Multipoint methods for solving nonlinear equations: a survey

Abstract

Multipoint iterative methods belong to the class of the most efficient methods for solving nonlinear equations. Recent interest in the research and development of this type of meth- ods has arisen from their capability to overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. This survey paper is a mixture of theoretical results and algorithmic aspects and it is intended as a review of the most efficient root-finding algorithms and developing techniques in a general sense. Many existing methods of great efficiency appear as special cases of presented general iter- ative schemes. Special attention is devoted to multipoint methods with memory that use already computed information to considerably increase convergence rate without addi- tional computational costs. Some classical results of the 1970s which have had a great influence to the topic, often neglected or unknown to many readers, are also included not only as historical notes but also as genuine sources of many recent ideas. To a certain degree, the presented study follows in parallel main themes shown in the recently pub- lished book (Petkovic ́ et al., 2013) [53], written by the authors of this paper.