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Author's personal copy Applied Mathematics and Computation 218 (2011) 88–95

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A comparison between iterative methods by using the basins of attraction Gheorghe Ardelean North 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

a b s t r a c t There exists a real competition between authors to construct improved iterative methods for solving nonlinear equations. In this paper, by using computer experiment, we study the basins 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 their performances (the area of convergence and theirs speed). The beauty fractal pictures generated by these methods are presented too. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Newton’s method Iterative methods Order of convergence Efﬁciency index Basin of attraction

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 sufﬁciently close to a and under some conditions, the classical Newton’s method deﬁned by

xnþ1 ¼ xn 

f ðxn Þ f 0 ðxn Þ

n ¼ 0; 1; 2; . . .

ð1Þ

generates a sequence fxn g1 n¼0 that converges to a. The mapping

UðxÞ ¼ x 

f ðxÞ f 0 ðxÞ

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 good monograf on numerical analysis is [5]. In this paper we present the results obtained by computer experiments for some of these improved methods applied to complex polynomial equations. 2. Fundamental concepts and description of the methods Order of convergence [17]. Let fxn g1 n¼0 be a sequence converging to a. Let ei = xi  a. If three exists a real number p and a nonzero constantC such that

jenþ1 j !C jen jp

E-mail address: ardelean_g@yahoo.com 0096-3003/\$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.055

ð3Þ

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then p is called order of the sequence and C is called the asymptotic error constant. If the sequence fxn g1 n¼0 is generated by an I. F. of an iterative method

xnþ1 ¼ Uðxn Þ;

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 corresponding method) is 1

Ei ¼ pd ;

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 

Pðzn Þ P0 ðzn Þ

n ¼ 0; 1; 2; . . .

ð6Þ

then the sequence fzn g1 n¼0 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 for which Newton’s iteration does not converge. Gaston Julia gives the answer to the question (1918). The basin of attraction corresponding 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 complex polynomial 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 efﬁ1 ciency index 22 ¼ 1:414. (b) Arithmetic mean Newton’s method (AN) [19] This method is deﬁned by

znþ1 ¼ zn 

2Pðzn Þ ; P0 ðzn Þ þ P0 ðyn Þ

n ¼ 0; 1; 2; . . . ;

ð7Þ 1

nÞ where yn ¼ zn  PPðz . The method is third-order of convergence and is of efﬁciency index 33 ¼ 1:442. 0 ðzn Þ

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

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

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

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

znþ1 ¼ zn 

Pðzn ÞðP0 ðzn Þ þ P0 ðyn ÞÞ 2P0 ðzn ÞP0 ðyn Þ

n ¼ 0; 1; 2; . . .

ð8Þ 1

nÞ where yn ¼ zn  PPðz . This method is third-order of convergence and efﬁciency index is 33 ¼ 1:442. 0 ðzn Þ

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91

Fig. 3. The basins of attraction for the roots of the polynomial P2(z).

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

znþ1 ¼ zn 

Pðzn Þ P ððzn þ yn Þ=2Þ 0

n ¼ 0; 1; 2; . . . ;

ð9Þ

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

Fig. 4. The basins of attraction for the roots of the polynomial P3(z).

where yn ¼ zn 

Pðzn Þ . P 0 ðzn Þ

1

This method is third-order of convergence and of efﬁciency index 33 ¼ 1:442 too. (e) Halley’s method (HM) [17] This method is deﬁned by

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

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

znþ1

! 1 LP ðzn Þ Pðzn Þ ¼ zn  1 þ ; 2 1  12 LP ðzn Þ P0 ðzn Þ

n ¼ 0; 1; 2; . . . ;

ð10Þ

00

n ÞPðzn Þ where LP ðzn Þ ¼ P ½Pðz0 ðz . Þ2 n

1

This method is third-order of convergence and of efﬁciency index 33 ¼ 1:442. (f) Traub–Ostrowski method (TOM) [17] The method is deﬁned by

znþ1 ¼ zn 

Pðyn Þ  Pðzn Þ Pðzn Þ 2Pðyn Þ  Pðzn Þ P0 ðzn Þ

nÞ where yn ¼ zn  PPðz . 0 ðzn Þ

n ¼ 0; 1; 2; . . .

ð11Þ

1

This method is of fourth-order of convergence and of efﬁciency index 43 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 in1 2 dex 43 ¼ 23 .

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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 can be 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 a grid 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 every z0 2 D. If the sequence generated by iterative method attempt a zero a of the polynomial with a tolerance e = 105 and a maximum of 14 iterations, we decide that z0 is in the basin of attraction of these zero and we can mark this point with a color associated to these zero, like in Fig. 1. We mark with white the points z0 2 D for which the corresponding iterative method 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 a color 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 the rectangles 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 employed 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 all informations 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 method AN – arithmetic mean Newton’s method HN – Harmonic mean Newton’s method MN – midpoint Newton’s method HM – Halley’s method TOM – Traub–Ostrowski’s method

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

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We think that the most relevant and important results are those in the diagram from Fig. 5 which show the computer time 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 for the 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 this group (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] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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