Applied Mathematics and Computation

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1 Applied Mathematics and Computation 245 (2014) Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: An analysis of a new family of eighth-order optimal methods Changbum Chun a, Beny Neta b, a Department of Mathematics, Sungkyunkwan University, Suwon , Republic of Korea b Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, United States article info abstract Keywords: Newton s method Iterative methods Nonlinear equations Order of convergence Extraneous fixed points Basin of attraction A new family of eighth order optimal methods is developed and analyzed. Numerical experiments show that our family of methods perform well and in many cases some members are superior to other eighth order optimal methods. It is shown how to choose the parameters to widen the basin of attraction. Published by Elsevier Inc. 1. Introduction There are many multistep methods for the solution of nonlinear equations, see e.g. Traub [1], and the recent book by Petković et al. [2]. The idea of optimality in such methods was introduced by Kung and Traub [3] who also developed optimal multistep method of increasing order. For example, the fourth-order optimal method given in [3] is 8 >< >: w n ¼ x n f ðxnþ ; f 0 ðx nþ x nþ1 ¼ w n f ðwnþ f 0 ðx nþ h i 2 : 1 1 f ðwnþ f ðxnþ ð1þ Based on this method Chun and Neta [4] constructed and analyzed the sixth order method 8 w n ¼ x n f ðxnþ ; f 0 ðx nþ s n ¼ w n f ðwnþ 1 >< h i 2 ; f 0 ðx nþ 1 f ðwnþ f ðxnþ x nþ1 ¼ s n f ðsnþ 1 f 0 ðx nþ h i 2 : >: 1 f ðwnþ ðsnþ f f ðxnþ f ðxnþ ð2þ In this paper we will use the idea of weight function to develop a family of optimal eighth order methods and show how to choose the parameters to obtain the best basins of attraction. Corresponding author. addresses: cbchun@skku.edu (C. Chun), bneta@nps.edu (B. Neta) /Published by Elsevier Inc.

2 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) An optimal eighth-order method We consider here a generalization of the Chun Neta sixth order scheme (2). The new family is constructed using the idea of weight functions. The multistep method is given by 8 w n ¼ x n f ðxnþ ; f 0 ðx nþ >< s n ¼ w n f ðwnþ 1 f 0 ðx nþ ; ½1 r n ð3þ Š 2 >: x nþ1 ¼ s n f ðsnþ 1 f 0 ðx nþ ; ½1 Hðr nþjðt nþpðq n ÞŠ 2 where r n ¼ f ðwnþ ; t f ðxnþ n ¼ f ðsnþ ; q f ðxnþ n ¼ f ðsnþ and HðrÞ; JðtÞ; PðqÞ are real-valued weight functions to be determined later. f ðwnþ For the method defined by (3), we have the following analysis of convergence. Theorem 2.1. Let n 2 I be a simple zero in an open interval I of a sufficiently differentiable function f : I! R. Let e n ¼ x n n. Then the new family of methods defined by (3) is of optimal eighth-order when Hð0ÞJð0ÞPð0Þ ¼2; H 0 ð0þpð0þjð0þ ¼ 1; H 00 ð0þpð0þjð0þ ¼ 1; H 000 ð0þpð0þjð0þ ¼3; jh ð4þ ð0þj < 1; J 0 ð0þ ¼ 3Jð0Þ=8; jj 00 ð0þj < 1; P 0 ð0þ ¼ Pð0Þ=4; jp 00 ð0þj < 1: The error at the ðn þ 1Þth step, e nþ1, satisfies the relation 2P 00 ð0þ e nþ1 ¼ c 2 P 0 ð0þ 1 c c 2c 3 c 4 þ 4 3P00 ð0þ P 0 c 2 2 ð0þ c2 3 þ 2c3 2 c 4 þ 1 3 P0 ð0þjð0þh ð4þ ð0þþ 6P00 ð0þ P 0 ð0þ 29 c c 3 þ P0 ð0þjð0þh ð4þ ð0þ 4P00 ð0þ P 0 c 6 2 e 8 n ð0þ þ Oðe9 n Þ; where c i are given by ð4þ c i ¼ f ðiþ ðnþ i!f 0 ; i P 1: ð5þ ðnþ Proof. Let e n ¼ x n n; e w n ¼ w n n and e s n ¼ s n n. Using the Taylor expansion of f ðxþ around x ¼ n and taking f ðnþ ¼0 into account, we get and f ðx n Þ¼f 0 ðnþ e n þ c 2 e 2 n þ c 3e 3 n þ c 4e 4 n þ c 5e 5 n þ c 6e 6 n þ c 7e 7 n þ c 8e 8 n þ Oðe9 n Þ f 0 ðx n Þ¼f 0 ðnþ 1 þ 2c 2 e n þ 3c 3 e 2 n þ 4c 4e 3 n þ 5c 5e 4 n þ 6c 6e 5 n þ 7c 7e 6 n þ Oðe7 n Þ : ð7þ Dividing (6) by (7) gives u n ¼ f ðx nþ f 0 ðx n Þ ¼ e n c 2 e 2 n þð 2c 3 þ 2c 2 2 Þe3 n þð 3c 4 þ 7c 2 c 3 4c 3 2 Þe4 n þð10c 2c 4 4c 5 þ 6c c 3c 2 2 þ 8c4 2 Þe5 n þð17c 4c 3 28c 4 c 2 2 þ 13c 2c 5 5c 6 33c 2 c 2 3 þ 52c 3c c5 2 Þe6 n þð 92c 3c 2 c 4 þ 22c 3 c 5 18c 3 3 þ 126c2 3 c c 3c 4 2 ð6þ þ 12c 2 4 þ 72c 4c c 5c 2 2 6c 7 þ 16c 2 c 6 þ 32c 6 2 Þe7 n þð 7c 8 118c 5 c 2 c 3 þ 348c 4 c 3 c 2 2 þ 19c 2c 7 64c 2 c 2 4 þ 31c 4 c 5 75c 4 c c 4c 4 2 þ 92c 5c 3 2 þ 27c 6c 3 44c 6 c 2 2 þ 135c 2c c2 3 c3 2 þ 304c 3c c7 2 Þe8 n þ Oðe9 n Þ: ð8þ

3 88 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) From (8), we have e w n ¼ c 2e 2 n ð2c2 2 2c 3Þe 3 n ð3c 4 7c 2 c 3 þ 4c 3 2 Þe4 n þð 10c 2c 4 þ 4c 5 6c 2 3 þ 20c 3c 2 2 8c4 2 Þe5 n þð 17c 4c 3 þ 28c 4 c c 2 c 5 þ 5c 6 þ 33c 2 c c 3c 3 2 þ 16c5 2 Þe6 n þð92c 3c 2 c 4 22c 3 c 5 þ 18c c2 3 c2 2 þ 128c 3c c2 4 72c 4 c 3 2 þ 36c 5c 2 2 þ 6c 7 16c 2 c 6 32c 6 2 Þe7 n þð64c7 2 þ 7c 8 þ 118c 5 c 2 c 3 348c 4 c 3 c c 2c 7 þ 64c 2 c c 4 c 5 þ 75c 4 c 2 3 þ 176c 4c c 5c c 6c 3 þ 44c 6 c c 2c 3 3 þ 408c2 3 c c 3c 5 2 Þe8 n þ Oðe9 n Þ: ð9þ Writing the Taylor s expansion for f ðw n Þ and using (9), we obtain f ðw n Þ¼f 0 ðnþ½e w n þ c 2ðe w n Þ2 þ c 3 ðe w n Þ3 þ c 4 ðe w n Þ4 þ Oððe w n Þ5 ÞŠ ¼ f 0 ðnþ½c 2 e 2 n þð2c 3 2c 2 2 Þe3 n þð3c 4 7c 2 c 3 þ 5c 3 2 Þe4 n þð 10c 2c 4 þ 4c 5 6c 2 3 þ 24c 3c c4 2 Þe5 n þð 17c 4 c 3 þ 34c 4 c c 2c 5 þ 5c 6 þ 37c 2 c c 3c 3 2 þ 28c5 2 Þe6 n þð18c3 3 64c6 2 þ 6c 7 þ 104c 3 c 2 c 4 16c 2 c 6 22c 3 c 5 160c 2 3 c2 2 þ 206c 3c c c 4c 3 2 þ 44c 5c 2 2 Þe7 n þð144c7 2 þ 7c 8 þ 134c 5 c 2 c 3 455c 4 c 3 c c 2c 7 þ 73c 2 c c 4c 5 þ 75c 4 c 2 3 þ 297c 4c c 5c c 6c 3 þ 54c 6 c c 2c 3 3 þ 582c 2 3 c c 3c 5 2 Þe8 n þ Oðe9 n ÞŠ: ð10þ Dividing (10) by (7) gives f ðw n Þ f 0 ðx n Þ ¼ c 2e 2 n þð2c 3 4c 2 2 Þe3 n þð3c 4 14c 2 c 3 þ 13c 3 2 Þe4 n þð 20c 2c 4 þ 4c 5 12c 2 3 þ 64c 3c c4 2 Þe5 n þð104c5 2 þ 5c 6 26c 2 c 5 34c 4 c 3 þ 90c 4 c 2 2 þ 103c 2c c 3c 3 2 Þe6 n þ Oðe7 n Þ: ð11þ Dividing (10) by (6) gives r n ¼ f ðw nþ f ðx n Þ ¼ c 2 e n þð2c 3 3c 2 2 Þe2 n þð3c 4 10c 2 c 3 þ 8c 3 2 Þe3 n þð 14c 2c 4 þ 4c 5 8c 2 3 þ 37c 3c c4 2 Þe4 n þð 22c 3 c 4 þ 51c 4 c c 2c 5 þ 55c 2 c c 3c 3 2 þ 5c 6 þ 48c 5 2 Þe5 n þð6c 7 þ 150c 4 c 2 c 3 22c 2 c 6 15c c 4c c 5 c 3 þ 65c 5 c c2 3 c2 2 þ 344c 3c 4 2 þ 26c c6 2 Þe6 n þ Oðe7 n Þ: ð12þ Using (8), (11) and (12), we find e s n ¼ ew n f ðw nþ 1 f 0 ðx n Þ ½1 rš 2 ¼ð c 2 c 3 þ 2c 3 2 Þe4 n þð 2c 2c 4 þ 14c 3 c 2 2 2c2 3 10c4 2 Þe5 n þð31c5 2 3c 2c 5 7c 4 c 3 þ 21c 4 c 2 2 þ 30c 2c c 3c 3 2 Þe6 n þð20c c6 2 þ 88c 3c 2 c 4 4c 2 c 6 10c 3 c 5 188c 2 3 c2 2 þ 246c 3c 4 2 6c c 4c 3 2 þ 28c 5c 2 2 Þe7 n þ Oðe8 n Þ; ð13þ so that, after elementary calculation, f ðs n Þ¼f 0 ðnþ½e s n þ c 2ðe s n Þ2 þ c 3 ðe s n Þ3 þ Oððe s n Þ4 ÞŠ ¼ f 0 ðnþ½ð c 2 c 3 þ 2c 3 2 Þe4 n þð 2c 2c 4 þ 14c 3 c 2 2 2c2 3 10c4 2 Þe5 n þð31c5 2 3c 2c 5 7c 4 c 3 þ 21c 4 c 2 2 þ 30c 2c c 3c 3 2 Þe6 n þð20c c6 2 þ 88c 3c 2 c 4 4c 2 c 6 10c 3 c 5 188c 2 3 c2 2 þ 246c 3c 4 2 6c c 4c 3 2 þ 28c 5c 2 2 Þe7 n þ Oðe8 n ÞŠ: ð14þ An easy calculation then produces f ðs n Þ f 0 ðx n Þ ¼ð c 2c 3 þ 2c 3 2 Þe4 n þð 2c 2c 4 þ 16c 3 c 2 2 2c2 3 14c4 2 Þe5 n þð59c5 2 3c 2c 5 7c 4 c 3 þ 25c 4 c 2 2 þ 37c 2c c 3c 3 2 Þe6 n þð26c c6 2 þ 112c 3c 2 c 4 4c 2 c 6 10c 3 c 5 310c 2 3 c2 2 þ 508c 3c 4 2 6c c 4c 3 2 þ 34c 5c 2 2 Þe7 n þ Oðe8 n Þ; ð15þ t n ¼ f ðs nþ f ðx n Þ ¼ð c 2 c 3 þ 2c 3 2 Þe3 n þð 2c 2c 4 2c 2 3 þ 15c 3c c4 2 Þe4 n þð 3c 2c 5 7c 3 c 4 þ 33c 2 c c 3c 3 2 þ 23c 4c 2 2 þ 43c5 2 Þe5 n þð98c 4 c 2 c 3 4c 2 c 6 6c c 4c c 5c 3 þ 31c 5 c c2 3 c2 2 þ 347c 3c 4 2 þ 22c c6 2 Þe6 n þð130c 5 c 2 c 3 651c 4 c 3 c 2 2 5c 2c 7 17c 5 c 4 162c 5 c c 6c 3 þ 39c 6 c 2 2 þ 72c 2c 2 4 þ 95c 4c 2 3 þ 468c 4c c 2c 3 3 þ 1087c 2 3 c c 3c 5 2 þ 266c7 2 Þe7 n þ Oðe8 n Þ ð16þ and

4 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) q n ¼ f ðs nþ f ðw n Þ ¼ð c 3 þ 2c 2 2 Þe2 n þð 2c 4 þ 8c 2 c 3 6c 3 2 Þe3 n þð 3c 5 þ 7c c 3c 2 2 þ 11c 2c 4 þ 9c 4 2 Þe4 n þð18c 3c 4 4c 6 30c 4 c 2 2 þ 14c 2c 5 32c 2 c 2 3 þ 36c 3c 3 2 2c5 2 Þe5 n þð22c 5c 3 71c 4 c 2 c 3 5c 7 36c 5 c 2 2 þ 17c 2c 6 þ 11c 2 4 þ 37c 4 c c3 3 þ 41c2 3 c2 2 þ 26c 3c c6 2 Þe6 n þ Oðe7 n Þ: ð17þ We now expand Hðr n Þ; Jðt n Þ; Pðq n Þ into Taylor series about 0 to obtain Hðr n Þ¼Hð0ÞþH 0 ð0þr n þ H00 ð0þ 2 r2 n þ H000 ð0þ r 3 n 6 þ Oðr4 n Þ; Jðt n Þ¼Jð0ÞþJ 0 ð0þt n þ J00 ð0þ 2 t2 n þ Oðt3 n Þ; Pðq n Þ¼Pð0ÞþP 0 ð0þq n þ P00 ð0þ 2 q2 n þ Oðq3 n Þ: Upon using the values Hð0ÞJð0ÞPð0Þ ¼2; H 0 ð0þpð0þjð0þ ¼ 1; H 00 ð0þpð0þjð0þ ¼ 1; H 000 ð0þpð0þjð0þ ¼3; J 0 ð0þ ¼ 3Jð0Þ=8; P 0 ð0þ ¼ Pð0Þ=4 ð18þ and Eqs. (11), (16) and (17) we obtain 1 ½1 Hðr n ÞJðt n ÞPðq n ÞŠ ¼ 1 þ 2c 2e 2 n þ 3c 3 e 2 n þ 4c 4e 3 n þ 2 P00 ð0þ P 0 ð0þ þ 1 3 P0 ð0þjð0þh ð4þ ð0þ 9 c þ 5c 5 c 2 c 4 þ 9 2 ð0þ 2P00 P 0 c 3 c 2 2 ð0þ þ 1 P 00 ð0þ 2 P 0 ð0þ 1 c þ Oðe 5 n Þ: Therefore, from (13), (15) and (19), we obtain e nþ1 ¼ e s n f ðs nþ 1 f 0 ðx n Þ ½1 HðrÞJðtÞPðqÞŠ 2 2P 00 ð0þ ¼ c 2 P 0 ð0þ 1 c c 2c 3 c 4 þ 4 3P00 ð0þ P 0 c 2 2 ð0þ c2 3 þ 2c3 2 c 4 þ 1 3 P0 ð0þjð0þh ð4þ ð0þþ 6P00 ð0þ P 0 ð0þ 29 c c 3 þ P0 ð0þjð0þh ð4þ ð0þ 4P00 ð0þ P 0 c 6 2 e 8 n ð0þ þ Oðe9 n Þ; this completing the proof. Now we can choose a þ br þ cr2 HðrÞ ¼ 1 þ dr þ gr ; 2 JðtÞ ¼ a þ bt 1 þ ct ; PðqÞ ¼ A þ Bq 1 þ Cq : h It is easy to see that B ¼ AðC 1=4Þ; a ¼ Aa 2 ; b ¼ 1 4g Aa ; b ¼ aðc 3=8Þ; c ¼ 1 2 H; J and P, we have HðrÞ ¼ 1 2 JðtÞ ¼ þð2 8gÞr þð8g 3Þr 2 ; Aað1 þð1 2gÞr þ gr 2 Þ að8 þð8c 3ÞtÞ ; 1 þ ct e 4 n ð19þ ð20þ 8g 3 Aa, and d ¼ 1 2g. Substituting these values in

5 90 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) PðqÞ ¼ 1 4 Að4 þð4c 1ÞqÞ : 1 þ Cq Therefore we have 5 parameters a; c; A; C; g. We can choose C ¼ 2=3 and g ¼ 41=36 by requiring that 3 4P 0 ð0þjð0þh ð4þ ð0þ 24P00 ð0þ P 0 ð0þ HðrÞ ¼ 2ð36 64r þ 55r2 Þ Aað36 46r þ 41r 2 Þ ; JðtÞ ¼ 1 8 PðqÞ ¼ 1 4 að8 þð8c 3ÞtÞ ; 1 þ ct Að11q 12Þ : 2q 3 ¼ 0 and 4 3P00 ð0þ ¼ 0, which eliminate the terms of c 6 P 0 ð0þ 2 and c2 2 c2 3 in the error Eq. (4). Thus we have This family of methods is denoted by OM. Since we only use the product HðrÞJðtÞPðqÞ, it is easy to see that Aa cancels out. HðrÞJðtÞPðqÞ ¼ þð2 8gÞr þð8g 3Þr 2 Aað1 þð1 2gÞr þ gr 2 Þ 1 að8 þð8c 3ÞtÞ þ ct 4 Að4 þð4c 1ÞqÞ : 1 þ Cq Thus we have the 3 free parameters g; c and C. This family is denoted OMN. Clearly OM is a special case since it has a specific choice of g and C. In the next section we give a comparative numerical study of several members of our families OM and OMN as well as other well known optimal eighth order methods. We will also discuss the basins of attraction of our families. 3. Numerical examples In this section we present some numerical experiments using our newly found methods and compare these results to other schemes. All computations were done using MAPLE using 128 digit floating point arithmetic (Digits :¼ 128). Given an initial guess x 0 we decide that the method converges if the sequence fx n g generated by the iterative method has a residual jf ðx n Þj less than a tolerance ¼ in a maximum of 100 iterations, otherwise we consider the method to be divergent. We used the following test functions and display the approximate zero x found up to the 28th decimal places. Index Test function Root 1 x 3 þ 4x sin 2 ðxþ x 2 þ ðx 1Þ x xe x2 sin 2 ðxþþ3cosðxþþ e x2 þ7x sinðxþ x x 5 þ x pffiffi 9 x 1 x e x þ x p 11 lnðxþþ ffiffi x x 2 e x 3x þ e x sinðxþþlnð1 þ x 2 Þ 0 14 e x2þxþ x 5 þ x 4 þ 4x Compared to our methods OM and OMN with various parameters we have taken the following methods Kung Traub eighth-order method (KT8) [3] defined by y n ¼ x n f ðx nþ f 0 ðx n Þ ; s n ¼ y n f ðx nþ f ðy n Þf ðx n Þ f 0 ðx n Þ ½f ðx n Þ fðy n ÞŠ ; 2 x nþ1 ¼ s n f ðx nþ f ðx n Þf ðy n Þf ðs n Þ f f 0 ðx n Þ ½f ðx n Þ fðy n ÞŠ 2 ½ 2 ðx n Þþfðy n Þ½f ðy n Þ fðs n ÞŠ f ðx n Þ fðs n ÞŠ 2 f ðy n Þ fðs n Þ ½ Š : ð21þ

6 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) The method based on Kung Traub optimal fourth-order method and Hermite interpolating polynomial (HKT8) [2] where y n ¼ x n f ðx nþ f 0 ðx n Þ ; s n ¼ y n f ðy nþ 1 f 0 ðx n Þ ½1 f ðy n Þ=f ðx n ÞŠ ; 2 x nþ1 ¼ s n f ðs nþ H 0 3 ðs nþ ; ð22þ H 0 3 ðs nþ¼2ðf ½x n ; s n Š f ½x n ; y n ŠÞ þ f ½y n ; s n Šþ y n s n y n x n ðf ½x n ; y n Š f 0 ðx n ÞÞ: ð23þ The method based on Kung Traub optimal fourth-order method and Hermite interpolating polynomial replacing the function (HKN8) [5] y n ¼ x n f ðx nþ f 0 ðx n Þ ; s n ¼ y n f ðy nþ 1 f 0 ðx n Þ ½1 f ðy n Þ=f ðx n ÞŠ ; 2 x nþ1 ¼ s n H 3ðs n Þ f 0 ðs n Þ ; ð24þ Table 1 Comparison of eighth-order iterative schemes. f KT8 HKT8 HKN8 N8 WM8 OM1 OM2 f 1 IT x 0 ¼ 1:5 f ðx Þ 3.5e e e-72 1e e e e 83 f 2 IT x 0 ¼ 1:37 f ðx Þ 1.1e e e e e e e 97 f 3 IT x 0 ¼ 2:5 f ðx Þ 0 9.5e e e e 31 f 4 IT x 0 ¼ 4 f ðx Þ 1.1e e e f 5 IT x 0 ¼ 1:5 f ðx Þ 1.2e e e e e e e 28 f 6 IT x 0 ¼ 4 f ðx Þ 3.7e e e 33 2e e 42 f 7 IT x 0 ¼ 2 f ðx Þ 4,4e 71 3e e e e e e 86 f 8 IT x 0 ¼ 4 f ðx Þ 1e e e e17 0 7e 124 f 9 IT x 0 ¼ 1 f ðx Þ 1e e e e e 127 f 10 IT x 0 ¼ 2:1 f ðx Þ e e e 29 2e f 11 IT x 0 ¼ 1 f ðx Þ 1.8e e e e e 96 1e f 12 IT x 0 ¼ 0:5 f ðx Þ 1.2e e e e e 84 6e e 84 f 13 IT x 0 ¼ 1 f ðx Þ 9.4e e e e e e e 107 f 14 IT x 0 ¼ 0:85 f ðx Þ 2.3e e e e e e e 83 f 15 IT x 0 ¼ 1:2 f ðx Þ 1.4e e e e 38 5e e e 49

7 92 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Table 2 Comparison of eighth-order iterative schemes. f OMN1 OMN2 OMN3 OMN6 f 1 IT x 0 ¼ 1:5 f ðx Þ 1.0e e e f 2 IT x 0 ¼ 1:37 f ðx Þ 9.5e e e e 89 f 3 IT x 0 ¼ 2:5 f ðx Þ 8.9e e e e 127 f 4 IT x 0 ¼ 4 f ðx Þ 2.1e e e94 f 5 IT x 0 ¼ 1:5 f ðx Þ 5.5e e e e 75 f 6 IT x 0 ¼ 4 f ðx Þ 6.2e e e 38 0 f 7 IT x 0 ¼ 2 f ðx Þ 2.8e e e e 68 f 8 IT x 0 ¼ 4 f ðx Þ 0 4.0e e e20 f 9 IT x 0 ¼ 1 f ðx Þ 2.0e e e e 84 f 10 IT x 0 ¼ 2:1 f ðx Þ 1.9e e e 121 f 11 IT x 0 ¼ 1 f ðx Þ 6.2e e e 45 f 12 IT x 0 ¼ 0:5 f ðx Þ 2.4e e e e 78 f 13 IT x 0 ¼ 1 f ðx Þ 4.3e e e e 34 f 14 IT x 0 ¼ 0:85 f ðx Þ 2.5e e e e 73 f 15 IT x 0 ¼ 1:2 f ðx Þ 1.9e e e e 41 Table 3 The parameters for each member. The first 2 belong to OM and the last 6 are OMN members. Case g c C 1 41/36 2 2/3 2 41/36 1 2/ where ðs H 3 ðs n Þ¼fðx n Þþf 0 n y ðx n Þ n Þ 2 ðs n x n Þ ðy n x n Þðx n þ 2y n 3s n Þ þ f 0 ðs n Þ ðs n y n Þðx n s n Þ x n þ 2y n 3s n f ½x n ; y n Š ðs n x n Þ 3 ðy n x n Þðx n þ 2y n 3s n Þ : An eighth order (N8) optimal method proposed by Neta [6] and based on King s fourth order optimal method [7] with b ¼ 2 given by y n ¼ x n f ðx nþ f 0 ðx n Þ ; s n ¼ y n f ðy nþ f ðx n Þþbfðy n Þ f 0 ðx n Þ f ðx n Þþðb 2Þfðy n Þ ; x nþ1 ¼ x n f ðx nþ f 0 ðx n Þ þ cf 2 ðx n Þ qf 3 ðx n Þ; ð25þ ð26þ

8 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Table 4 Number of EFPs, minimum and maximum values of the absolute value of the real parts of EFPs. Case Number of EFPs Min. value Max. value e e e e e e e Fig. 1. Our method with g ¼ 41=36; c ¼ 2; C ¼ 2=3; a ¼ 1, and A ¼ 3 for the roots of the polynomial z 2 1. Fig. 2. Our method with g ¼ 41=36; c ¼ 1; C ¼ 2=3; a ¼ 2, and A ¼ 1 for the roots of the polynomial z 2 1.

9 94 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 3. Our method with g ¼ 4; c ¼ 0; C ¼ 4, and any a and A for the roots of the polynomial z 2 1. Fig. 4. Our method with g ¼ 4; c ¼ 0; C ¼ 0, and any a and A for the roots of the polynomial z 2 1. where q ¼ / y / t F y F t ; c ¼ / y qf y ; F y ¼ f ðy n Þ f ðx n Þ; F t ¼ f ðs n Þ f ðx n Þ; / y ¼ y n x n F 2 y 1 F y f 0 ðx n Þ ; / t ¼ s n x n F 2 t 1 F t f 0 ðx n Þ : ð27þ A weight function based eighth order (WM8) optimal method [2] (using the fourth order Maheshwari s method [8]) given by

10 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 5. Our method with g ¼ 0; c ¼ 4; C ¼ 0, and any a and A for the roots of the polynomial z 2 1. Fig. 6. Our method with g ¼ 2:9; c ¼ 4; C ¼ 4, and any a and A for the roots of the polynomial z 2 1. y n ¼ x n f ðx nþ f 0 ðx n Þ ; " s n ¼ x n f ðy 2 # nþ f ðx n Þ f ðx n Þ f ðx n Þ f ðy n Þ fðx n Þ f 0 ðx n Þ ; x nþ1 ¼ s n / f ðy nþ f ðs n Þ þ f ðx n Þ f ðy n Þ af ðs n Þ þ 4f ðs nþ f ðsn Þ f ðx n Þ f 0 ðx n Þ ; ð28þ where / is an arbitrary real function satisfying the conditions /ð0þ ¼1; / 0 ð0þ ¼2; / 00 ð0þ ¼4; / 000 ð0þ ¼ 6: We have taken a ¼ 1 and /ðtþ ¼1 þ 2t þ 2t 2 t 3.

11 96 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Table 5 Average number of iterations per point. Case Ex1 Ex2 EX3 Ex4 Ex5 Total Fig. 7. Our method with g ¼ 41=36; c ¼ 2; C ¼ 2=3; a ¼ 1, and A ¼ 3 for the roots of the polynomial ðz 1Þ 3 1. Fig. 8. Our method with g ¼ 41=36; c ¼ 1; C ¼ 2=3; a ¼ 2, and A ¼ 1 for the roots of the polynomial ðz 1Þ 3 1.

12 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 9. Our method with g ¼ 4; c ¼ 0; C ¼ 4, and any a and A for the roots of the polynomial ðz 1Þ 3 1. Fig. 10. Our method with g ¼ 4; c ¼ 0; C ¼ 0, and any a and A for the roots of the polynomial ðz 1Þ 3 1. In Table 1 we presented the results for KT8 (21), HKT8 (22), HKN8 (24),N8(26) with b ¼ 2, WM8 (28) and our new methods OM (OM1 is with a ¼ 1; A ¼ 3; c ¼ 2 and OM2 is with a ¼ 2; A ¼ 1; c ¼ 1). In Table 2 we also presented the results for our methods OMN1 (case 3, see Table 3), OMN2 (case 4), OMN3 (case 5) and OMN6 (case 8). The number of iterations IT required to converge is given along with the value of the function at the last iteration f ðx Þ. It can be observed that for most of the considered test functions our methods show as good performance as the other methods in their convergence speed and also have reasonable smallness of the residuals. In fact, in one case (f 8 ) our methods converged even though WM8 diverged. We also found that in that example OMN6 diverged even though the other methods converged. We will show later that OMN6 is not a good choice. Therefore we can conclude that the new methods (OM1, OM2, OMN1 OMN3) are competitive with other eighth-order schemes being considered for solving nonlinear equations.

13 98 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 11. Our method with g ¼ 0; c ¼ 4; C ¼ 0, and any a and A for the roots of the polynomial ðz 1Þ 3 1. Fig. 12. Our method with g ¼ 2:9; c ¼ 4; C ¼ 4, and any a and A for the roots of the polynomial ðz 1Þ 3 1. Fig. 13. Our method with g ¼ 41=36; c ¼ 2; C ¼ 2=3; a ¼ 1, and A ¼ 3 for the roots of the polynomial z 4 1.

14 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 14. Our method with g ¼ 41=36; c ¼ 1; C ¼ 2=3; a ¼ 2, and A ¼ 1 for the roots of the polynomial z 4 1. Fig. 15. Our method with g ¼ 4; c ¼ 0; C ¼ 4, and any a and A for the roots of the polynomial z 4 1. In the next section, we analyze the basin of attraction of our eighth order family of methods to find out what is the best choice for the parameters. The idea of using basins of attraction was initiated by Stewart [9] and followed by the works of Amat et al. [10 13], Scott et al. [14], Chun et al. [15], Chicharro et al. [16], Cordero et al. [17], Neta et al. [18] and Chun et al. [19]. The only papers comparing basins of attraction for methods to obtain multiple roots is due to Neta et al. [20] and Neta and Chun [21 23]. 4. Basins of attraction In this section we give the basins of attraction of various members of the families OM and OMN. The 8 members are listed with their parameters in Table 3. The first 2 cases are of OM type and the last 6 cases are of OMN type.

15 100 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 16. Our method with g ¼ 4; c ¼ 0; C ¼ 0, and any a and A for the roots of the polynomial z 4 1. Fig. 17. Our method with g ¼ 0; c ¼ 4; C ¼ 0, and any a and A for the roots of the polynomial z 4 1. The first 2 cases were chosen so that we annihilate two terms in the error constant. The first of those was arbitrarily picked. In order to understand the choice of the parameters in the other cases we discuss the extraneous fixed points. In solving a nonlinear equation iteratively we are looking for fixed points which are zeros of the given nonlinear function. Many multipoint iterative methods have fixed points that are not zeros of the function of interest. Thus, it is imperative to investigate the number of extraneous fixed points, their location and their properties. In the family of methods described in this paper, the parameters g; c, and C can be chosen to position the extraneous fixed points on or close to the imaginary axis. This idea is due to Neta et al. [18] where they have shown an improvement in King s method by choosing the parameter that will position the extraneous fixed points on the imaginary axis. The second case was chosen so that we also have the extraneous fixed points as close as possible to the imaginary axis. Similarly, in the next 5 cases we have chosen the extraneous fixed

16 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 18. Our method with g ¼ 2:9; c ¼ 4; C ¼ 4, and any a and A for the roots of the polynomial z 4 1. Fig. 19. Our method with g ¼ 41=36; c ¼ 2; C ¼ 2=3; a ¼ 1, and A ¼ 3 for the roots of the polynomial zðz 2 þ 1Þðz 2 þ 4Þ. points to be close to the imaginary axis. The last case violates that condition and one can see later that this violation increases the average number of iterations. The eighth order family of methods discussed here can be written as where x nþ1 ¼ x n f ðx nþ f 0 ðx n Þ H f ðx n Þ; ð29þ r n H f ðx n Þ¼1þ ð1 r n Þ þ 2 ð1 Hðr n ÞJðt n ÞPðq n ÞÞ : 2 t n ð30þ

17 102 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 20. Our method with g ¼ 41=36; c ¼ 1; C ¼ 2=3; a ¼ 2, and A ¼ 1 for the roots of the polynomial zðz 2 þ 1Þðz 2 þ 4Þ. Fig. 21. Our method with g ¼ 4; c ¼ 0; C ¼ 4, and any a and A for the roots of the polynomial zðz 2 þ 1Þðz 2 þ 4Þ. Clearly the root n of f ðxþ is a fixed point of the method. The points a n at which H f ðaþ ¼0 are also fixed points of the family. We have searched the parameter space and found that there are no point on the imaginary axis, but there are 5 cases with the smallest real part (these are denoted cases 3 7). To convince that this is a good choice we have taken a case where the real part of the extraneous roots is the largest (case 8). In Table 4 we have listed the number of extraneous fixed points (EFPs) and range of absolute value of the real parts of the EFPs for each case. Example 1. In our first example we have used the polynomial z 2 1. The basins of attraction are given in Figs All the results are good. The best one is case 5 (Fig. 5) and the worst is case 3 (Fig. 3). The difference is barely noticeable. One can see it only when computing the average number of iterations per point (see Table 5.) All cases require between 3.1 and 3.3 iterations per point. Notice that cases 6 and 7 were not shown since we found that they yield identical results to cases 3 and 4, respectively.

18 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 22. Our method with g ¼ 4; c ¼ 0; C ¼ 0, and any a and A for the roots of the polynomial zðz 2 þ 1Þðz 2 þ 4Þ. Fig. 23. Our method with g ¼ 0; c ¼ 4; C ¼ 0, and any a and A for the roots of the polynomial zðz 2 þ 1Þðz 2 þ 4Þ. Example 2. In our next example we used the polynomial p 2 ðzþ ¼ðz 1Þ 3 1: The basins are given in Figs In this case the best performer is case 4 (Fig. 10) and the worst is case 8 (Fig. 12). Example 3. In our third example we have taken the polynomial p 3 ðzþ ¼z 4 1: The basins are given in Figs Again the best performer is case 4 (Fig. 16) and the worst is case 8 (Fig. 18).

19 104 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 24. Our method with g ¼ 2:9; c ¼ 4; C ¼ 4, and any a and A for the roots of the polynomial zðz 2 þ 1Þðz 2 þ 4Þ. Fig. 25. Our method with g ¼ 41=36; c ¼ 2; C ¼ 2=3; a ¼ 1, and A ¼ 3 for the roots of the polynomial ðz 2 1Þðz 2 þ 1Þðz 2 þ 2iÞ. Example 4. In our fourth example we have taken the polynomial p 4 ðzþ ¼zðz 2 þ 1Þðz 2 þ 4Þ; whose roots are 0; i; 2i. The basins are given in Figs In this example the best is case 5 (Fig. 23) and the worst is case 1 (Fig. 19). Example 5. In our last example we have taken the polynomial p 5 ðzþ ¼ðz 2 1Þðz 2 þ 1Þðz 2 þ 2iÞ

20 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 26. Our method with g ¼ 41=36; c ¼ 1; C ¼ 2=3; a ¼ 2, and A ¼ 1 for the roots of the polynomial ðz 2 1Þðz 2 þ 1Þðz 2 þ 2iÞ. Fig. 27. Our method with g ¼ 4; c ¼ 0; C ¼ 4, and any a and A for the roots of the polynomial ðz 2 1Þðz 2 þ 1Þðz 2 þ 2iÞ. whose roots are 1; i; 1 þ i, and 1 i. The basins are given in Figs The best is case 3 (Fig. 28) and the worst is case 8(Fig. 30). Based on Table 5, the best case overall is case 4 for which g ¼ 4 and C ¼ c ¼ 0 and the worst is case 8 for which g ¼ 2:9 and C ¼ c ¼ 4. In general cases 3 5 (OMN) are better than cases 1 2 (OM). Case 8 is also OMN but there we picked parameters that lead to largest (in absolute value) real part of the extraneous fixed points. On the other hand, cases 3 5 have the smallest real part. Remarks. (1) Note that cases 3 and 6 yield identical results. Similarly cases 4 and 7 are identical. For this reason, we have not shown the results for cases 6 and 7.

21 106 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 28. Our method with g ¼ 4; c ¼ 0; C ¼ 0, and any a and A for the roots of the polynomial ðz 2 1Þðz 2 þ 1Þðz 2 þ 2iÞ. Fig. 29. Our method with g ¼ 0; c ¼ 4; C ¼ 0, and any a and A for the roots of the polynomial ðz 2 1Þðz 2 þ 1Þðz 2 þ 2iÞ. (2) Out of the first 2 cases, were we annihilated two terms in the error constant, the best is case 2 were we also chose the extraneous fixed points to be closest to the imaginary axis. 5. Conclusions In this paper we have developed a new family of optimal eighth order iterative method. The scheme is optimal in the sense that it satisfies the Kung Traub conjecture. We have compared several members of our family to existing optimal eighth order schemes and found that some members of our family are competitive. We have shown how to choose the parameters of the family to find the best members by evaluating all the extraneous fixed points. We have shown that the

22 C. Chun, B. Neta / Applied Mathematics and Computation 245 (2014) Fig. 30. Our method with g ¼ 2:9; c ¼ 4; C ¼ 4, and any a and A for the roots of the polynomial ðz 2 1Þðz 2 þ 1Þðz 2 þ 2iÞ. best members have extraneous fixed point close to the imaginary axis. One member (case 8) for which this is not true was the worst perform. Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A ). References [1] J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, [2] M.S. Petković, B. Neta, L.D. Petković, J. Dz unić, Multipoint Methods for Solving Nonlinear Equations, Elsevier, [3] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iterations, J. Assoc. Comput. Mach. 21 (1974) [4] C. Chun, B. Neta, A new sixth-order scheme for nonlinear equations, Appl. Math. Lett. 25 (2012) [5] B. Neta, C. Chun, M. Scott, Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations, Appl. Math. Comput. 227 (2014) [6] B. Neta, On a family of multipoint methods for nonlinear equations, Int. J. Comput. Math. 9 (1981) [7] R.F. King, A family of fourth-order methods for nonlinear equations, SIAM Numer. Anal. 10 (1973) [8] A.K. Maheshwari, A fourth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 211 (2009) [9] B.D. Stewart, Attractor basins of various root-finding methods (M.S. thesis), Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA, June [10] S. Amat, S. Busquier, S. Plaza, Iterative root-finding methods, unpublished report, [11] S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Scientia 10 (2004) [12] S. Amat, S. Busquier, S. Plaza, Dynamics of a family of third-order iterative methods that do not require using second derivatives, Appl. Math. Comput. 154 (2004) [13] S. Amat, S. Busquier, S. Plaza, Dynamics of the King and Jarratt iterations, Aeq. Math. 69 (2005) [14] M. Scott, B. Neta, C. Chun, Basin attractors for various methods, Appl. Math. Comput. 218 (2011) [15] C. Chun, M.Y. Lee, B. Neta, J. Dz unić, On optimal fourth-order iterative methods free from second derivative and their dynamics, Appl. Math. Comput. 218 (2012) [16] F. Chircharro, A. Cordero, J.M. Gutiérrez, J.R. Torregrosa, Complex dynamics of derivative-free methods for nonlinear equations, Appl. Math. Comput. 219 (2013) [17] A. Cordero, J. García-Maimó, J.R. Torregrosa, M.P. Vassileva, P. Vindel, Chaos in King s iterative family, Appl. Math. Lett. 26 (2013) [18] B. Neta, M. Scott, C. Chun, Basin of attractions for several methods to find simple roots of nonlinear equations, Appl. Math. Comput. 218 (2012) [19] C. Chun, B. Neta, Sujin Kim, On Jarratt s family of optimal fourth-order iterative methods and their dynamics, in press, S X [20] B. Neta, M. Scott, C. Chun, Basin attractors for various methods for multiple roots, Appl. Math. Comput. 218 (2012) [21] B. Neta, C. Chun, On a family of Laguerre methods to find multiple roots of nonlinear equations, Appl. Math. Comput. 219 (2013) [22] B. Neta, C. Chun, Basins of attraction for several optimal fourth order methods for multiple roots, Math. Comput. Simul. 103 (2014) [23] B. Neta, C. Chun, Basins of attraction for Zhou Chen Song fourth order family of methods for multiple roots, Math. Comput. Simul., submitted for publication.