In this paper, two effective methods based on the simpler block GMRES method are established in order to solve the linear systems of equations with multiple right-hand sides. The first method is derived from the simpler block GMRES method with vector deflation restarting (SBGMRES-DR). The second method is constructed from a combination of SBGMRES-DR with the eigenvalue deflation technique, which is called the deflated simpler block GMRES method with vector deflation restarting (D-SBGMRES-DR). To be more specific, SBGMRES-DR is capable of removing linearly or almost linearly dependent vectors created by the block Arnoldi process. On the other hand, D-SBGMRES-DR not only deletes linearly or almost linearly dependent vectors but also retains harmonic Ritz vectors associated with the smallest harmonic Ritz values in magnitude, and adds them to the new search subspace at the time of restart. Finally, a wide range of practical experiments are carried out to assess the efficiency of the proposed methods. The numerical results indicate that the D-SBGMRES-DR method outperforms the compared methods with respect to the number of matrix–vector products and the computational time.

为了解决具有多个右手边的方程组的线性系统，本文建立了两种基于简单块GMRES方法的有效方法。第一种方法是从带有向量缩小重新启动的更简单的块GMRES方法（SBGMRES-DR）中派生的。第二种方法是通过将SBGMRES-DR与特征值放气技术相结合而构造的，该方法称为带有向量放气重新启动的放气简单块GMRES方法（D-SBGMRES-DR）。更具体地说，SBGMRES-DR能够删除由Arnoldi块过程创建的线性或几乎线性相关的向量。另一方面，D-SBGMRES-DR不仅删除线性或几乎线性相关的向量，而且还保留与幅度最小的谐波Ritz值相关的谐波Ritz向量，并在重新启动时将它们添加到新的搜索子空间中。最后，进行了广泛的实际实验，以评估所提出方法的效率。数值结果表明，就矩阵向量乘积的数量和计算时间而言，D-SBGMRES-DR方法优于比较方法。