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

Document Type : Full Length Article


1 Hamadan Branch, Islamic Azad University



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