SeMA DOI 10.1007/s40324-016-0085-x Improved Newton-like methods for solving systems of nonlinear equations Janak Raj Sharma 1 · Himani Arora 1 Received: 16 November 2015 / Accepted: 2 May 2016 © Sociedad Española de Matemática Aplicada 2016 Abstract We present the iterative methods of fifth and eighth order of convergence for solving systems of nonlinear equations. Fifth order method is composed of two steps namely, Newton’s and Newton-like steps and requires the evaluations of two functions, two first derivatives and one matrix inversion in each iteration. The eighth order method is composed of three steps, of which the first two steps are that of the proposed fifth order method whereas the third is Newton-like step. This method requires one extra function evaluation in addition to the evaluations of fifth order method. Computational efficiency of proposed techniques is discussed and compared with the existing methods. Some numerical examples are considered to test the performance of the new methods. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in numerical examples. Numerical results have confirmed the robust and efficient character of the proposed techniques. Keywords Systems of nonlinear equations · Newton’s method · Order of convergence · Computational efficiency Mathematics Subject Classification 65H10 · 65Y20 · 41A58 1 Introduction The development of iterative methods for approximating the solution of systems of nonlinear equations is an important and interesting task in numerical analysis and applied scientific branches. With the advancement of computers, the problem of solving systems of nonlinear equations by numerical methods has gained more importance than before. This problem can be precisely stated as to find a vector α = (α 1 ,α 2 ,...,α m ) T such that F(α) = 0, where F(x ) : D ⊂ R m −→ R m is the given nonlinear system, F(x ) = ( f 1 (x ), f 2 (x ),..., f m (x )) T and x = (x 1 , x 2 ,..., x m ) T . Newton’s method [20, 21], is the most widely used algorithm for B Janak Raj Sharma jrshira@yahoo.co.in 1 Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, 148106 Sangrur, India 123