The basins of attraction of Murakami's fifth order family of methods

doi:10.1016/j.apnum.2016.07.012

Applied Numerical Mathematics

Volume 110, December 2016, Pages 14-25

https://doi.org/10.1016/j.apnum.2016.07.012 Get rights and content

Abstract

In this paper we analyze Murakami's family of fifth order methods for the solution of nonlinear equations. We show how to find the best performer by using a measure of closeness of the extraneous fixed points to the imaginary axis. We demonstrate the performance of these members as compared to the two members originally suggested by Murakami. We found several members for which the extraneous fixed points are on the imaginary axis, only one of these has 6 such points (compared to 8 for the other members). We show that this member is the best performer.

Introduction

There is a vast literature on the solution of nonlinear equations, see for example Ostrowski [19], Traub [23], Neta [16] and Petković et al. [20]. In this paper we consider a fifth-order family of methods and show how to choose the best parameters. We will compare the performance of the two originally suggested members to two new ones by using the idea of basin of attraction and analyzing the extraneous fixed points.

Murakami [15] has developed a fifth order family of methods $x_{n + 1} = x_{n} - a_{1} u_{n} - a_{2} w_{2} (x_{n}) - a_{3} w_{3} (x_{n}) - ψ (x_{n}),$ where $\begin{matrix} u_{n} & = & \frac{f (x_{n})}{f^{'} (x_{n})}, \\ w_{2} (x_{n}) & = & \frac{f (x_{n})}{f^{'} (x_{n} - u_{n})}, \end{matrix}$ $\begin{matrix} w_{3} (x_{n}) & = & \frac{f (x_{n})}{f^{'} (x_{n} + β u_{n} + γ w_{2} (x_{n}))}, \\ ψ (x_{n}) & = & \frac{f (x_{n})}{b_{1} f^{'} (x_{n}) + b_{2} f^{'} (x_{n} - u_{n})} . \end{matrix}$

This family is of order five when we take $\begin{matrix} a_{1} = \frac{1}{6} (1 + \frac{4 γ + 1}{θ}), a_{2} = \frac{1}{θ - 1} (\frac{1}{6} θ - \frac{2}{3} γ - \frac{1}{3}), a_{3} = \frac{2}{3}, \\ b_{1} = - \frac{6 θ {(θ - 1)}^{2}}{4 γ + 1}, b_{2} = \frac{6 θ^{2} (θ - 1)}{4 γ + 1}, β = - γ - \frac{1}{2}, \end{matrix}$ and $θ = \frac{16 γ + 5}{4 (4 γ + 1)} .$

Murakami suggested the following two possibilities: $\begin{matrix} γ = 0, a_{1} = 0.3, a_{2} = - 0.5, a_{3} = \frac{2}{3}, \\ b_{1} = - \frac{15}{32}, b_{2} = \frac{75}{32}, β = - \frac{1}{2} \end{matrix}$ and $\begin{matrix} γ = - 0.5, a_{1} = - \frac{1}{18}, a_{2} = - \frac{1}{2}, a_{3} = \frac{2}{3}, \\ b_{1} = \frac{9}{32}, b_{2} = \frac{27}{32}, β = 0 . \end{matrix}$ The idea there probably to choose one of the parameters to be zero, i.e. either $γ = 0$ or $β = 0$ . As it turns out these parameters are not far from the best.

In this paper, we find the best possible value of the parameter γ. We will use two criteria we have developed in previous work [7] based on the location of the extraneous fixed points. In the next section, we discuss the extraneous fixed points. In section 3 we will discuss the two criteria and give the best parameter based on these criteria. In section 4 we describe the basins of attraction for the best members of the family for 7 different examples. We close with conclusions.

Section snippets

Extraneous fixed points

For the Murakami family $z_{n + 1} = M_{f} (z_{n})$ , where $\begin{matrix} M_{f} (z) & = & z - a_{1} u_{f} (z) - a_{2} w_{2, f} (z) - a_{3} w_{3, f} (z) - ψ_{f} (z), \\ u_{f} (z) & = & \frac{f (z)}{f^{'} (z)}, \\ w_{2, f} (z) & = & \frac{f (z)}{f^{'} (z - u_{f} (z))}, \\ w_{3, f} (z) & = & \frac{f (z)}{f^{'} (z + β u_{f} (z) + γ w_{2, f} (z))}, \\ ψ_{f} (z) & = & \frac{f (z)}{b_{1} f^{'} (z) + b_{2} f^{'} (z - u_{f} (z))}, \end{matrix}$ we explore its conjugacy on quadratic polynomials. We begin with a preliminary result.

Lemma 1

Let $f (z)$ be an analytic function on the Riemann sphere, and let $T (z) = α z + β, α \neq 0$ , be an affine map. If $g (z) = (f \circ T) (z)$ , then we have $\begin{matrix} u_{f} (T (z)) = α u_{g} (z), \\ w_{2, f} (T (z)) = α w_{2, g} (z), \\ w_{3, f} (T (z)) = α w_{3, g} (z), \\ ψ_{f} (T (z)) = α ψ_{g} (z) . \end{matrix}$

Proof

We have $g^{'} (z) = α f^{'} (T (z))$

Best possible parameters

The parameters can be chosen to position the extraneous fixed points on the imaginary axis or, at least, close to that axis, (see, for example, Chun and Neta [7] and [10]).

We have searched the parameter space (γ) and found that the extraneous fixed points are on the imaginary axis for certain values of the parameter γ. We have considered one measure of closeness to the imaginary axis, denoted by d, and another measure of averaged stability of the extraneous fixed points, denoted by A. These

Numerical experiments

The Basin of Attraction is a method to visually understand how a method behaves as a function of the various starting points. This idea was started by Stewart [22] and continued in the work of Amat et al. [1], [2], [3], Argyros and Magreñan [4], Chicharro et al. [5], Chun et al. [6], [9], [11], Cordero et al. [12], Geum et al. [13], Magreñan [14], Neta et al. [17], [18], and Scott et al. [21].

We have used the 4 members of the Murakami family for 7 different polynomials. The choice of the

Conclusions

In order to decide which member is best overall, we have averaged the numbers across the examples. We now find that Murakamid requires the least number of iterations per point (3.61) followed by Murakami2 (3.90). MurakamiA requires the most (4.11). Same conclusion for the average CPU time across examples, i.e. Murakamid is the fastest (426 seconds) and MurakamiA is the slowest (491 seconds). In terms of the number of points requiring 40 iterations, the highest is 2138 for Murakamid and the

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005012). The first author thanks the Applied Mathematics Department at the Naval Postgraduate School for hosting him during the years.

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The basins of attraction of Murakami's fifth order family of methods

Abstract

Introduction

Section snippets

Extraneous fixed points

Best possible parameters

Numerical experiments

Conclusions

Acknowledgements

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