Amc a comparison between iterative methods

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A comparison between iterative methods by using the basins of attraction

Gheorghe ArdeleanNorth University of Baia Mare, Department of Mathematics and Computer Science, Victoriei 76, 430122 Baia Mare, Romania

a r t i c l e i n f o

Keywords:Newton’s methodIterative methodsOrder of convergenceEfficiency indexBasin of attraction

a b s t r a c t

There exists a real competition between authors to construct improved iterative methodsfor solving nonlinear equations. In this paper, by using computer experiment, we study thebasins of attraction for some of the iterative methods for solving the equation P(z) = 0,where P : C! C is a complex coefficients polynomial, and this allows us to compare theirperformances (the area of convergence and theirs speed). The beauty fractal pictures gen-erated by these methods are presented too.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Let f be a function f : R! R and a a root of the equation f(x) = 0, that is f(a) = 0. It is known that if f0(a) – 0, x0 is suffi-ciently close to a and under some conditions, the classical Newton’s method defined by

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

n ¼ 0;1;2; . . . ð1Þ

generates a sequence fxng1n¼0 that converges to a.The mapping

UðxÞ ¼ x� f ðxÞf 0ðxÞ ð2Þ

is called the Iteration function (abrev. I. F.) [17] of the method defined by (1).It is known that Newton’s method converges quadratically in a sufficiently small neighborhood of the root a.To increase the order of convergence of the iterative methods for solving nonlinear equations, many authors developed

new methods [2,3,6,8,10,11,13,19]. Other papers generating the basins of attraction are [1,4,7,12,14,15,18]. A very goodmonograf on numerical analysis is [5].

In this paper we present the results obtained by computer experiments for some of these improved methods applied tocomplex polynomial equations.

2. Fundamental concepts and description of the methods

Order of convergence [17].Let fxng1n¼0 be a sequence converging to a. Let ei = xi � a. If three exists a real number p and a nonzero constantC such thatjenþ1jjenjp

! C ð3Þ

0096-3003/$ - see front matter � 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2011.05.055

E-mail address: [email protected]

Applied Mathematics and Computation 218 (2011) 88–95

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Applied Mathematics and Computation

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then p is called order of the sequence and C is called the asymptotic error constant.If the sequence fxng1n¼0 is generated by an I. F. of an iterative method

xnþ1 ¼ UðxnÞ; n ¼ 0;1;2; . . . ð4Þ

then p is called the order of I. F. (and of the corresponding method too).Efficiency index [17].Let U be an I. F. A piece of information is any evaluation of a function or one of its derivatives. The informational usage d

of I. F. is the number of new pieces of information required per iteration. The efficiency index of U (and of the correspondingmethod) is

Ei ¼ p1d; ð5Þ

where p is the order of U and d is the informational usage.Basin of attraction [9]Let P : C! C be a complex polynomial of degree at least two, and let a be one of its zeros. If z0 2 C is a starting point for

the Newton’s method defined by

znþ1 ¼ zn �PðznÞP0ðznÞ

n ¼ 0;1;2; . . . ð6Þ

then the sequence fzng1n¼0 converges, or not, to a zero of P. If the sequence converges to a then we say that z0 is attracted to a.

Arthur Cayley in ‘‘The Newton–Fourier imaginary problem’’ (1879) recognized that if we know the roots of a function,Newton’s method suggests another problem: which starting point to which roots and what about the staring points forwhich Newton’s iteration does not converge. Gaston Julia gives the answer to the question (1918). The basin of attractioncorresponding to a zero a of the polynomial P is the set of all starting points z0 which are attracted to a.

In our study we investigate six iterative Newton-type methods and we present the results obtained for three complexpolynomial equations.

The methods are:

(a) Classical Newton’s method (CN).The method is defined by the relation (6). It is well known that this method is quadratically convergent and of effi-ciency index 2

12 ¼ 1:414.

(b) Arithmetic mean Newton’s method (AN) [19]This method is defined by

znþ1 ¼ zn �2PðznÞ

P0ðznÞ þ P0ðynÞ; n ¼ 0;1;2; . . . ; ð7Þ

where yn ¼ zn � PðznÞP0 ðznÞ. The method is third-order of convergence and is of efficiency index 3

13 ¼ 1:442.

Fig. 1. Gridpoints of the rectangle D (480 � 480 points).

G. Ardelean / Applied Mathematics and Computation 218 (2011) 88–95 89


(c) Harmonic mean Newton’s method (HN) [13]The method is defined by

znþ1 ¼ zn �PðznÞðP0ðznÞ þ P0ðynÞÞ

2P0ðznÞP0ðynÞn ¼ 0;1;2; . . . ð8Þ

where yn ¼ zn � PðznÞP0 ðznÞ. This method is third-order of convergence and efficiency index is 3

13 ¼ 1:442.

Fig. 2. The basins of attraction for the roots of the polynomial P1(z).

90 G. Ardelean / Applied Mathematics and Computation 218 (2011) 88–95


(d) Midpoint Newton’s method (MN) [13]The method is defined by

znþ1 ¼ zn �PðznÞ

P0 ðzn þ ynÞ=2ð Þn ¼ 0;1;2; . . . ; ð9Þ




where yn ¼ zn � PðznÞP0 ðznÞ.

This method is third-order of convergence and of efficiency index 313 ¼ 1:442 too.

(e) Halley’s method (HM) [17]This method is defined by




znþ1 ¼ zn � 1þ 12

LPðznÞ1� 1

2 LPðznÞ

!PðznÞP0ðznÞ

; n ¼ 0;1;2; . . . ; ð10Þ

where LPðznÞ ¼ P00 ðznÞPðznÞ½P0 ðznÞ�2

.

This method is third-order of convergence and of efficiency index 313 ¼ 1:442.

(f) Traub–Ostrowski method (TOM) [17]The method is defined by

znþ1 ¼ zn �PðynÞ � PðznÞ

2PðynÞ � PðznÞPðznÞP0ðznÞ

n ¼ 0;1;2; . . . ð11Þ

where yn ¼ zn � PðznÞP0 ðznÞ.

This method is of fourth-order of convergence and of efficiency index 413 ¼ 1:587.

By Kung–Traub Conjecture, a method has the optimal order equals to 2d�1 and optimal efficiency index equals to 2(d�1)/d,where d is the informational usage. So, this method has optimal order 4 = 23�1 (in this case d = 3) and optimal efficiency in-dex 4

13 ¼ 2

23.

Fig. 5. Computer time needed to generate the basins of attraction (Newton = 1).

Fig. 6. Mean number of iterations for convergent points.



The test polynomials used in our computer experiments are as follows:

(1) P1(z) = z7 � 1 whose zeros are: 1, 0.623 ± 0.782i, � 0.223 ± 0.975i, �0.901 ± 0.434i. This example is like in [16].(2) P2(z) = z8 + (1 + 8i)z7 + (�22 + 27i)z6 + (�105 + 70i)z5 + (� 271 + 185i)z4 + (�346 + 872i)z3 + (1282 + 1658i)z2 +

(3060 � 2820i)z � 3600. The zeros are: 1, 1 + i, 2i, �2 + 2i, �3, �3 � 3i, � 5i, 5 � 5i (in a ‘‘spiral’’ placement that canbe seen in Fig. 3. This example is from [6].

(3) P3(z) = z3 + (2 � 3i)z2 � (5 + 5i)z � 8 � 12i with zeros: �3 + 2i, � 1 � i, 2 + 2i.

3. Fractal pictures for the basins of attraction

To generate the basins of attraction for the zeros of a polynomial and an iterative method we take a rectangle D � C and agrid points system covering D, like in Fig. 1.

In our numerical experiments, we take a grid of 480 � 480 points in D and we apply the iterative method starting in everyz0 2 D. If the sequence generated by iterative method attempt a zero a of the polynomial with a tolerance e = 10�5 and amaximum of 14 iterations, we decide that z0 is in the basin of attraction of these zero and we can mark this point with acolor associated to these zero, like in Fig. 1. We mark with white the points z0 2 D for which the corresponding iterativemethod starting in z0 does not reach any zero of the polynomial, with tolerance e = 10�5 in a maximum of 14 iterations.

In practice, if the iterative method starting in z0 2 D reach a zero in k iterations (k < 15), then we mark this point z0 with acolor depending on k. For example, for k = 14 the associated color is red. For our polynomials P1(z), P2(z) and P3(z) we take therectangles D1, D2 and respectively D3 as follows: D1 = [2,2] � [�2,2]; D2 = [�15,20] � [�20,15]; D3 = [�300,300] � [�300,300]. In each of these cases, the rectangle contains all zeros of corresponding polynomial. To generate the pictures we em-ployed Mathcad 14.0 and a computer laptop DELL Inspiron 1501, 1.6 GHz.

For the basins of attraction the following Figs. 2–4 was obtained:

4. Numerical results

The results of our computer experiments are presented in three suggestive diagrams (in Figs. 5–7), that concentrate allinformations concerning the behaviour of the investigated methods on the three example polynomials P1, P2 and P3.

In them, the methods are identified as follows:

CN – classical Newton’s methodAN – arithmetic mean Newton’s methodHN – Harmonic mean Newton’s methodMN – midpoint Newton’s methodHM – Halley’s methodTOM – Traub–Ostrowski’s method

Fig. 7. Number of convergent poins/ Total number of starting points evaluated.



We think that the most relevant and important results are those in the diagram from Fig. 5 which show the computertime needed by each method to generate the basins of attraction for each polynomial P1, P2 and P3.

This time is presented as a relative time to Newton’s method, to not depend on the computer used.The absolute values of time for Newton’s method, on our computer was:220 s for the polynomial P1; 332 s for the polynomial P2; 206 s for the polynomial P3.

5. Conclusions

In each of shown diagrams, we included for each method a green bar corresponding to the mean of values presented forthe three polynomials P1, P2 and P3. This shows us that ‘‘the best’’ from all points of view is Traub–Ostrowski’s method(TOM). It’s normally, because this method has the greatest efficiency index, that is 1.587, and is an optimal efficiency index.We observe too that this method is only of fourth-order between the methods investigated.

About the methods of third-order (AN, HN, MN and HM) we consider that Halley’s method (HM) to be the best of thisgroup (is the best as the time parameter and percentage of convergent points). The Arithmetic mean Newton’s method(AN) seems to be ‘‘the last’’ from all points of view.

We specify that classical Newton’s method (CN) is not included in the ‘‘competition’’. This is only used as a standard.

References

[1] G. Ardelean, Generating the basins of attraction for Newton’s method, Creative Math. Inf. 17 (2008) 285–292.[2] D. Basu, From third order to fourth order variant of Newton’s method for simple roots, Appl. Math. Comp. 202 (2008) 886–892.[3] C. Chun, Y. Ham, Some forth-order modifications of Newton’s method, Appl. Math. Comp. 197 (2008) 654–658.[4] O. Cira, Lect�ii de Mathcad 2001 Professional, Ed. Albastra, Cluj-Napoca, 2003.[5] W. Gautschi, Numerical Analysis: An Introduction, Birkhäuser, 1997.[6] Y. Ham, C. Chun, S.G. Lee, Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comp. Appl. Math. 222 (2008) 477–

486.[7] B. Kalantari, Polinomiography and applications in art, education and science, Comput. Graph. 28 (2004) 417–430.[8] Y.I. Kim, A new two-step biparametric family of sixth-order iterative methods free from second derivatives for solving nonlinear algebraic equations,

Appl. Math. Comput. 215 (2010) 3418–3424.[9] D. Kinkaid, W. Chenney, Numerical analysis: mathematics of scientific computing, AMS (2009).

[10] J. Kou, Y. Li, X. Wang, A family of fifth-order iterations composed of Newton and third-order methods, Appl. Math. Comput. 186 (2007) 1258–1262.[11] L. Liu, X. Wang, Eigth-order methods with high efficiency index for solving nonlinear equations, Appl. Math. Comput. 215 (2010) 3449–3454.[12] S�t. Marus�ter, Numerical experiments on attraction basins, Ann. Comput. Sci. Ser. 2nd Tome 1st Fasc. (2004) 167–171.[13] A.Y. Ozban, Some new variant of Newton’s Method, Appl. Math. Lett. 17 (2004) 677–682.[14] M.L. Sahari, I. Djellit, Fractal Newton basins, Discrete Dyn. Nature Soc. (2006) 28756 (Article ID).[15] R.A. Salajan (Pomian), The convergency order of some methods generalized from Newton method, Studii s�i Cercetari S�tiint�ifice, seria. Matem. Univ.

Bacau 18 (2008) 267–282.[16] H. Susanto, N. Karjanto, Newton’s method basin of attraction revisited, Appl. Math. Comput. 215 (2009) 1084–1090.[17] J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea publishing company, New York, 1997.[18] J.L. Varona, Graphic and numerical comparison between iterative methods, Math. Intell. 24 (2002) 37–46.[19] S. Weerakoon, T.G.I. Fernando, A Variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93.


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