求解非对称线性方程组的QMRGCGS方法

1 引言 求解非对称线性方程组Ax=b的双共轭梯度方法(BCG)[3]和它的变形共轭梯度平方方法(CGS)[6]都有典型的不规则收敛行为,后来Freund和Nachtigal提出一种BCG类方法,即拟极小剩余方法(QMR)[7],用来补救BCG方法的收敛性并且产生了光滑的收敛曲线。然而,象BCG方法一样,QMR方法要用到系数矩阵A及其转置A~T与向量的乘积,为了解决这一问题,Freund提出TFQMR方法,此方法具有拟极小剩余性,同时不需用到A~T与向量的乘积。

A QMRGCGS METHOD FOR NONSYMMETRIC LINEAR SYSTEM

Recently, Fokkema, Sleijpen and Vorst propose to consider products of two nearly BICG polynomials which leads to generalized CGS method(GCGS method). Just like CGS method and BiCG method, GCGS has no quasi-minimal residual property, so they all typically exhibit a rather irregular convergence behavior with wild oscillations in the residual norm. In this note, it is demonstrated that the quasi-minimal residual approach can also be used to obtain a smoothly convergent GCGS-like algorithm, called QMRGCGS method. QMRGCGS method contains the well known TFQMR and QMRCG STAB method, and at the same time gives two new methods, called QMRGCGS2 and QM RSCGS method. Finally, numerical experiments are reported.