A simple algorithm for high order Newtoniteration formulae and some new variants

Abstract

The high order Newton iteration formulas are revisited in this paper. Translating thenonlinear root finding problem into a fixed point iteration involving an unknown generalfunction whose root is searched, a double Taylor series is undertaken regarding the rootand the root finding function. Based on the error analysis of the expansion, a simple algo-rithm is later proposed to construct Newton iteration formulae of any order commencingfrom the traditional linearly convergent fixed point iteration method and quadraticallyconvergent Newton-Raphson method of frequently at the disposal of the scientific commu-nity. It is shown that the well-known variants like the Halley’s method or Haouseholder’smethods of high order can be reproduced from the general case outlined here. Somefurther rare single-step classes of any order are shown to be derivable from the presentedalgorithm. Finally, some new higher order accurate variants are also offered taking into ac-count multi-step compositions which demand less computation of higher derivatives. Theefficiency, accuracy and performance of the proposed methods and also their potentialadvantages over the classical ones are numerically demonstrated and discussed on somewell-documented examples from the open literature.