Open AccessFamilies of optimal multipoint methods for solving nonlinear equations: A survey
Author(s) -
Stanko Petković,
Dalibor Petković
Publication year - 2010
Publication title -
applicable analysis and discrete mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.69
H-Index - 26
eISSN - 2406-100X
pISSN - 1452-8630
DOI - 10.2298/aadm100217015p
Subject(s) - mathematics , convergence (economics) , nonlinear system , class (philosophy) , root finding algorithm , function (biology) , order (exchange) , iterative method , root (linguistics) , mathematical optimization , point (geometry) , rate of convergence , computer science , geometry , key (lock) , linguistics , philosophy , physics , finance , quantum mechanics , artificial intelligence , evolutionary biology , economics , biology , economic growth , computer security
Multipoint iterative root-solvers belong to the class of the most powerful methods for solving nonlinear equations since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. Although the construction of these methods has occurred in the 1960s, their rapid development have started in the first decade of the 21-st century. The most important class of multipoint methods are optimal methods which attain the convergence order 2n using n + 1 function evaluations per iteration. In this paper we give a review of optimal multipoint methods of the order four (n = 2), eight (n = 3) and higher (n > 3), some of which being proposed by the authors. All of them possess as high as possible computational e±ciency in the sense of the Kung-Traub hypothesis (1974). Numerical examples are included to demonstrate a very fast convergence of the presented optimal multipoint methods