Third-order modifications of Newton’s method based on Stolarsky and Gini means

Abstract

In this paper we consider third-order modifications of Newton’s method for solving nonlinear equations. Considered methods are based on Stolarsky and Gini means, Stolarsky (1975) [13], Stolarsky (1980) [14], Chen (2008) [12] and depend on two parameters. Some known methods are special cases of our methods, for example, the power mean Newton’s method Zhou (2007) [10], the arithmetic mean Newton’s method Weerakoon and Fernando (2000) [5], the harmonic mean Newton’s method, Özban (2004) [8] and the geometric mean Newton’s method, Lukić and Ralević (2008) [9]. Third order convergence of the considered methods is proved, and corresponding asymptotic error constants are expressed in terms of two parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, support the theoretical results. Some numerical tests were performed, and it was shown that our methods yield better numerical results (i.e. a smaller error |x4−α|) when compared to Halley, Euler, Hansen-Patrick, Ostrowski and inverse interpolation methods.