Applied Mathematics and Computation

Alied Mathematics and Comutation 218 (2012) 10548 10556 Contents lists available at SciVerse ScienceDirect Alied Mathematics and Comutation journal homeage: www.elsevier.com/locate/amc Basins of attraction for several methods to find simle roots of nonlinear equations Beny Neta a,, Melvin Scott b, Changbum Chun c a Naval Postgraduate School, Deartment of Alied Mathematics, Monterey, CA 994, United States b 494 Carlton Court, Ocean Isle Beach, NC 28469, USA c Deartment of Mathematics, Sungkyunkwan University, Suwon 440-746, Reublic of Korea article info abstract Keywords: Basin of attraction Simle roots Nonlinear equations Halley method Suer Halley method Modified suer Halley method King s family of methods There are many methods for solving a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its deendence on the order. We discuss several third and fourth order methods to find simle zeros. The relationshi between the basins of attraction and the corresonding conjugacy mas will be discussed in numerical exeriments. The effect of the extraneous roots on the basins is also discussed. Published by Elsevier Inc. 1. Introduction There is a vast literature for the numerical solution of nonlinear equations. The methods are classified by their order of convergence,, and the number, d, of function (and derivative) evaluations er ste. There are two efficiency measures (see Traub [1]) defined as I ¼ =d (informational efficiency) and E ¼ 1=d (efficiency index). Another measure, introduced recently, is the basin of attraction. See Stewart [2], Scott et al. [], Amat et al. [4,5], Chun et al. [6], and for methods to find multile roots, see Neta et al. [7]. Chun et al. [6] have develoed a new family of methods for simle roots free from second derivative. The family is of order four and includes Jarratt s method (see [8]) as a secial case. They have discussed the dynamics of the family and comared its basin of attraction to three other fourth order methods. Amat et al. [9] discuss the dynamics of a family of third-order methods that do not require second derivatives. In another aer [10] they discuss the dynamics of King and Jarratt s schemes. They do not discuss the best choice of the arameter in King s method as we will do here. In recent years there has been considerable interest in develoing new algorithms with high order convergence. Normally, these high order convergence algorithms contain higher derivatives of the function or multi-ste. In the former case, various techniques can be used to eliminate the derivatives. However, the resulting iteration function may be more comlex than the original, for examle, it can introduce extraneous zeroes. Our study considers several methods of various orders. We include Halley s method, suer Halley, modified suer Halley, Jarratt s method, and King s family of methods. Newton s method of order = 2 was discussed by Stewart [2] and Scott et al. [] and thus will not be given here. Halley s method of order three was discussed by Stewart [2] and we included it for comarison with suer Halley and modified suer Halley (both of order four). Neta et al. [11] have shown that the modified suer Halley method is a rediscovered Jarratt s scheme. We also include two other fourth order methods, namely Jarratt s method and King s family. In this study, we will find the Corresonding author. E-mail addresses: bneta@ns.edu (B. Neta), mscott822@atmc.net (M. Scott), cbchun@skku.edu (C. Chun). 0096-00/$ - see front matter Published by Elsevier Inc. htt://dx.doi.org/10.1016/j.amc.2012.04.017

B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 10549 extraneous fixed oints, if any. We will also show how to choose a arameter (in the case of King s family of methods) to get best results. In the next section we describe the methods to be considered in this comarative study. Section will give the conjugacy mas for each method and discuss the ossibility of extraneous fixed oints [12]. We will show the relationshi between these mas, extraneous fixed oints, and the basins of attraction in our numerical exeriments detailed in Section 4. 2. Methods for the comarative study First we list the methods we consider here with their order of convergence. (1) Halley s method ( = ). (2) Suer Halley otimal method ( = 4). () Modified suer Halley otimal method ( = 4). (4) King s family of methods ( = 4). (5) Jarratt s method ( = 4). King s family of fourth order methods did not erform well in our revious study []. We will show how to choose the family member to get best results based on the location of the extraneous fixed oints. We now detail the methods studied here. Halley s third order (H) method [1] is given by 1 x nþ1 ¼ x n u n 1 1 u ; ð1þ fn 00 2 n where f 0 n u n ¼ f n ; ð2þ fn 0 and f n ¼ f ðx n Þ and similarly for the derivatives. Suer Halley fourth order (SH4) method [14] is given by y n ¼ x n 2 u n; x nþ1 ¼ x n 1 þ 1 L f u n ; 2 1 L f ðþ where L f ¼ f nf 00 n ðf 0 n Þ2 : ð4þ A modified suer Halley fourth order (MSH4) otimal method [15] is given by y n ¼ x n 2 u n;! x nþ1 ¼ x n 1 þ 1 ^Lf 2 1 ^L u n ; f where ð5þ f n f 0 ðy ^Lf ¼ n Þ fn 0 : ð6þ ðfn 0Þ2 y n x n King s family of fourth order methods (K4) [16] is given by y n ¼ x n u n ; x nþ1 ¼ y n f ðy nþ f 0 n f n þ bf ðy n Þ f n þðb 2Þf ðy n Þ : ð7þ Jarratt s fourth order (J4) method [17] is given by y n ¼ x n 2 u n; u n x nþ1 ¼ x n 1 2 u n 1 : 2 1 þ f 0 ðy n Þ 1 2 fn 0 ð8þ

10550 B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 Fig. 1. To row: Halley s (left), suer Halley s method (middle) and modified suer Halley (right). Bottom row: King s (left), and Jarratt s (right). The results are for the olynomial z 2 1. Note that this is a different method than the one discussed by Amat et al. [10].. Corresonding conjugacy mas for quadratic olynomials For Newton s method the following is well known [4]. Theorem.1 (Newton s method). For a rational ma R ðzþ arising from Newton s method alied to ðzþ ¼ðz aþðz bþ; a b; R ðzþ is conjugate via the Möbius transformation given by MðzÞ ¼ z a SðzÞ ¼z 2 : Theorem.2 (Halley s method). For a rational ma R ðzþ arising from Halley s method alied to ðzþ ¼ðz aþðz bþ; a b; R ðzþ is conjugate via the Möbius transformation given by MðzÞ ¼ z a SðzÞ ¼z : Proof. Let ðzþ ¼ðz aþðz bþ; a b and let M be the Möbius transformation given by MðzÞ ¼ z a with its inverse z b M 1 ðuþ ¼ ub a, which may be considered as a ma from C [f1g. We then have u 1 M R M 1 ub a ðuþ ¼M R ¼ u : u 1 The roof for the other methods is similar. In the sequel we resent only the result. Theorem. (Suer Halley s method). For a rational ma R ðzþ arising from suer Halley s method alied to ðzþ ¼ðz aþðz bþ; a b; R ðzþ is conjugate via the Möbius transformation given by MðzÞ ¼ z a SðzÞ ¼z 4 :

B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 10551 Fig. 2. To row: Halley s (left), suer Halley s method (middle) and modified suer Halley (right). Bottom row: King s (left), and Jarratt s (right). The results are for the olynomial z 1. Fig.. To row: Halley s (left), suer Halley s method (middle) and modified suer Halley (right). Bottom row: King s (left), and Jarratt s (right). The results are for the olynomial z z.

10552 B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 Fig. 4. To row: Halley s (left), and suer Halley s method (right). Bottom row: King s (left), and Jarratt s (right). The results are for the olynomial z 4 10z 2 þ 9. Theorem.4 (Modified suer Halley method). For a rational ma R ðzþ arising from the method (5) alied to ðzþ ¼ðz aþðz bþ; a b; R ðzþ is conjugate via the Möbius transformation given by MðzÞ ¼ z a SðzÞ ¼z 4 : Theorem.5 (King s fourth-order family of methods). For a rational ma R ðzþ arising from the method (7) alied to ðzþ ¼ðz aþðz bþ; a b; R ðzþ is conjugate via the Möbius transformation given by MðzÞ ¼ z a 1 þ 2b þð2þbþzþz2 SðzÞ ¼ 1 þð2þbþzþð1þ2bþz 2 z4 : For Jarratt s method we have the following result. Theorem.6 (Jarratt s fourth order otimal method). For a rational ma R ðzþ arising from the method (8) alied to ðzþ ¼ðz aþðz bþ; a b; R ðzþ is conjugate via the Möbius transformation given by MðzÞ ¼ z a SðzÞ ¼z 4 : Note that the mas are of the form SðzÞ ¼z RðzÞ where RðzÞ is either unity or a rational function and is the order of the method..1. Extraneous fixed oints Note that all these methods can be written as x nþ1 ¼ x n u n H f ðx n ; y n Þ:

B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 1055 Fig. 5. To row: Halley s (left), and suer Halley s method (right). Bottom row: King s (left), and Jarratt s (right). The results are for the olynomial z 5 1. Clearly the root a is a fixed oint of the method. The oints n a at which H f ðnþ ¼0 are also fixed oints of the method, since the second term on the right vanishes. These oints are called extraneous fixed oints (see [12]). Theorem.7. There are no extraneous fixed oints for Halley s method. Proof. For Halley s method (1) we have 1 H f ¼ 1 1 u 2 n : fn 00 fn 0 This function does not vanish and therefore there are no extraneous fixed oints. h Theorem.8. The extraneous fixed oints of suer Halley s, modified suer Halley s and Jarratt s methods are i ffiffi. Proof. For suer Halley s method, the extraneous fixed oint can be found by solving L f ¼ 2. This leads to the equation 1 z 2 þ 1 2 z 2 þ 1 ¼ 0 ffiffi for which the roots are i. These fixed oints are reulsive. The oles are at z ¼ ffiffi 2 2 i. For modified suer Halley s method, the extraneous fixed oints are the solution of ^L f ¼ 2. This leads to the same equation as above and therefore the same extraneous fixed oints. These fixed oints are reulsive. For Jarratt s method the extraneous fixed oints are those for which

10554 B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 Fig. 6. To row: Halley s (left), and suer Halley s method (right). Bottom row: King s (left), and Jarratt s (right). The results are for the olynomial z 6 1 2 z5 þ 11ðiþ1Þ z 4 iþ19 z þ 5iþ11 z 2 þ i 11 z þ i. 4 4 4 4 2 1 1 þ ¼ 0 1 þ f 0ðyðzÞÞ 1 2 f 0ðzÞ where yðzþ ¼z 2 uðzþ. Uon substituting f ðzþ ¼z2 1, we get the equation 1 1 þ 1 þ ð1 z2 1Þ ¼ 0: 2 2 z 2 ffiffi The solution is again z ¼i. These fixed oints are reulsive. The oles are at z ¼ ffiffi 2 2 i. h Theorem.9. There are four extraneous fixed oints of King s family of methods. For b ¼ 2 ffiffiffi 2 we get the roots very close to the imaginary axis. Proof. The extraneous fixed oint of King s family of methods are those for which 1 þ yðzþ2 1 vðzþ ¼0 where z 2 1 vðzþ ¼ z2 1þbðyðzÞ 2 1Þ f ðzþ. Uon substituting yðzþ ¼z we get the equation z 2 1þðb 2ÞðyðzÞ 2 1Þ f 0ðzÞ ð5b þ 12Þz 4 þð4 6bÞz 2 þ b ¼ 0: 4z 2 ððb þ 2Þz 2 b þ 2Þ In order to get roots on the imaginary axis, we choose b ¼ 2 ffiffiffi 2 and then the four roots are :2074660892e 4 :98755690i. Note that the real arts are negligible, so the roots are not on the imaginary axis but very close. These four fixed oints are slightly reulsive (the derivative at those oint is 1:000000172 :0005549846764i and its magnitude is 1.00000026.

B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 10555 Fig. 7. To row: Halley s (left), and suer Halley s method (right). Bottom row: King s (left), and Jarratt s (right). The results are for the olynomial z 7 1. qffiffiffiffiffiffi b 2 The oles are at z ¼ 0 (double) and z ¼ bþ2. For b ¼ 2 ffiffiffi q ffiffiffiffiffiffiffiffiffiffiffi 2 the oles are at z ¼i 2 2 1 ffiffi all on the imaginary 5 2 2 axis. h 4. Numerical exeriments In our first exeriment, we have run all the methods to obtain the real simle zeros of the quadratic olynomial z 2 1. The results of the basin of attractions are given in Fig. 1. It is clear that Halley, suer Halley and Jarratt are the best followed by King s method. The member of King s family of methods we have used is the one with b ¼ 2 ffiffiffi 2 so that the extraneous fixed oints are very close to the imaginary axis. In our revious study [] we have used other values and the results were not as good as these. The modified suer Halley does not give good results even though we roved that it is a rediscovered Jarratt s scheme. The mas are identical and the extraneous roots are identical, of course. This means that it does matter how one organizes the calculation. In our next exeriment we have taken the cubic olynomial z 1. The results are given in Fig. 2. Again the modified suer Halley s method did not give the same results as Jarratt s method. Halley s method is best followed by suer Halley s and Jarratt s methods. In the next examles we have taken olynomials of increasing degree. The results for the cubic olynomial z z are given in Fig.. Here Jarratt s scheme is best followed by Halley s and King s methods. We have decided not to show the results for the modified suer Halley method for the rest of the exeriments. Fig. 4 shows the results for the olynomial z 4 10z 2 þ 9. In this case Jarratt s method is best followed by Halley s and King s methods. The fifth order olynomial, z 5 1, results are shown in Fig. 5. Halley s and Jarratt s methods are best. The next examle is for a olynomial of degree 6 with comlex coefficients, z 6 1 2 z5 þ 11ðiþ1Þ z 4 iþ19 z þ 5iþ11 z 2 þ i 11 z þ i. The results are 4 4 4 4 2 resented in Fig. 6. Again Halley s and Jarratt s methods erformed better than the other two. The last examle for a olynomial of degree 7, z 7 1. In all these Figs. 2 7 we find that Halley s and Jarratt s methods are better than the other schemes.

10556 B. Neta et al. / Alied Mathematics and Comutation 218 (2012) 10548 10556 5. Conclusions In this aer we have considered several third and fourth order methods for finding simle zeros of a nonlinear equation. Note that the conjugacy mas do not tell the whole story as one can see from comaring Jarrat s method to the suer Halley method. We have studied all of the extraneous fixed oints and they are reulsive. We have shown how to find the best arameter for the King s family of methods so that its erformance is imroved. Unfortunately, Halley s third order method and Jarratt s fourth order methods erformed even better. We should also mention that since Ostrowski s method [18] is a secial case of King s family with b ¼ 0, then we cannot exect it to erform better than King s method with the choice of b ¼ 2 ffiffiffi 2. Acknowledgements Professor Chun s research was suorted by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0025877). References [1] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964. [2] B.D. Stewart, Attractor Basins of Various Root-finding Methods, M.S. Thesis, Naval Postgraduate School, Deartment of Alied Mathematics, Monterey, CA, June 2001. [] M. Scott, B. Neta, C. Chun, Basin attractors for various methods, Al. Math. Comut. 218 (2011) 2584 2599, htt://dx.doi.org/10.1016/ j.amc.2011.07.076. [4] S. Amat, S. Busquier, S. Plaza, Iterative Root-finding Methods, Unublished Reort, 2004. [5] S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical oint of view, Scientia 10 (2004) 5. [6] C. Chun, M.Y. Lee, B. Neta, J. Dz unić, On otimal fourth-order iterative methods free from second derivative and their dynamics, Al. Math. Comut. 218 (2012) 6427 648. [7] B. Neta, M. Scott, C. Chun, Basin attractors for various methods for multile roots, Al. Math. Comut. 218 (2012) 504 5066, htt://dx.doi.org/ 10.1016/j.amc.2011.10.071. [8] P. Jarratt, Some fourth-order multioint iterative methods for solving equations, Math. Comut. 20 (1966) 44 47. [9] S. Amat, S. Busquier, S. Plaza, Dynamics of a family of third-order iterative methods that do not require using second derivatives, Al. Math. Comut. 154 (2004) 75 746. [10] S. Amat, S. Busquier, S. Plaza, Dynamics of the King and Jarratt iterations, Aeq. Math. 69 (2005) 212 226. [11] B. Neta, C. Chun, M. Scott, A note on the modified suer-halley method, Al. Math. Comut. 218 (2012) 9575 9577, htt://dx.doi.org/10.1016/ j.amc.2012.0.046. [12] E.R. Vrscay, W.J. Gilbert, Extraneous fixed oints, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions, Numer. Math. 52 (1988) 1 16. [1] E. Halley, A new, exact and easy method of finding the roots of equations generally and that without any revious reduction, Philos. Trans. Roy. Soc. Lond. 18 (1694) 16 148. [14] J.M. Gutiérrez, M.A. Hernández, An acceleration of Newton s method: suer-halley method, Al. Math. Comut. 117 (2001) 22 29. [15] C. Chun, Y. Ham, Some second-derivative-free variants of suer-halley method with fourth-order convergence, Al. Math. Comut. 195 (2008) 57 541. [16] R.F. King, A family of fourth-order methods for nonlinear equations, SIAM Numer. Anal. 10 (197) 876 879. [17] P. Jarratt, Multioint iterative methods for solving certain equations, Comut. J. 8 (1966) 98 400. [18] A.M. Ostrowski, Solution of Equations and Systems of Equations, third ed., Academic Press, New York, London, 197.