A Modified Simpler GMRES Algorithm

In practical application, GMRES(m) is one of the most popular methods for solving the linear system of equations. Many scholars have done a lot of work on the convergence and calculation quantity of GMRES(m) method. The restarted Simpler GMRES (SGMRES(m)) algorithm, as a kind of modifications to the GMRES(m) algorithms, does not require upper Hessenberg factorization. In order to improve the convergence of SGMRES(m) method, much of the work is based on the choice of the Krylov subspace basis.In this paper, we consider the Simpler GMRES (SGMRES(m)) with varying the restarted parameter and modifying the restarted vector. The advantage is that the SGMRES(m) variant has less computational demand of work than GMRES (m) with varying the restarted parameter, and then the total time for solving the linear system is reduced. Numerical examples are conducted to illustrate the efficiency of the new method.