Sharif Digital Repository

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... 

We show how to update the general solution of linear Diophantine systems obtained by the integer ABS method when the coefficient matrix is subjected to a rank one change  

ABS methods are a large class of algorithms for solving continuous and integer linear algebraic equations, and nonlinear continuous algebraic equations, with applications to optimization. Recent work by Chinese researchers led by Zunquan Xia has extended these methods also to stochastic, fuzzy and infinite systems, extensions not considered here. The work on ABS methods began almost thirty years. It involved an international collaboration of mathematicians especially from Hungary, England, China and Iran, coordinated by the university of Bergamo. The ABS method are based on the rank reducing matrix update due to Egerváry and can be considered as the most fruitful extension of such technique....