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An iterative method LSMR is presented for solving linear systems Ax=b and least-squares problems \|Ax-b\|₂, with A being sparse or a fast linear operator. LSMR is based on the Golub–Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation AT\! Ax = AT\! b, so that the quantities \|AT\! rₖ\| are monotonically decreasing (where rₖ = b - Axₖ is the residual for the current iterate xₖ). We observe in practice that \|rₖ\| also decreases monotonically, so that compared to LSQR (for which only \|rₖ\| is monotonic) it is safer to terminate LSMR early. We also report some experiments with reorthogonalization.
Fong et al. (Sat,) studied this question.