The simpler block CMRH method for linear systems

Abstract

The block changing minimal residual method based on the Hessenberg reduction algorithm (in short BCMRH) is a recent block Krylov method that can solve large linear systems with multiple right-hand sides. This method uses the block Hessenberg process with pivoting strategy to construct a trapezoidal Krylov basis and minimizes a quasi-residual norm by solving a least squares problem. In this paper, we describe the simpler BCMRH method which is a new variant that avoids the QR factorization to solve the least-squares problem. Another major difference between the classical and simpler variants of BCMRH is that the simpler one allows to check the convergence within each cycle of the block Hessenberg process by using a recursive relation that updates the residual at each iteration. This is not possible with the classical BCMRH where we can only compute an estimate of the residual norm. Experiments are described to compare the behavior of the new proposed method with that of the classical and simpler versions of the block GMRES method. These numerical experiments show the good performances of the simpler BCMRH method.