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 parallel to Ax - b parallel to(2) that is analytically equivalent to the MINRES method applied to a normal equation A(T)Ax = A(T) b so that the quantities parallel to A(T)r(k)parallel to(2) are minimised (where r(k) = b - Ax(k) is the residual for current iterate x(k)). 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.