LSMB: Minimizing the Backward Error for Least-Squares Problems
Eric Hallman, Ming Gu · SIAM Journal on Matrix Analysis and Applications · 2018
An iterative method, LSMB, is given for solving $\min_x \|Ax-b\|_2$. LSMB is based on the Golub--Kahan bidiagonalization process and is constructed so that an objective function closely related to the backward error for the least-squares problem is minimized with every iteration. We find that at every step the iterate $x_k$ produced by LSMB is a convex combination of those produced by LSQR (which minimizes $\|r_k\|_2 = \|b-Ax_k\|_2$ over a Krylov subspace) and LSMR (which minimizes $\|A^T r_k\|_2$ over the same subspace). Experiments on test cases from the University of Florida Sparse Matrix Collection show that in practice LSMB performs at least as well as both LSQR and LSMR, although never by more than a small margin. This suggests that LSMB could replace both solvers when stopping rules are based on the backward error.