Preconditioned conjugate gradient method for generalized least squares problems (Q1919957)

\textit{C. C. Paige} [SIAM J. Numer. Anal. 16, 165-171 (1979; Zbl 0402.65006); Math. Comput. 33, 171-183 (1979; Zbl 0405.65018)] transformed the generalized least squares problem into an equality-constrained least squares problem and proposed a direct method to solve it. However, as Paige's method is based on some matrix decompositions, for large sparse problems it is not appropriate. This paper presents a preconditioned conjugate gradient method. Theoretical comparison results show that the new method is better than SOR-type methods. By the way, the new method needs only a preconditioner, but not the inverse of the weight matrix or of any of its submatrices.