Successive overrelaxation methods for solving the rank deficient linear squares problem

Abstract

We develop successive overrelaxation (SOR) methods for finding the least squares solution of minimal norm to the system Ax = b , (∗) where A is an m × n matrix of rank r. The methods are obtained by first augmenting the system (∗) to a block 4 × 4 consistent system of linear equations. The augmented coefficient matrix is then split by a subproper SOR splitting. An interval for the relaxation parameter in which the subproper SOR iteration matrix is semiconvergent is determined along with the optimal relaxation parameter which minimizes the modulus of the controlling eigenvalue of the SOR matrix. Since the scheme computes at first only a solution to the augmented system, it is subsequently shown how to transform such a solution to the unique solution of minimal 2-norm. Analysis of the practical implementation of the algorithms developed here will be given elsewhere.