A flexible and adaptive simpler block GMRES with deflated restarting for linear systems with multiple right-hand sides

Abstract

Block GMRES is one of the most popular algorithms for solving large non-Hermitian linear systems with multiple right-hand sides. The simpler block GMRES algorithm is a variation of block GMRES, which avoids the factorization of a block upper Hessenberg matrix; and it is much simpler to program and requires less work than block GMRES. However, theoretical results given in this paper indicate that it is less stable due to the ill-conditioning of the basis used. In order to overcome this difficulty, we propose an adaptive simpler block GMRES algorithm for large linear systems with multiple right-hand sides. Theoretical analysis is made to show the reason why the new algorithm is superior to its original counterpart. Moreover, we consider how to restart and precondition the adaptive simpler block GMRES algorithm, and propose a flexible and adaptive simpler block GMRES algorithm with deflated restarting. Numerical experiments show the numerical behavior of our new algorithm.