A right preconditioner for the LSMR method

Abstract

Abstract The LSMR (Least Squares Minimal Residual) method is an absorbing solver that can solve linear system Ax = b and least squares problem min ||Ax = b|| where A is a sparse and large matrix. This method is based on the Golub-Kahan bidiagonalization process and sometimes it may converge slowly like other methods. I n order to prevent this event, a right preconditioner for LSMR method is presented to solve large and sparse linear system which used for LSQR (Least Squares with QR factorization) method before. Numerical examples and comparing the preconditioned LSMR method to unpreconditioned LSMR method would show the effectiveness of the preconditioner. I t is obtained from this paper that PLSMR (Preconditioned LSMR) method has a better performance in reducing the number of iterations and relative residual norm in comparing with the original LSMR method.