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

Subject Areas : International Journal of Mathematical Modelling & Computations

Leila Asadbeigi ¹ (Hamadan Branch, Islamic Azad University)
Majid Amirfakhrian ² (IAUCTB)

Received: 2017-03-08 Accepted : 2017-11-12 Published : 2017-08-01

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

Abstract :

‎The ABS methods‎, ‎introduced by Abaffy‎, ‎Broyden and Spedicato‎, ‎are‎‎direct 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 this‎‎paper‎, ‎we introduce a class of ABS-type methods for solving a full row‎‎rank 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 the‎‎the general solution of each iteration‎.