求解非对称线性方程组的总体拟极小向后扰动方法

在利用QPMR方法求解非对称线性方程组(尤其是病态方程组)的Lanczos过程中通常会发生算法中断或数值不稳定的情况．为解决这个问题，将求解非对称线性方程组的QMR方法与总体向后扰动范数拟极小化的技巧相结合，给出求解非对称线性方程组的总体拟极小向后扰动方法(TQMBACK方法)，同时，为减少存储量和运算量，新算法将采用重新开始的循环格式，通常人们采用残量范数作为判断算法终止的准则，但是，当近似解非常接近真值时，残量范数是小的，而反过来不一定，为克服残量范数作为算法终止准则的不足，将总体向后扰动范数作为判断算法终止的准则，得到求解非对称线性方程组的循环总体拟极小向后扰动方法(RTQMBAK方法)，数值实验表明，新算法比Lanczos方法和QMR方法收敛速度更快．而且，新算法对求解病态的非对称线性方程组很有效。

The Total Quasi Minimal Backward Perturbation Method for Nonsymmetric Linear Systems

The Lanczos process is susceptible to possible breakdown and numerical instabilities of nonsymmetric linear systems. The total quasi-minimal backward perturbation (TQMBACK) method is used for nonsymmetric linear systems in this paper. This new method is composed of the Lanczos method and the technique which can minimize the norm of the total backward perturbation. At the same time, in order to reduce the computations and the memories, the restarted version is used in the TQMBACK method. A drawback of using residual norm as the termination in an iterative process is that the residual norm must be small if the approximation is accurate. However, the converse nees not to be true. By using the norm of the total quasi-minimal backward perturbation as the termination of the restarted version, the new method overcomes the problems of using residual norm as the termination, and then the restarted total quasi-minimal backward perturbation (RTQMBACK) method is proposed. At last, the experiments show that the new methods are much better than the old ones for nonsymmetric linear systems. Furthermore, the RTQMBACK method is suitable for ill-conditioned nonsymmetric linear systems.