linear complementarity sub-problems exactly. In this paper, we present a two-stage multisplitting iteration method, in which the modulus-based matrix splitting iteration and its relaxed variants are employed as inner iterations to solve the linear complementarity sub-problems approximately. The convergence theorems of these two-stage multisplitting iteration methods are established. Numerical experiments show that the two-stage multisplitting relaxation methods are superior to the matrix multisplitting iteration methods in computing time, and can achieve a satisfactory parallel efficiency.     

A two-step modulus-based multisplitting iteration method for the nonlinear complementarity problem

In this paper, we construct a two-step modulus-based multisplitting iteration method based on multiple splittings of the system matrix for the nonlinear complementarity problem. And we prove its convergence when the system matrix is an $H$-matrix with positive diagonal elements. Numerical experiments show that the proposed method is efficient.     

On multisplitting methods for band matrices

Summary. We present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively. These classes are based on overlapping blocks of the underlying coefficient matrix which is assumed to be a band matrix. We show that under suitable conditions the spectral radius of the iteration matrix does not depend on the weights of the method even if these weights are allowed to be negative. For a certain class of splittings we prove an optimality result for with respect to the weights provided that is an M–matrix. This result is based on the fact that the multisplitting method can be represented by a single splitting which in our situation surprisingly turns out to be a regular splitting. Furthermore we show by numerical examples that weighting factors may considerably improve the convergence. Received July 18, 1994 / Revised version received November 20, 1995     

Two-Step Modulus-Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems

To reduce the communication among processors and improve the computing time for solving linear complementarity problems, we present a two-step modulus-based synchronous multisplitting iteration method and the corresponding symmetric modulus-based multisplitting relaxation methods. The convergence theorems are established when the system matrix is an $H_+$-matrix, which improve the existing convergence theory. Numerical results show that the symmetric modulus-based multisplitting relaxation methods are effective in actual implementation.     

Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems

In order to solve large sparse linear complementarity problems on parallel multiprocessor systems, we construct modulus-based synchronous two-stage multisplitting iteration methods based on two-stage multisplittings of the system matrices. These iteration methods include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, SOR and AOR of the modulus type as special cases. We establish the convergence theory of these modulus-based synchronous two-stage multisplitting iteration methods and their relaxed variants when the system matrix is an H _?+?-matrix. Numerical results show that in terms of computing time the modulus-based synchronous two-stage multisplitting relaxation methods are more efficient than the modulus-based synchronous multisplitting relaxation methods in actual implementations.     

Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods

Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied. Received September 5, 1995 / Revised version received April 3, 1996     

Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems

Abstract

In this paper, the convergence conditions of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems of H-matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones.     

Two-step modulus-based matrix splitting iteration method for linear complementarity problems

Bai has recently presented a modulus-based matrix splitting iteration method, which is a powerful alternative for solving the large sparse linear complementarity problems. In this paper, we further present a two-step modulus-based matrix splitting iteration method, which consists of a forward and a backward sweep. Its convergence theory is proved when the system matrix is an H ₊-matrix. Moreover, for the two-step modulus-based relaxation iteration methods, more exact convergence domains are obtained without restriction on the Jacobi matrix associated with the system matrix, which improve the existing convergence theory. Numerical results show that the two-step modulus-based relaxation iteration methods are superior to the modulus-based relaxation iteration methods for solving the large sparse linear complementarity problems.