A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases of the search space and its image are maintained orthonormal that increases the robustness of the method. Moreover there is no need to store both bases since they are effectively represented within a common basis. Along with it our method is theoretically equivalent to the GCR method extended for a case of multiple right-hand sides but is more numerically robust and requires less memory. The main result of the paper is a mechanism of adding an arbitrary direction vector to the search space that can be easily adopted for flexible GMRES or GMRES with deflated restarting.

翻译：本文提出了一种GMRES方法的新变体，用于求解具有相同矩阵及后续获得的多重右端项的线性系统。新方法保持了经典GMRES算法的以下特性：搜索空间基与其像空间基均保持正交归一化，这增强了方法的鲁棒性。此外，由于两个基可通过公共基有效表示，无需同时存储两者。与此同时，本方法在理论上等价于针对多重右端项情形扩展的GCR方法，但具有更好的数值鲁棒性和更低的内存需求。本文的主要贡献在于提出了一种向搜索空间添加任意方向向量的机制，该机制可轻松适配于灵活GMRES或带收缩重启的GMRES方法。