Abstract
The conventional GPBi-CG method involves inherently instability of convergence rate. Even if the so-called preconditioning technique is adopted in the GPBi-CG method for reduction of difficulties of solving problems, the situation remains as it is. The fact that the GPBi-CG method computed in finite arithemetics has two variants of residual causes the instability. We refer to this situation as “split of residual”. On the other hand, in our proposed GPBiCG_AR method we does not almost meet with the above “split of residual” because of consistency of residual. In this paper, we introduce two kinds of the preconditioned GPBi-CG and GPBiCG_AR methods. Through numerical experiments, we demonstrate that the preconditioned GPBiCG_AR method performs well as compared with the preconditioned GPBi-CG method in view of stability of convergence and robustness of convergence for several problems.
