Lagrange方程组的直接解法  

DIRECT SOLUTION OF LAGRANGE EQUATION SET

作　　者：郑宏[1] 刘德富[1] 黄哲聪[1] 

机构地区：[1]三峡大学三峡库区地质灾害教育部重点实验室

出　　处：《岩石力学与工程学报》2006年第10期2079-2085,共7页Chinese Journal of Rock Mechanics and Engineering

基　　金：国家自然科学基金重点项目(90510019)

摘　　要：由Lagrange乘子法所导致的线性方程组(称之为Lagrange方程组)的系数矩阵(称之为Lagrange矩阵)通常是一对称不定矩阵。当其中的主子阵(也就是刚度矩阵)亏秩时,求解会遇到许多困难,这些困难往往是导致许多程序员放弃Lagrange乘子法而选择罚函数法的根本原因。基于Sherman-Morrison公式和对称正定矩阵的LDLT分解,提出了一个稳定、高效并特别适用于并行求解的直接解法。最后,将所建议的方法用于采用移动最小二乘(MLS)插值的无单元Galerkin法(EFGM)的方程组的求解。The coefficient matrices are called the Lagrange matrix, and associated with the system of linear equations. The system of linear equations is referred to the Lagrange equation set in this study, which is deduced by the Lagrange multiplier method, and is in general symmetric indefinite matrices. Solving such a system would encounter some intricacies if its leading principal submatrix, i.e. the stiffness matrix, is rank deficient. This is believed to be one of the main reasons that many programmers would unwillingly give up the Lagrange multiplier method but select the penalty function method. Based on the Sherrnan-Morrison formula and the conventional LDL^T decomposition for symmetric positive definite matrices, a robust direct solution is proposed, which is efficient and particularly suitable for parallel computation. As a paradigm, the proposed procedure is used to solve the set of linear equations derived by the element-free Galerkin method（EFGM） with the moving least squares interpolation.

关 键 词：算法 LAGRANGE乘子法 对称不定矩阵 无单元Galerkin法 

分 类 号：O242[理学—计算数学]