Preconditioned conjugate gradient method for generalized least squares problems

摘要

A variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented. If the problem is min (Ax - b)TW-1(Ax - b) with A ∈ Rm*n and W ∈ Rm*m symmetric and positive definite, the method needs only a preconditioner A1 ∈ Rn*n, but not the inverse of matrix W or of any of its submatrices. Freund's comparison result for regular least squares problems is extended to generalized least squares problems. An error bound is also given.