On accelerate overrelaxation methods for rank deficient linear systems

Tian [H. Tian, Accelerate overrelaxation methods for rank deficient linear systems, Appl. Math. Comput. 140 (2003) 485–499] presented an accelerated overrelaxation (AOR) method for solving the rank deficient linear least squares problem equation(∗)Ax=b,Ax=b,by augmenting the system to a block 4 × 4 consistent system and splitting the coefficient matrix into a subproper splitting, where A∈Cm×nA∈Cm×n is a matrix of rank r   and b∈Cmb∈Cm. With this splitting, one set of sufficient conditions for the semiconvergence of the AOR iteration matrix are presented by Tian. In this paper, we further investigate the AOR method for solving such linear problem by using an new subproper splitting and the necessary and sufficient conditions for the semiconvergence of the AOR iteration matrix are explicitly given out. A numerical example shows that the method is applicable and efficient for solving such rank deficient linear system.