Ball convergence theorems for eighth-order variants of Newton s method under weak conditions

Abstract

We present a local convergence analysis for eighth-order variants of Newton s method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as Amat et al. (Appl Math Comput 206(1):164 174, 2008), Amat et al. (Aequationes Math 69:212 213, 2005), Chun et al. (Appl Math Comput. 227:567 592, 2014), Petkovic et al. (Multipoint methods for solving nonlinear equations. Elsevier, Amsterdam, 2013), Potra and Ptak (Nondiscrete induction and iterative processes. Pitman Publ, Boston, 1984), Rall (Computational solution of nonlinear operator equations. Robert E. Krieger, New York, 1979), Ren et al. (Numer Algorithms 52(4):585 603, 2009), Rheinboldt (An adaptive continuation process for solving systems of nonlinear equations. Banach Center, Warsaw, 1975), Traub (Iterative methods for the solution of equations. Prentice Hall, Englewood Cliffs, 1964), Weerakoon and Fernando (Appl Math Lett 13:87 93, 2000), Wang and Kou (J Differ Equ Appl 19(9):1483 1500, 2013) using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. 2015, The Author(s).