Accurate Solution of Weighted Least Squares by Iterative Methods

We consider the weighted least-squares (WLS) problem with a very ill-conditioned weight matrix. WLS problems arise in many applications including linear programming, electrical networks, boundary value problems, and structures. Because of roundoff errors, standard iterative methods for solving a WLS problem with ill-conditioned weights may not give the correct answer. Indeed, the difference between the true and computed solution (forward error) may be large. We propose an iterative algorithm, called MINRES-L, for solving WLS problems. The MINRES-L method is the application of MINRES, a Krylov-space method due to Paige and Saunders [SIAM J. Numer. Anal., 12 (1975), pp. 617--629], to a certain layered linear system. Using a simplified model of the effects of roundoff error, we prove that MINRES-L ultimately yields answers with small forward error. We present computational experiments for some applications.