OICC Press International Journal of Mathematical Modelling & Computations 2228-6233 7 3 2025 07 14 ABS-Type Methods for Solving $m$ Linear Equations in $\frac{m}{k}$ Steps for $k=1,2,\cdots,m$ 10.71932/ EN Leila Asadbeigi Department of Mathematics, Hamadan Branch, Islamic Azad University, Hamadan, Iran. Majid Amirfakhrian Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran. Journal Article 2025 07 14 ‎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.