The block LSMR method: a novel efficient algorithm for solving non-symmetric linear systems with multiple right-hand sides

It is well known that if the coefficient matrix in a linear system is large and sparse or sometimes not readily available, then iterative solvers may become the only choice. The block solvers are an attractive class of iterative solvers for solving linear systems with multiple right-hand sides. In general, the block solvers are more suitable for dense systems with preconditioner. In this paper, we present a novel block LSMR (least squares minimal residual) algorithm for solving non-symmetric linear systems with multiple right-hand sides. This algorithm is based on the block bidiagonalization and LSMR algorithm and derived by minimizing the 2-norm of each column of normal equation. Then, we give some properties of the new algorithm. In addition, the convergence of the stated algorithm is studied. In practice, we also observe that the Frobenius norm of residual matrix decreases monotonically. Finally, some numerical examples are presented to show the efficiency of the new method in comparison with the traditional LSMR method.