LSMR (Least Squares Minimal Residual) is an iterative method for the solution of the linear system of equations and least-squares problems. This paper presents a block version of the LSMR algorithm for solving linear systems with multiple right-hand sides. The new algorithm is based on the block bidiagonalization and derived by minimizing the Frobenius norm of the residual matrix of normal equations. In addition, the convergence of the proposed algorithm is discussed. In practice, it is also observed that the Frobenius norm of the residual matrix decreases monotonically. Finally, numerical experiments from real applications are employed to verify the effectiveness of the presented method.

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