On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems

In the present paper, we are concerned by weighted Arnoldi like methods for solving large and sparse linear systems that have different right-hand sides but have the same coefficient matrix. We first give detailed descriptions of the weighted Gram-Schmidt process and of a Ruhe variant of the weighted block Arnoldi algorithm. We also establish some theoretical results that links the iterates of the weighted block Arnoldi process to those of the non weighted one. Then, to accelerate the convergence of the classical restarted block and seed GMRES methods, we introduce the weighted restarted block and seed GMRES methods. Numerical experiments that are done with different matrices coming from the Matrix Market repository or from the university of Florida sparse matrix collection are reported at the end of this work in order to compare the performance and show the effectiveness of the proposed methods.