Modified families of Newton, Halley and Chebyshev methods

Abstract

This paper presents new families of Newton-type iterative methods (Newton, Halley and Chebyshev methods) for finding simple zero of univariate non-linear equation, permitting f ′ ( x ) = 0 in the vicinity of the root. Newton-type iterative methods have well-known geometric interpretation and admit their geometric derivation from a parabola. These algorithms are comparable to the well-known powerful classical methods of Newton, Halley and Chebyshev respectively, and these can be seen as special cases of these families. The efficiency of the presented methods is demonstrated by numerical examples.