ABS-Type Methods for Solving m Linear Equations in \frac{m}{k} Steps for k=۱,۲,\cdots,m

‎The ABS methods‎, ‎introduced by Abaffy‎, ‎Broyden and Spedicato‎, ‎are‎‎direct iteration methods for solving a linear system where the‎‎i-th iteration satisfies the first i equations‎, ‎therefore a‎ ‎system of m equations is solved in at most m steps‎. ‎In this‎‎paper‎, ‎we introduce a class of ABS-type methods for solving a full row‎‎rank linear equations‎, ‎where the i-th iteration solves the first‎‎۳i equations‎. ‎We also extended this method for k steps‎. ‎So‎,‎termination is achieved in at most \left[\frac{m+(k-۱)}{k}\right]‎‎steps‎. ‎Morever in our new method in each iteration, we have the‎‎the general solution of each iteration‎.

Keywords:

ABS methods‎ , ‎rank k update‎ , ‎linear system‎ , ‎general‎ ‎solution of a system‎ , ‎general solution of an iteration