ABS-Type Methods for Solving $m$ Linear Equations in $frac{m}{k}$ Steps for $k=1,2,cdots,m$

Abstract

‎The ABS methods‎, ‎introduced by Abaffy‎, ‎Broyden and Spedicato‎, ‎are‎<br />‎direct iteration methods for solving a linear system where the‎<br />‎$i$-th iteration satisfies the first $i$ equations‎, ‎therefore a‎ ‎system of $m$ equations is solved in at most $m$ steps‎. ‎In this‎<br />‎paper‎, ‎we introduce a class of ABS-type methods for solving a full row‎<br />‎rank linear equations‎, ‎where the $i$-th iteration solves the first‎<br />‎$3i$ equations‎. ‎We also extended this method for $k$ steps‎. ‎So‎,<br />‎termination is achieved in at most $left[frac{m+(k-1)}{k}right]$‎<br />‎steps‎. ‎Morever in our new method in each iteration, we have the‎<br />‎the general solution of each iteration‎.