Accuracy improvement of the shifted block BiCGGR method for linear systems with multiple shifts and multiple right-hand sides

Abstract

We consider solving linear systems with multiple shifts and multiple right-hand sides. In order to solve these linear systems efficiently, we develop the Shifted Block BiCGGR method. This method is based on the shift-invariance property of Block Krylov subspaces. Using this property, the Shifted systems can be solved in the process of solving the Seed system without matrix-vector multiplications. The Shifted Block BiCGGR method can generate high accuracy approximate solutions of the Seed system. However, the accuracy of the approximate solutions of the Shifted systems may deteriorate due to the error of the matrix multiplications appearing in the algorithm. In this paper, we improve the accuracy of the approximate solutions of the Shifted systems generated by the Shifted Block BiCGGR method.