复线性方程组的预处理MCG算法  

Preconditioned MCG Method for Complex Linear Systems

作　　者：张迎春 吕全义[1] 肖曼玉[1] ZHANG Yingchun,LV Quanyi,XIAO Manyu(Department of Applied Mathematics,Northwestern Polytechnical University,Xi＇an 71007)

机构地区：[1]西北工业大学应用数学系,西安710072

出　　处：《工程数学学报》2018年第3期308-318,共11页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11302173);陕西省自然科学基金(2017JQ1037);研究生培养过程保障计划-研究生高水平课程(17GH020213)~~

摘　　要：复线性方程组在科学与工程计算的诸多领域中有着重要的应用价值,如何高效的求解复线性方程组,一直是人们所关心的问题.目前对于复线性方程组,常用的处理方式有以下两种:一种是直接对方程组迭代求解,另外一种是将其转化为实线性方程组后进行求解.本文主要从两种处理方式讨论了共轭梯度法(CG法),并理论上证明了两种处理方式下的CG法具有相同的收敛性.之后基于变形共轭梯度法(MCG法)收敛速度的本质与CG法类似,只需将MCG法推广到复线性方程组进行研究,并且为了提高MCG法的收敛速度,提出了一种预处理MCG法.最后,通过数值算例验证了算法与理论分析的一致性,以及预处理算法的有效性.Complex linear equations have a wide application in science and engineering, and an important issue is how to solve it with high efficiency. Until now, complex linear equations are usually solved by either iteration methods or the solution of the real equations transformed from the original equations. Conjugate gradient method （CG method） is discussed from two different viewpoints, and it is proved theoretically that these two kinds of CG methods have the same convergence. Because the convergence speed of the modified conjugate gradient method （MCG method） and conjugate gradient method are essentially similar, MCG method is extended to solve complex linear equations. Besides, a preconditioned MCG method is proposed in order to improve the convergence speed. Finally, the consistency of algorithms and theoretical analysis and effectiveness of the proposed precondition algorithm are validated by numerical examples.

关 键 词：复线性方程组 变形共轭梯度法(MCG法) 预处理方法 收敛性 

分 类 号：O241.6[理学—计算数学]