对称正定矩阵多级分裂预处理子

本文基于对称正定矩阵的多级分裂,讨论了这类方程组的多级分裂预处理子的构造,证明了预处理子的合理性.数值试验显示我们的预处理子是有效的.     

A High-Quality Preconditioning Technique for Multi-Length-Scale Symmetric Positive Definite Linear Systems

We study preconditioning techniques used in conjunction with the conjugate gradient method for solving multi-length-scale symmetric positive definite linear systems originating from the quantum Monte Carlo simulation of electron interaction of correlated materials. Existing preconditioning techniques are not designed to be adaptive to varying numerical properties of the multi-length-scale systems. In this paper, we propose a hybrid incomplete Cholesky (HIC) preconditioner and demonstrate its adaptivity to the multi-length-scale systems. In addition, we propose an extension of the compressed sparse column with row access (CSCR) sparse matrix storage format to efficiently accommodate the data access pattern to compute the HIC preconditioner. We show that for moderately correlated materials, the HIC preconditioner achieves the optimal linear scaling of the simulation. The development of a linear-scaling preconditioner for strongly correlated materials remains an open topic.     

Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems

A class of modified block SSOR preconditioners is presented for the symmetric positive definite systems of linear equations, whose coefficient matrices come from the hierarchical-basis finite-element discretizations of the second-order self-adjoint elliptic boundary value problems. These preconditioners include a block SSOR iteration preconditioner, and two inexact block SSOR iteration preconditioners whose diagonal matrices except for the (1,1)-block are approximated by either point symmetric Gauss–Seidel iterations or incomplete Cholesky factorizations, respectively. The optimal relaxation factors involved in these preconditioners and the corresponding optimal condition numbers are estimated in details through two different approaches used by Bank, Dupont and Yserentant (Numer. Math. 52 (1988) 427–458) and Axelsson (Iterative Solution Methods (Cambridge University Press, 1994)). Theoretical analyses show that these modified block SSOR preconditioners are very robust, have nearly optimal convergence rates, and especially, are well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

对称正定Toeplitz型方程组的混合预处理

本文提出一个新的预条件子,用共轭梯度法求解对称正定的Teoplitz型线性方程组.该预处理子构造简单,易于实施快速傅里叶变换.理论和数值实验显示,我们的预处理子与T.Chan预处理子收敛性相近.     

Skew-Hermitian Triangular Splitting Iteration Methods for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts

By further generalizing the skew-symmetric triangular splitting iteration method studied by Krukier, Chikina and Belokon (Applied Numerical Mathematics, 41 (2002), pp. 89–105), in this paper, we present a new iteration scheme, called the modified 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 new method in depth. Moreover, when it is applied to precondition the Krylov subspace methods like GMRES, the preconditioning property of the modified skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that, as both solver and preconditioner, the modified skew-Hermitian triangular splitting iteration method is very effective for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

The Use of Pre-conditioning in Iterative Methods for Solving Linear Equations with Symmetric Positive Definite Matrices

The asymptotic convergence rates of many standard iterativemethods for the solution of linear equations can be shown todepend inversely on the P-condition number of the co-efficientmatrix. The notion of minimizing the P-condition number andhence maximizing the convergence rate by the introduction ofa new pre-conditioning factor is shown to be computationallyfeasible. The application of this idea to the method of SimultaneousDisplacement, Richardson's method and other iterative methods,are discussed and numerical examples given to illustrate itseffectiveness.     

关于对称共轭点的有界正实部亚纯星象函数类

引入了一类关于对称共轭点的有界正实部亚纯p叶星象函数类,并获得了函数类的积分表达式,充分条件,Fekete-Szeg?问题和卷积性质.特别地,得到了亚纯p叶函数类的另一种积分表示方法.     

Multisplitting Iteration Schemes for Solving a Class of Nonlinear Complementarity Problems

We consider several synchronous and asynchronous multisplitting iteration schemes for solving aclass of nonlinear complementarity problems with the system matrix being an H-matrix.We establish theconvergence theorems for the schemes.The numerical experiments show that the schemes are efficient forsolving the class of nonlinear complementarity problems.     

Szasz不等式在亚正定矩阵和拟广义正定矩阵上的推广

实对称正定矩阵的Szasz不等式是Hadamard不等式的加细;本文将Szasz不等式推广到一类亚正定矩阵和拟广义正定矩阵上去,从而推广了关于实对称正定矩阵的Szasz不等式和Hadamard不等式.     

Stability of the Matrix Factorization for Solving Block Tridiagonal Symmetric Indefinite Linear Systems

In this paper, we propose a new factorization method for block tridiagonal symmetric indefinite matrices. We also discuss the stability of the factorization method. As a measurement of stability, an effective condition number is derived by using backward error analysis and perturbation analysis. It shows that under some suitable assumptions, the solution obtained by this factorization method is acceptable. Numerical results demonstrate that the factorization is stable if its condition number is not too large. This revised version was published online in July 2006 with corrections to the Cover Date.     

具有线性约束条件的Hermite型是否正定的一种判别法

具有线性约束条件的Ｈｅｒｍｉｔｅ型是否正定的一种判别法周伟光（南京教育学院数学系，南京２１００１７）在经济数学中，有一个对具有某种线性约束条件的实二次型是否正定的判别法。文「２］对此判别法作了严格的叙述与证明．按文［２］，该判别法叙述如下：＂设Ｈ一】...     

Asynchronous Spectrum Assignment for Linear Almost Periodic Systems with Diagonal Mean of the Coefficient Matrix

Differential Equations - We consider a linear control system with an almost periodic coefficient matrix whose mean is a diagonal matrix and with a linear state feedback control. The control matrix...     

一类常系数非齐次线性微分方程的特解公式

给出了一类常系数非齐次线性微分方程的特解的计算公式.     

线性Hamilton系数的几个辛格式

本文利用二次Pad啨逼近的几个结果构造了线性Ｈａｍｉｌｔｏｎ系数的几个辛格式     

一类区间数系数矩阵模糊线性方程组的迭代算法

讨论模糊线性方程组X=A-X+U解的存在条件及其迭代算法(其中,A-是区间数为元素的n阶矩阵,未知量X和常量U都是以模糊数为元素的n维向量,并且其加法和乘法均由Zadeh的扩张原理定义).首先研究解的存在条件,尔后探讨求解的迭代算法及误差估计.     

正线性方程组正解的判别

本文给出了正线性方程组正解的概念，给出了正线性方程组正解的若干判别方法     

A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems

Recently, the study of symmetric cone complementarity problems has been a hot topic in the literature. Many numerical methods have been proposed for solving such a class of problems. Among them, the problems concerned are generally monotonic. In this paper, we consider symmetric cone linear complementarity problems with a class of non-monotonic transformations. A smoothing Newton algorithm is extended to solve this class of non-monotonic symmetric cone linear complementarity problems; and the algorithm is proved to be well-defined. In particular, we show that the algorithm is globally and locally quadratically convergent under mild assumptions. The preliminary numerical results are also reported.     

Alternating Direction Method for a Class of Sylvester Matrix Equations with Linear Matrix Inequality Constraint

This paper introduces an alternating direction method of multipliers (ADMM) for finding solutions to a class of Sylvester matrix equation AXB = E subject to a linear matrix inequality constraint CXD≥G. Preliminary convergence properties of ADMM are presented. Numerical experiments are performed to illustrate the feasibility and effectiveness of ADMM. In addition, some numerical comparisons with a recent algorithm are also given.     

对“关于复正定矩阵行列式的不等式”一文的注记

本文指出文 [1 ]中的错误 ,并把文 [1 ]中关于复正定矩阵与正定 Hermite矩阵的行列式不等式推广到较为广泛的复矩阵类