Numerical methods for generalized least squares problems

Usually generalized least squares problems are solved by transforming them into regular least squares problems which can then be solved by well-known numerical methods. However, this approach is not very effective in some cases and, besides, is very expensive for large scale problems. In 1979, Paige suggested another approach which consists of solving an equivalent equality-constrained least squares problem by the orthogonal decomposition, the BNP algorithm or the Jamesʹ implicit nullspace iterative methods. In this paper, we present some new developments of the numerical methods, for example, 2-cycle SOR method and preconditioned conjugate gradient method, for generalized least squares problems. Some numerical comparisons are included as well.