How good are methods with memory for the solution of nonlinear equations?

Abstract

Multipoint methods for the solution of a single nonlinear equation allow higher order of convergence without requiring higher derivatives. Such methods have an order barrier as conjectured by Kung and Traub. To overcome this barrier, one constructs multipoint methods with memory, i.e. use previously computed iterates. We compare multipoint methods with memory to the best methods without memory and show that the use of memory is computationally more expensive and the methods are not competitive.