A new form of LSMR for solving linear systems and least-squares problems

The least squares minimal residual (LSMR) method of Fong and Saunders (2011) is an algorithm for solving linear systems Ax = b and least-squares problems min∥Ax - b∥2 that is analytically equivalent to the MINRES method applied to a normal equation ATAx = AT b so that the quantities ∥ATrk∥2 are minimised (where rk = b - Axk is the residual for current iterate xk). This method is based on the Golub-Kahan bidiagonalisation 1 process, which generates orthonormal Krylov basis vectors. Here, the Golub-Kahan bidiagonalisation 2 process is implemented in the LSMR algorithm. This substitution makes the algorithm simpler than the standard algorithm. Also, numerical results show the new form to be competitive.