Extended reduced rank two Abaffian update schemes in the ABS-type methods

The ABS methods, introduced by Abaffy, Broyden and Spedicato, are direct iteration methods for solving a linear system where the ith iterate satisfies the first i equations, therefore a system of m equations is solved in at most m steps. Recently, we have presented a new approach to devise a class of ABS-type methods for solving full row rank systems [K. Amini, N. Mahdavi-Amiri, M. R. Peyghami, ABS-type methods for solving full row rank linear systems using a new rank two update, Bulletin of the Australian Mathematical Society 69 (2004) 17-31], the ith iterate of which solves the first 2i equations. Here, to reduce the space and computation time, we use a new extended rank two update formula for the Abaffian matrix so that the number of rows of the Abaffian matrix is reduced by two in every iteration. This extension along with the reduction offer more flexibility for the definition of the Abaffian matrix.