An improvement to double-step Newton method and its multi-step version for solving system of nonlinear equations and its applications

In this work, we have improved the order of the double-step Newton method from four to five using the same number of evaluation of two functions and two first order Fr,chet derivatives for each iteration. The multi-step version requires one more function evaluation for each step. The multi-step version converges with order 3r+5, r >= 1. Numerical experiments are done comparing the new methods with some existing methods. Our methods are also tested on Chandrasekhar's problem and the 2-D Bratu problem to illustrate the applications.