An analysis of the properties of the variants of Newton’s method with third order convergence

Abstract

For the last five years, the variants of the Newton’s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we investigate about the relationship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited Taylor expansion. We also prove that they are different forms of the Halley method and are all contractive iterative methods in a common neighbourhood. We extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models.

Introduction

Iterative methods for finding approximate solutions to the single non-linear equations f(x) = 0 that have no solutions in closed form have various applications in physics and chemistry. Newton’s method [1] is often the method of choice for approximating such solutions. The method is of quadratic convergence at a simple root, that is, the number of good digits is doubled at each iteration. The convergence of the iterates and the rate of convergence play an important role in the design of new iterative methods [2]. There are higher order methods that allow for faster convergence. For instance, Halley’s method [3], Householder’s method [4], double convex acceleration of Whittaker’s method [5], super-Newton’s method (Chebyshev) [6], super-Halley’s method or the Chebyshev–Halley’s family of iterative methods [7], [8] has third order convergence, but require the second derivative which is difficult to evaluate and thus time consuming for complicated functions. To avoid the calculation of second derivatives, some variants of Newton’s method with third order convergence at simple root have been developed. In these methods, Newton’s method is used as an intermediate step. These variants are termed as multi-point methods because they involve the calculation of the function or its first order derivative at multi-points. They are easy to implement and only require function and first derivative evaluations. Traub [9] has introduced the mid-point Newton method (MP) as one of these variants. However, during the last five years, there has been a ’Newton boom’. In fact, many variants of Newton’s method have been introduced and developed. By using different approximations to the definite integral in the Newton’s theorem [10], different iterative formulas can be obtained. For instance, Newton’s method can be obtained using a rectangle as approximation in the Newton theorem. Weerakoon and Fernando [10] have used the trapezoid rule instead and developed a new variant which was then termed as the Arithmetic Mean third order Newton method (AM) [11] because the denominator is the arithmetic mean of the derivatives at k iterate and at the Newton iterate. In 2002, Hasanov and al. [12] have developed another variant (VS) from the Newton theorem making use of the Simpson’s rule instead. Nedzhibov [13] has shown that approximating the integral in the Newton theorem using different quadrature formulae yields different iterative functions including those we have mentioned before. Furthermore, Kasturiarachi [2] independently introduced and developed the Leap-frogging Newton method. This method is termed as the Newton–Secant method (NS) [13] because it is a composition of Newton’s and Secant methods. It can also be derived from the discrete modifications of the AM method or the MP method. In 2003, Sormarni and Frontini [14], [15] have studied some of these variants mentioned and their applications to multiple roots. Homeier [16], [17] independently derived one of the variants and extended it to the multivariate case. More recently, in 2005, Homeier [18] have shown that one can modify the Weerakoon–Fernando [10] approach by using Newton’s theorem for the inverse function and derive a new class of cubically convergent Newton-type methods. New variants of Newton’s method can also be obtained using the harmonic and geometric mean instead of the arithmetic mean in the Weerakoon’s variant. In this way, Ozban [11] and Lukic [19] have derived the Harmonic Mean Newton method (HM) and the Geometric Mean third order Newton method (GM) respectively. These variants of Newton’s method have been proved to converge locally with third order of convergence assuming that the starting point is close to the root. However, approximating the Newton theorem using different quadrature formulae does not clearly explain why the variants are third order accurate iterative method. Also, the contractive property of these methods have not been studied. The contraction theorem is a sufficient condition to ensure convergence for the choice of the starting points. In this work, we study and analyze the properties of one of the variants (AM’s method) and explain the origin of the third order accuracy via the Taylor expansion. In Section 4, we shall prove that the method is a contraction in a neighbourhood of the root in which the value of the function is assumed to be small enough. We show that the method is in fact a form of the Halley method while considering a limited Taylor expansion and both methods contract in the same neighbourhood. We also extend AM’s method to multivariate case and prove that it converges locally with cubic convergence from its linear model. In Section 5, we show that the third order accurate variants are in fact different forms of the Halley method in the neighbourhood of the root and their contraction follows from that of the Halley method. We also extend some of the other variants to the multivariate case. In the last section, we conduct a comparative study of these variants via numerical experiments and add concluding remarks.