A simple algorithm for high order Newton iteration formulae and some new variants

Abstract

The high order Newton iteration formulas are revisited in this paper. Translating the nonlinear root finding problem into a fixed point iteration involving an unknown general function whose root is searched, a double Taylor series is undertaken regarding the root and the root finding function. Based on the error analysis of the expansion, a simple algorithm is later proposed to construct Newton iteration formulae of any order commencing from the traditional linearly convergent fixed point iteration method and quadratically convergent Newton-Raphson method of frequently at the disposal of the scientific community. It is shown that the well-known variants like the Halley's method or Haouseholder's methods of high order can be reproduced from the general case outlined here. Some further rare single-step classes of any order are shown to be derivable from the presented algorithm. Finally, some new higher order accurate variants are also offered taking into account multi-step compositions which demand less computation of higher derivatives. The efficiency, accuracy and performance of the proposed methods and also their potential advantages over the classical ones are numerically demonstrated and discussed on some well-documented examples from the open literature.