关键词: 预处理, 共轭梯度法, 随机初值, 迭代次数

Abstract: It was shown in literatures of the preconditioned conjugate gradient (PCG) that the initial guess gives little influence upon the number of PCG iterations when the stopping criterion requests to reduce the residual norm by a factor. However, the examples reported here show that using a zero initial gress or different random initial guess causes the number of PCG iterations varies seriously. Moreover, the number of iterations still varies seriously for the different parameter of the modeb when the same random initial guess is used. This variation should be avoided since it may cause confusions when different methods are compared. This paper shows that if the preconditioner is given, the series {r_k} determined uniquely by the linear systems provided a zero initial guess is used. To avoid the serious variation in the number of PCG iterations, using the zero initial guess is a good choice.

Key words: preconditioned conjugate gradient, random initial guess, number of iterations