Construction of optimal order nonlinear solvers using inverse interpolation

Abstract

There is a vast literature on finding simple roots of nonlinear equations by iterative methods. These methods can be classified by order, by the information used or by efficiency. There are very few optimal methods, that is methods of order 2 m requiring m + 1 function evaluations per iteration. Here we give a general way to construct such methods by using inverse interpolation and any optimal two-point method. The presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence.