New Fourth-Order Iterative Solver And Its Multi-Point Solver For Nonlinear Systems

This manuscript presents a new two-step iterative algorithm having order of convergence four for approximating solutions of nonlinear system of equations. It requires one vector function evaluation and two Frechet derivative evaluations per iteration. Also, the fourth order algorithm is extended into a general multi-point method with an additional vector function evaluation per step, having 2k + 4 order of convergence, k >= 1. It is proved that the root of the nonlinear system is a point of attraction for the new iterative algorithms. Convergence analysis for the iterative process is derived from which order of convergence of the methods are obtained. Computational efficiency of the methods are provided based on the cost of computation. Numerical experimentation through some suitable examples are given and some known methods are compared with presented methods. Further, an application of these methods to solve boundary value problems for ordinary differential equations is also given. The presented algorithms perform better than many existing algorithms and equivalent to few available algorithms.