On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems

The problem of shifted linear systems is an important and challenging issue in a number of research applications. Krylov subspace methods are effective techniques for different kinds of this problem due to their advantages in large and sparse matrix problems. In this paper, two new block projection methods based on respectively block FOM and block GMRES are introduced for solving sequences of shifted linear systems. We first express the original problem explicitly by a sequence of Sylvester matrix equations whose coefficient matrices are obtained from the shifted linear systems. Then, we show the restarted shifted block FOM (rsh-BFOM) method and derive some of its properties. We also present a framework for the restarted shifted block GMRES (rsh-BGMRES) method. In this regard, we describe two variants of rsh-BGMRES, including (1) rsh-BGMRES with an unshifted base system that applies a fixed unshifted base system and (2) rsh-BGMRES with a variable shifted base system in which the base block system can change after restart. Furthermore, we consider the use of deflation techniques for improving the performance of the rsh-BFOM and rsh-BGMRES methods. Finally, some numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.