Direct Iterative Methods for Rank Deficient Generalized Least Squares Problems (Journal of Computational Mathematics, 2000)
The generalized least squares (LS) problem $$\min_{x\in R^n} (Ax-b)^T W^{-1} (Ax-b)$$ appears in many application areas. Here $W$ is an $m × m$ symmetric positive definite matrix and $A$ is an $m × n$ matrix with $m ≥ n$. Since the problem has many solutions in rank deficient case, some special preconditioned techniques are adapted to obtain the minimum 2-norm solution. A block SOR method and the preconditioned conjugate gradient (PCG) method are proposed here. Convergence and optimal relaxation parameter for the block SOR method are studied. An error bound for the PCG method is given. The comparison of these methods is investigated. Some remarks on the implementation of the methods and the operation cost are given as well.
Keywords: Rank deficient generalized LS problem; block SOR method; PCG method; convergence; optimal parameter.