Weighted and flexible versions of block CMRH method for solving nonsymmetric linear systems with multiple right-hand sides

Block Krylov subspace methods are the most popular algorithms for solving large nonsymmetric linear systems with multiple right-hand sides. One of them is the block CMRH method. This method generates a (non orthogonal) basis of the Krylov subspace through the block Hessenberg process. To accelerate the convergence of the block CMRH method, we will introduce two new methods. First, we present the block CMRH method with weighting strategy. In this method, the block CMRH method uses a different product at each restart. Second, we introduce a flexible version of the block CMRH algorithm that allows varying preconditioning at every step of the algorithm. Numerical experiments illustrate the benefits of the presented methods.