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In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and twofirst order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r+4, where r is a positive integer and r ≥ 1. We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we improve the order of the Midpoint iterative method from three to four with same number of function evaluations using weight functions to obtain a class of 2-point fourth order Jarratt-type methods. Then we prove a general result to further improve the order of the old methods through an additional function evaluation using weight functions. In this way, we develop two 3-point sixth order midpoint methods using different weights. A family of n-point I.F.s of order 2n is proposed. Using weight functions, we further improve the methods to obtain a 5-point sixteenth order midpoint method with same efficiency index as the optimal 2-point fourth order Jarratt-type methods. Two numerical examples using different starting are provided to test the robustness of the methods. [Copyright &y& Elsevier]