一类弱非线性方程组的Picard-MHSS迭代方法  

ON PICARD-MHSS METHODS FOR WEAKLY NONLINEAR SYSTEMS

作　　者：王洋[1] 伍渝江[2] 付军[1] 

机构地区：[1]吉林师范大学数学学院,吉林四平136000 [2]兰州大学数学与统计学院,兰州730000

出　　处：《计算数学》2014年第3期291-302,共12页Mathematica Numerica Sinica

基　　金：国家基础研究规划973项目(2011CB706903);吉林省自然科学基金(201115222);吉林省教育厅十二五科技项目(20130578);吉林师范大学博士启动项目资助(2011033);四平市科技发展计划项目(20111006);吉林师范大学研究生创新项目(201114);吉林师范大学高等教育教学研究项目;吉林省科技发展计划资助项目(20130420085FG);四平市科技发展计划资助项目(四科合字2011003)

摘　　要：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.Modified Hermitian and skew-Hermitian splitting（MHSS） iteration method is an unconditionally convergent method for solving large sparse complex symmetric linear systems. Based on the special structure and properties of the nonlinear systems, choosing Picard iteration as outer iteration and MHSS iteration as the inner solver of Picard iteration, we present the Picard-MHSS and nonlinear MHSS-like iteration methods for solving large scale systems of weakly nonlinear equations. The advantage of these methods is that they do not require explicit construction and accurate computation of the Jacobian matrix, and only need to solve linear sub-systems of constant coefficient real symmetric positive definite matrices. Moreover, Under suitable conditions, we establish local convergence theorems for both Picar-MHSS and nonlinear MHSS-like iteration methods. Numerical results show that these iteration methods are feasible, effective and robust.

关 键 词：非线性方程组 Hermite反Hermite分裂 Picard方法 局部收敛性质 

分 类 号：O241.6[理学—计算数学]