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 solvingnonlinear equations. Considered methods are based on Stolarsky and Gini means, Stolarsky(1975) [13], Stolarsky (1980) [14], Chen (2008) [12] and depend on two parameters. Someknown methods are special cases of our methods, for example, the power mean Newton’smethod 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 meanNewton’s method, Lukić and Ralević (2008) [9]. Third order convergence of the consideredmethods is proved, and corresponding asymptotic error constants are expressed in termsof two parameters. Numerical examples, obtained using Mathematica with high precisionarithmetic, support the theoretical results. Some numerical tests were performed, and itwas 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 interpolationmethods.