An iterative method is presented for solving linear systems and
, with
being large and sparse, or a fast linear
operator. The method is based on the Golub-Kahan bidiagonalization
process. It is analytically equivalent to the standard method of MINRES
applied to the normal equation
, so that the quantities
are monotonically decreasing (where
is
the residual for the current iterate
). In practice we observe that
also decreases monotonically. Compared to LSQR, for which
only
is monotonic, it is safer to terminate LSMR early.