There are several ways that can be used to classify or compare iterative methods for nonlinear equations, for instance; order of convergence, informational efficiency, and efficiency index. In this work, we use another way, namely the basins of attraction of the method. The purpose of this study is to compare several iterative schemes for nonlinear equations. All the selected schemes are of the third-order of convergence and most of them have the same efficiency index. The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees. As a comparison, we determine the CPU time (in seconds) needed by each scheme to obtain the basins of attraction, besides, we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods. Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders, furthermore, they vary for iterative methods of the same order even if they have the same efficiency index. Consequently, this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.

The subject of finding the solutions of nonlinear equations is important; because many nonlinear equations result from applied sciences like physics, chemistry and engineering. This field has been studied widely, see for example [

The well-known Newton’s method and all root-finding methods depend on at least one initial guess _{0} for the root _{0} always converge to the same root if we use different iterative schemes?

The field of basins of attraction firstly considered and attributed by Cayley [

Having basins of attraction with smooth convergent pattern or basins of attraction with chaotic pattern does not mean that the iterative scheme with a smooth pattern has a larger area of convergence than the scheme with chaotic basins of attraction, although this leads sometimes the algorithm converges to unwanted zero. Very few researchers have worked on finding number of convergent and divergent points in a selected range for iterative schemes when applied to numerical examples. Some questions arise from this subject are:

Could the basins of attraction of the iterative schemes be affected by the number of steps needed in each scheme?

If the basins of attraction of a specific iterative scheme were better than others in one example, is it necessary to be the best in all test problems?

What are possible factors that affect the basins of attraction of the iterative schemes?

Based on the basins of attraction of different schemes, is the current efficiency index enough to make comparisons between iterative schemes with equal order of convergence and an equal number of functions that need to be evaluated per iteration?

We shall in this work find answers to the above questions. We will compare some iterative schemes of third-order of convergence by using their basins of attraction. Some of these schemes are second-derivative free. We find out the number of convergent and divergent points on a selected range for all schemes when applied on different polynomials. The work in this paper is divided as follows: Some definitions and preliminaries related to the subject were mentioned in Section 2. In Section 3, the basins of attractions were used to compare eight iterative schemes of order three on some numerical examples. Finally, the conclusion of the paper is given in Section 4.

Let’s start by stating some definitions and preliminaries which are related to the subject of basins of attraction.

If _{0}) = _{0}, then _{0} is called a fixed point. For ^{th} iterate of _{0} is called a periodic point of period _{0} is periodic of period _{0} is said to be attracting if

The Julia set

The complex polynomial of order

In this part, we study the area of convergence of eight iterative schemes of third-order of convergence by obtaining the basins of attraction of their zeros, and finding the number of convergent and divergent points in a selected region. All polynomials in the examples are of roots with multiplicity one. Some of the compared schemes were considered before, but without finding out the number of convergent and divergent points in a selected range. See Stewart [

The modified Halley method (MH) proposed by Said Solaiman et al. [

The well-known Halley’s method [

Potra-Pták (PP) method [

Weerakon-Fernando (WF) method [

Frontini-Sormani (FS) method [

Homeier method (HM) [

Kou-Wang (KW) method [

Chun method (CM) [

The idea of the basins of attraction of

For the purpose of comparison, the CPU time (in sec) needed to obtain the basins of attraction has been computed, see ^{−3} with a maximum of 100 iterations.

Method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

NCP | NDP | NCP | NDP | NCP | NDP | NCP | NDP | NCP | NDP | |

MH | 160800 | 1 | 160800 | 1 | 160445 | 356 | 160797 | 4 | 160800 | 1 |

Halley | 160799 | 2 | 160798 | 3 | 160689 | 112 | 160797 | 4 | 160799 | 2 |

PP | 160799 | 2 | 160799 | 2 | 159651 | 1150 | 160797 | 4 | 156074 | 4727 |

WF | 145688 | 15113 | 152133 | 8668 | 121909 | 38892 | 123942 | 36859 | 119569 | 41232 |

FS | 154203 | 6598 | 159891 | 910 | 143150 | 17651 | 150320 | 10481 | 139737 | 21064 |

HM | 160800 | 1 | 160800 | 1 | 160390 | 411 | 160797 | 4 | 160800 | 1 |

KW | 160800 | 1 | 160798 | 3 | 159925 | 876 | 160797 | 4 | 158881 | 1920 |

CM | 160800 | 1 | 160800 | 1 | 160797 | 4 | 160797 | 4 | 160773 | 28 |

All calculations have been performed on Intel Core i3-2330M CPU@2.20 GHz with 4 GB RAM, using Microsoft Windows 10, 64 bit based on X64-based processor. Mathematica 9 has been used to produce all graphs and computations.

The basins of attraction for the eight iterative schemes have been showed in

The last set in

We have compared several iterative schemes for nonlinear equations by visualizing their basins of attraction and finding out the number of convergent and divergent points for the iterative schemes in a selected region. Although all iterative schemes in this work have been selected of equal order of convergence and most of them have an equal number of function evaluations at each iteration, but clear differences have been noted in their behaviors. One can easily note that being an iterative scheme with smooth basins of attraction does not mean that the scheme has a larger area of convergence. In addition, we can conclude that it’s not necessary that a one-step iterative scheme is better than a two-step iterative scheme of the same order. Hence, it is not easy to determine if a specific iterative scheme is better than the other. Finally, even though all the iterative schemes used in this work have the same efficiency index, however, the results show that there are sometimes big differences in their basins of attraction and hence their area of convergence. These results force the need of proposing another index that reflects the real accuracy and efficiency of the iterative schemes.