Journal of the Vol. 35, pp. 71-83, 2017 Nigerian Mathematical Society c Nigerian Mathematical Society BARZILAI-BORWEIN-LIKE METHOD FOR SOLVING LARGE-SCALE NON-LINEAR SYSTEMS OF EQUATIONS HASSAN MOHAMMAD ABSTRACT. In this paper, a derivative-free Barzilai-Borwein- like algorithm is developed for solving large-scale non-linear sys- tems of equations. The algorithm is based on approximating the Jacobian matrix in quasi-Newton manner using a scalar multi- ple of an identity matrix. Under suitable conditions, we show that the proposed algorithm is locally superlinearly convergent. Numerical results show that the proposed method is efficient for large-scale problems (up to 10 6 ) variables. Keywords and phrases: Non-linear equations, Large-scale prob- lems, Barzilai and Borwein method, Superlinear convergence 2010 Mathematical Subject Classification: 65K05 , 90C06, 90C52, 90C56, 49M30 1. INTRODUCTION We present an iterative method for solving system of non-linear equations of the form F (x)=0, (1) where F : R n → R n is continuously differentiable function, F = (f 1 ,f 2 , ..., f n ) T ,f i : R n → R(i =1, 2, ..., n). Newton’s method is the most popular method used to solve (1), it converges locally with a quadratic rate of convergence [1]. The main drawback of Newton’s method for large-scale problems is the need of computing and storing Jacobian matrix and solving system of linear equations in every iteration. As a remedy of these drawbacks, quasi-Newton methods have been introduced [2]. These methods are derivative- free, and enjoys superlinear rate of convergence [3]. For excellent review of quasi-Newton methods see [4, 5]. A suitable quasi-Newton method for solving (1) is Broyden method, it is given by x k+1 = x k − B −1 k F (x k ), k =0, 1, 2, ... (2) Received by the editors February 10, 2016; Revised: June 08, 2016; Accepted: January 18, 2017 www.nigerianmathematicalsociety.org 71