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2 Applied Matematics and Computation 218 (2012) Contents lists available at SciVerse ScienceDirect Applied Matematics and Computation journal omepage: On optimal fourt-order iterative metods free from second derivative and teir dynamics Cangbum Cun a, Mi Young Lee a, Beny Neta b,, Jovana Džunić c a Department of Matematics, Sungkyunkwan University, Suwon , Republic of Korea b Naval Postgraduate Scool, Department of Applied Matematics, Monterey, CA 93943, United States c Faculty of Electronic Engineering, Department of Matematics, University of Nis, Nis, Serbia article info abstract Keywords: Iterative metods Order of convergence Rational maps Basin of attraction Julia sets Conjugacy classes In tis paper new fourt order optimal root-finding metods for solving nonlinear equations are proposed. Te classical Jarratt s family of fourt-order metods are obtained as special cases. We ten present results wic describe te conjugacy classes and dynamics of te presented optimal metod for complex polynomials of degree two and tree. Te basins of attraction of existing optimal metods and our metod are presented and compared to illustrate teir performance. Publised by Elsevier Inc. 1. Introduction In tis paper, we consider iterative metods and teir dynamics to find a simple root q, i.e., f(q) = 0 and f 0 (q) 0, of a nonlinear equation f(x) = 0. Newton s metod [1] is te best known metod for finding a real or complex root q of te nonlinear equation f(x) = 0, wic is given by f 0 ðx n Þ : Tis metod converges quadratically in some neigborood of q. It is also well-known (see [2]) tat for any function H wit H(0) = 1, H 0 (0) = 1/2 and jh 00 (0)j < 1, te iterative metod f 0 ðx n Þ Hðtðx nþþ; ð1þ tðx n Þ¼ f ðx nþf 00 ðx n Þ ½f 0 ðx n ÞŠ 2 ð2þ is of order 3 [2]. Te Scemes (1) and (2) include many well-known metods as particular cases; for example, wen HðtÞ ¼ 1 1 t 1 p ; HðtÞ ¼2 1þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1, 1 2t 2 and H(t)=(1 t) 0.5 it reduces to Halley s metod [2], Euler s formula [2], and Ostrowski s square root iteration [3], respectively. Te metods (1) and (2) require f(x n ), f 0 (x n ) and f 00 (x n ) per step, but it is of tird order, so it is not an optimal metod. By an optimal metod we mean a multipoint one witout memory wic requires n + 1 functional evaluations per iteration, but acieves te order of convergence 2 n [4]. It sould be observed tat Corresponding autor. addresses: cbcun@skku.edu (C. Cun), sisley9678@naver.com (M.Y. Lee), bneta@nps.edu (B. Neta), jovana.dzunic@elfak.ni.ac.rs (J. Džunić) /$ - see front matter Publised by Elsevier Inc. doi: /j.amc

3 6428 C. Cun et al. / Applied Matematics and Computation 218 (2012) te metod (1) and (2) even involves te computation of te second derivative of f per step, wic restricts its practical use. Optimal root-finding metods wic overcome te lack of optimality and practical utility tat (1) and (2) as are tus preferred. A new approximation to te second derivative wit arbitrarily given second-order metod is devised and applied to (1) and (2). Te metods derived in tis manner will be of order 4 and require one function- and two first derivativeevaluations per step, so tey are optimal metods. Te metods are also free of second derivative and contain te classical Jarratt s fourt-order metods. Our metod developed ere also contains Kou et al. s fourt-order family of metods free from second derivative proposed in [5]. Tus our work can be viewed as an extension of te results of Kou et al. [5]. Sarma and Goyal [6] ave developed two fourt-order one-parameter family of metods requiring no evaluation of derivatives. Tere are various criteria involved in coosing an iterative metod to approximate te root of an equation [7]. Tese include te initial value problem (for wat initial values will te metod converge? Will it converge to a root, and if so, wic root?), te rate of convergence (ow fast te convergence occurs near a root?) and te complexity of te calculation (do first or iger derivatives ave to be calculated?). Some of tese problems were investigated in [7] by sowing ow complex dynamics can sed ligt on tem wen using Newton s metod for finding te real or complex roots of polynomial. In order to investigate tese dynamics wit some iger order metods we will improve te metod (1) and (2) in order to increase te rate of convergence and reduce te complexity of te calculation, and ten study teir complex dynamics. Te dynamics of te König iteration metods [8], te super-newton metod, Caucy s metod, and Halley s metods [9] and a number of root-finding metods including Jarratt s and King s metods [10] were previously studied in detail. Scott et al. compared te dynamics of several metods for simple roots [11] and Neta et al. as performed similar comparison of metods for multiple roots [12]. See also Amat et al. [13]. Motivated by tese works tis paper tus may be considered as an extension of tem in various aspects. A precise analysis of convergence is given for te presented optimal metods. We present results wic describe te conjugacy classes and dynamics of one of te new optimal fourt order metods for complex polynomials of degree two and tree. Te fact tat our metod is not generally convergent for polynomials is also investigated by constructing a specific polynomial suc tat te rational map arising from our metod applied to te polynomial as an attracting periodic orbit of period 2. Te basins of attraction of some existing fourt order optimal metods and our metod are considered and presented. To tis end, we sall recall some preliminaries, see for example Milnor [14] and Plaza [10]. Let R : b C! b C be a rational map on te Riemann spere. Definition 1. For z 2 C b we define its orbit as te set n o orbðzþ ¼ z; RðzÞ; R 2 ðzþ;...; R n ðzþ;... : Definition 2. A point z 0 is a fixed point of R if R(z 0 )=z 0. Definition 3. A periodic point z 0 of period m is suc tat R m (z 0 )=z 0 m is te smallest suc integer. Te set of te m distinct points {z,r(z),r 2 (z),...,r m 1 (z)} is called a periodic cycle. Remark 1.1. If z 0 is periodic of period m ten it is a fixed point for R m. Definition 4. If z 0 is a periodic point of period m, ten te derivative (R m ) 0 (z 0 ) is called te eigenvalue of te periodic point z 0. Remark 1.2. By te cain rule, if z 0 is a periodic point of period m, ten its eigenvalue is te product of te derivatives of R at eac point on te orbit of z 0, and we ave ðr m Þ 0 ðz 0 Þ¼ðR m Þ 0 ðz 1 Þ¼¼ðR m Þ 0 ðz n 1 Þ; tat is, all te points of a cycle ave te same eigenvalue. We classify te fixed points of a map based on te magnitude of te derivative. Definition 5. A point z 0 is called attracting if jr 0 (z 0 )j < 1, repelling if jr 0 (z 0 )j > 1, and neutral if jr 0 (z 0 )j = 1. If te derivative is zero ten te point is called super-attracting. Definition 6. Te Julia set of a nonlinear map R(z), denoted J(R), is te closure of te set of its repelling periodic points. Te complement of J(R) is te Fatou set FðRÞ. By its definition, J(R) is a closed subset of b C. A point z 0 belongs to te Julia set if and only if dynamics in a neigborood of z 0 displays sensitive dependence on te initial conditions, so tat nearby initial conditions lead to wildly different beavior after a number of iterations. As a simple example, consider te map R(z)=z 2 on b C. Te entire open disk is contained in FðRÞ,

4 C. Cun et al. / Applied Matematics and Computation 218 (2012) since successive iterates on any compact subset converge uniformly to zero. Similarly te exterior is contained in FðRÞ. On te oter and if z 0 is on te unit circle ten in any neigborood of z 0 any limit of te iterates would necessarily ave a jump discontinuity as we cross te unit circle. Terefore J(R) is te unit circle. Suc smoot Julia sets are exceptional. Lemma 1.1 (Invariance Lemma Milnor [14]). Te Julia set J(R) of a olomorpic map R : b C! b C is fully invariant under R. Tat is, z belongs to J if and only if R(z) belongs to J. Lemma 1.2. Iteration LemmaFor any k > 0, te Julia set J(R k ) of te k-fold iterate coincides wit J(R). Definition 7. If O is an attracting periodic orbit of period m, we define te basin of attraction to be te open set A 2 b C consisting of all points z 2 b C for wic te successive iterates R m (z), R 2m (z),... converge towards some point of O. Te basin of attraction of a periodic orbit may ave infinitely many components. Definition 8. Te immediate basin of attraction of a periodic orbit is te connected component containing te periodic orbit. Lemma 1.3. Every attracting periodic orbit is contained in te Fatou set of R. In fact te entire basin of attraction A for an attracting periodic orbit is contained in te Fatou set. However, every repelling periodic orbit is contained in te Julia set. 2. New iterative metods Trougout tis work let / be an iteration function of order at least two. We let y n = x n [x n /(x n )] = (1 )x n + /(x n ), is a nonzero real parameter. Let us consider te approximation: f 00 ðx n Þ f 0 ðy n Þ f 0 ðx n Þ y n x n ¼ f 0 ðx n Þ f 0 ðy n Þ ½x n /ðx n ÞŠ from wic (2) can be approximated tðx n Þ¼ f ðx nþf 00 ðx n Þ ½f 0 ðx n ÞŠ 2 f ðx nþ½f 0 ðx n Þ f 0 ðy n ÞŠ ½x n /ðx n ÞŠ½f 0 ðx n ÞŠ 2 : Tis gives rise to a new iterative sceme f 0 ðx n Þ Hð ~tðx n ÞÞ; ~tðx n Þ¼ f ðx nþ½f 0 ðx n Þ f 0 ðy n ÞŠ ½x n /ðx n ÞŠ½f 0 ðx n ÞŠ 2 : ð3þ ð4þ We will sow tat in spite of not using as many function evaluations, we ave increased te order of convergence to 4. Te following teorem will prove tat te metod defined by (3) and (4) is of order 4 under additional conditions on H and on. Teorem 2.1. Let q 2 I be a simple zero of a sufficiently differentiable function f : I? R in an open interval I. Let H be any function wit H(0) = 1, H 0 (0) = 1/2 and jh 00 (0)j < 1, a nonzero real number and / any iteration function of order at least two. Let y n =x n [x n /(x n )]. Ten te metod defined by (3) and (4) as tird-order convergence, and its error equation is given as e nþ1 ¼ 2ð1 H 00 ð0þþc 2 2 þ c 3 e 3 n þ 14H 00 ð0þ 4 3 H000 ð0þ 9 c 3 2 þð6h00 ð0þ 12H 00 ð0þ 6þ12Þc 2 c / 2c 3 ð2 2 6 þ 3Þc 4 e n =x n q, e 4 n þ Oðe5 nþ; ð5þ c k ¼ð1=k!Þf ðkþ ðqþ=f 0 ðqþ; k ¼ 1; 2;... ð6þ c 0 =f(q) = 0, and /ðx n Þ¼qþ / 2 e 2 n þ Oðe3 n Þ. Furtermore, if we ave H00 (0) = 1 and ¼ 2, ten te order of te metod defined by 3 (3) and (4) is at least four. Proof. Let e n = x n q and d n = y n q, y n = x n w(x n ). Using Taylor expansion and taking into account f(q) = 0, we ave

5 6430 C. Cun et al. / Applied Matematics and Computation 218 (2012) and f ðx n Þ¼f 0 ðqþ e n þ c 2 e 2 n þ c 3e 3 n þ c 4e 4 n þ Oðe5 n Þ ; ð7þ f 0 ðx n Þ¼f 0 ðqþ 1 þ 2c 2 e n þ 3c 3 e 2 n þ 4c 4e 3 n þ Oðe4 n Þ ð8þ ½f 0 ðx n Þ Š 2 ¼ ½f 0 ðqþš 2 1 þ 4c 2 e n þð4c 2 2 þ 6c 3Þe 2 n þ 12c 2c 3 e 3 n þ Oðe4 n Þ ; ð9þ c k is given by (6). Dividing (7) by (8) gives f ðx n Þ f 0 ðx n Þ ¼ e n c 2 e 2 n þ 2ðc2 2 c 3Þe 3 n þð7c 2c 3 3c 4 4c 3 2 Þe4 n þ Oðe5 n Þ: ð10þ Since / is an iteration function of order at least two, it follows tat /ðx n Þ¼q þ / 2 e 2 n þ / 3e 3 n þ / 4e 4 n þ Oðe5 n Þ; / k ¼ 1 k! /ðkþ ðqþ; k ¼ 2; 3; 4 so tat x n /ðx n Þ¼e n / 2 e 2 n / 3e 3 n / 4e 4 n þ Oðe5 n Þ ð11þ and ence, we ave d n ¼ e n ½x n /ðx n ÞŠ ¼ ð1 Þe n þ / 2 e 2 n þ / 3e 3 n þ / 4e 4 n þ Oðe5 n Þ: ð12þ Expanding f 0 (y n ) about q, we ave i f 0 ðy n Þ¼f 0 ðqþ 1 þ 2c 2 d n þ 3c 3 d 2 n þ 4c 4d 3 n þ 5c 5d 4 n þ Oðd5 n Þ and ten from (12), we obtain i i f 0 ðy n Þ¼f 0 ðqþ 1 þ 2ð1 Þc 2 e n þ 2/ 2 c 2 þ 3ð1 Þ 2 c 3 e 2 n þ 2/ 3c 2 þ 6ð1 Þ/ 2 c 3 þ 4ð1 Þ 3 c 4 e 3 n i þ 2/ 4 c 2 þ 3ð/ 2 2 þ 2ð1 Þ/ 3Þc 3 þ 12ð1 Þ 2 / 2 c 4 þ 5ð1 Þ 4 c 5 ie 4n þ Oðe5n Þ : It is ten clear tat f 0 ðx n Þ f 0 ðy n Þ¼f 0 ðqþ 2c 2 e n þ½3c 3 2/ 2 c 2 3ð1 Þ 2 c 3 Še 2 n þ½4c 4 2/ 3 c 2 6ð1 Þ/ 2 c 3 4ð1 Þ 3 c 4 Še 3 n ½2/ 4 c 2 þ 3ð/ 2 2 þ 2ð1 Þ/ 3Þc 3 þ 12ð1 Þ 2 / 2 c 4 þ 5ð1 Þ 4 c 5 5c 5 Še 4 n þ Oðe5 n Þ i: ð13þ By a simple calculation, we ave from (7), (9), (11) and (13) tat ~tðx n Þ¼ f ðx nþ½f 0 ðx n Þ f 0 ðy n ÞŠ ½x n /ðx n ÞŠ½f 0 ðx n ÞŠ 2 ¼ 2c 2 e n þ 3ð2 Þc 3 6c 2 2 e 2 n þ 16c3 2 þð9 28Þc 2c 3 þ 3/ 2 c 3 þ 4ð 2 3 þ 3Þc 4 e 3 n þ Oe4 n and so, ~t 2 ðx n Þ¼4c 2 2 e2 n þ 4c 2 6c 3 3c 3 6c 2 2 e 3 n þ Oe4 n : ð15þ From (14) and (15), we ave upon using te values of H(0) and H 0 (0) Hð~tðx n ÞÞ ¼ 1 þ 1 2 ~ tðx n Þþ 1 2 H00 ð0þ~t 2 ðx n Þþ 1 6 H000 ð0þ~t 3 ðx n ÞþOð~t 4 ðx n ÞÞ ¼ 1 þ c 2 e n þ ð2h 00 ð0þ 3Þc 2 2 þ 3 2 ð2 Þc 3 e 2 n þ 8 12H00 ð0þþ 4 3 H000 ð0þ c 3 2 þ 12H 00 ð0þ 6H 00 ð0þþ 9 14 c 2 c 3 þ 32 2 / 2c 3 þ 2ð 2 3 þ 3Þc 4 e 3 n þ Oe4 n : ð16þ Hence, from (10) and (16), we obtain f 0 ðx n Þ Hð ~tðx n ÞÞ ¼ q þ 2ð1 H 00 ð0þþc 2 2 þ c 3 e 3 n þ 14H 00 ð0þ 4 3 H000 ð0þ 9 c 3 2 þð6h00 ð0þ 12H 00 ð0þ 6þ12Þc 2 c / 2c 3 ð2 2 6 þ 3Þc 4 e 4 n þ Oe5 n ; ð14þ

6 C. Cun et al. / Applied Matematics and Computation 218 (2012) terefore, e nþ1 ¼ 2ð1 H 00 ð0þþc 2 2 þ c 3 e 3 n þ 14H 00 ð0þ 4 3 H000 ð0þ 9 c 3 2 þð6h00 ð0þ 12H 00 ð0þ 6þ12Þc 2 c / 2c 3 ð2 2 6 þ 3Þc 4 e 4 n þ Oe5 n ; wic is te same one tat appears in (5). Now if we coose ¼ 2 3 and H00 (0) = 1, ten (2) becomes e nþ1 ¼ 14H 00 ð0þ 4 3 H000 ð0þ 9 c 3 2 þð6h00 ð0þ 12H 00 ð0þ 6þ12Þc 2 c / 2c 3 ð2 2 6 þ 3Þc 4 e 4 n þ Oe5 n ð17þ and we obtain te fourt-order class of metods y n ¼ x n 2 3 ½x n /ðx n ÞŠ; f 0 ðx n Þ Hð ~tðx n ÞÞ; ~tðx n Þ¼ 3 f ðx n Þ½f 0 ðx n Þ f 0 ðy n ÞŠ 2 ½x n /ðx n ÞŠ½f 0 ðx n ÞŠ 2 ð18þ ð19þ ð20þ and / is any iteration function of order at least two. Tis completes te proof. If we consider an iterative function / requiring f(x n ) and f 0 (x n ), ten our family of metods as an optimal order since it requires f(x n ), f 0 (x n ) and f 0 (y n ) per step. 3. New fourt order optimal metods For te sake of simplicity, we consider only Newton s iteration function /ðxþ ¼x f ðxþ, even toug oter coices for / f 0 ðxþ may provide us wit many oter optimal fourt-order metods. For te Newton iteration function, (18) (20) simplifies to y n ¼ x n 2 f ðx n Þ 3 f 0 ðx n Þ ; f 0 ðx n Þ Hð ~tðx n ÞÞ; ~tðx n Þ¼ 3 2 f 0 ðx n Þ f 0 ðy n Þ : ð23þ f 0 ðx n Þ If we take HðtÞ ¼1 þ 1 t, ten (21) (23) leads to te well-known Jarratt s fourt-order metod [15] 2 1 t x nþ1 ¼ x n 1 3 f 0 ðy n Þ f 0 ðx n Þ f ðxn Þ 2 3f 0 ðy n Þ f 0 ðx n Þ f 0 ðx n Þ ; f ðx nþ f 0 ðx nþ. y n ¼ x n 2 3 If we take anoter HðtÞ ¼1 þ t 6 2t x nþ1 ¼ x n w 1 ðx n Þ 3 2 w 3f ðx n Þ 2ðx n Þþ f 0 ðx n Þþf 0 ðz n Þ ;, ten (21) (23) leads to anoter optimal fourt-order Jarratt s metod [15] w 1 ðx n Þ¼ f ðxnþ ; w f 0 ðx nþ 2ðx n Þ¼ f ðxnþ and z f 0 ðz nþ n ¼ x n 2 w 3 1ðx n Þ. Tis metod is suggested by Jarratt in order to reduce te possibility of cancelation in te denominator. If we take HðtÞ ¼ 3 tðctþ 3 2Þ 4 ðatþ 3 2Þðbtþ 3 2Þ, c ¼ a þ b 3 ; a; b 2 R, ten (21) (23) leads to te optimal Kou et al. s fourt-order 2 family of metods [5] x nþ1 ¼ x n 1 3 ðf 0 ðy n Þ f 0 ðx n ÞÞðcf 0 ðy n Þþð1 cþf 0 ðx n ÞÞ f ðxn Þ 4 ðaf 0 ðy n Þþð1 aþf 0 ðx n ÞÞðbf 0 ðy n Þþð1 bþf 0 ðx n ÞÞ f 0 ðx n Þ : ð21þ ð22þ

7 6432 C. Cun et al. / Applied Matematics and Computation 218 (2012) In te case tat HðtÞ ¼1 þ t þ t2, (21) (23) gives a new fourt-order optimal metod 2 2 " f 0 ðx n Þ 1 þ 3 f 0 ðx n Þ f 0 ðy n Þ þ 9 f 0 ðx n Þ f 0 2 # ðy n Þ ; 4 f 0 ðx n Þ 8 f 0 ðx n Þ f ðx nþ f 0 ðx nþ. y n ¼ x n 2 3 In te case tat HðtÞ ¼1 þ 2 þ 4, we obtain from (21) (23) anoter new optimal fourt-order metod t 2 ðt 2Þ " 2 x nþ1 ¼ x n f ðx nþ f 0 ðx n Þ 1 4f 0 ðx n Þ 3f 0 ðy n Þþf 0 ðx n Þ þ 4f 0 2 # ðx n Þ ; 3f 0 ðy n Þþf 0 ðx n Þ f ðx nþ f 0 ðx nþ. y n ¼ x n 2 3 In te case tat HðtÞ ¼ t 4 1, (21) (23) reduces to te metod 2 t 2 x nþ1 ¼ x n f ðx nþ 3 f 0 ðy n Þ f 0 ðx n Þ 8f 0 ðx n Þ þ f 0 ðx n Þ 4 f 0 ðx n Þ 3f 0 ðy n Þþf 0 ðx n Þ 1 ; f ðx nþ f 0 ðx nþ. y n ¼ x n 2 3 In te case tat HðtÞ ¼ 4, we obtain from (21) (23) anoter new optimal fourt-order metod 4 2t t 2 16f ðx n Þf 0 ðx n Þ x nþ1 ¼ x n 5½f 0 ðx n ÞŠ 2 þ 30f 0 ðx n Þf 0 ðy n Þ 9½f 0 ðy n ÞŠ ; 2 ð24þ y n ¼ x n 2 3 f ðx nþ f 0 ðx nþ. 4. Conjugacy classes Trougout te remainder of tis paper we study te dynamics of te rational map R f arising from te metod (24) 16f ðzþf 0 ðzþ R f ðzþ ¼z þ 5½f 0 ðzþš 2 30f 0 ðzþf 0 ðyþþ9½f 0 ðyþš ; 2 y ¼ z 2 3 f ðzþ f 0 ðzþ applied to a generic polynomial wit simple roots. We tried oter possibilities and tey are not competitive. Let us first recall te definition of analytic conjugacy classes. Definition 9 [16]. Let f and g be two maps from te Riemann spere into itself. An analytic conjugacy between f and g is an analytic diffeomorpism from te Riemann spere onto itself suc tat f = g. R f as te following useful property for an analytic function f. Teorem 4.1 (Te Scaling Teorem). Let f(z) be an analytic function on te Riemann spere, and let T(z) = az+b,a 0, be an affine map. If g(z) = f T(z), ten T R g T 1 (z) = R f (z). Tat is, R f is analytically conjugate to R g by T. ð25þ Proof. Wit te iteration function R(z), we ave "! R g ðt 1 ðzþþ ¼ T 1 ðzþþ16g 0 ðt 1 ðzþþgðt 1 ðzþþ 5g 02 ðt 1 ðzþþ 30g 0 ðt 1 ðzþþg 0 T 1 ðzþ 2 gðt 1 ðzþþ!# þ9g 02 T 1 ðzþ 2 1 gðt 1 ðzþþ : Since g T 1 ðzþ ¼f ðzþ; ðg T 1 Þ 0 ðzþ ¼a 1 g0 ðt 1 ðzþþ, we get g 0 (T 1 (z)) = a (gt 1 ) 0 (z)=af 0 (z), ave g 00 (T 1 (z)) = a 2 f 00 (z). We terefore T R g T 1 ðzþ¼tðr g ðt 1 ðzþþþ ¼ ar g ðt 1 ðzþþ þ b ¼ at 1 ðzþþa16g 0 ðt 1 ðzþþgðt 1 ðzþþ "!!# 5g 02 ðt 1 ðzþþ 30g 0 ðt 1 ðzþþg 0 T 1 ðzþ 2 gðt 1 ðzþþ þ 9g 02 T 1 ðzþ 2 1 gðt 1 ðzþþ þ b "!!# ¼ z þ 16a 2 f 0 ðzþf ðzþ 5a 2 ½f 0 ðzþš 2 30af 0 ðzþg 0 T 1 ðzþ 2 gðt 1 ðzþþ þ 9g 02 T 1 ðzþ 2 1 gðt 1 ðzþþ : ð26þ

8 C. Cun et al. / Applied Matematics and Computation 218 (2012) On te oter and, we ave! g 0 T 1 ðzþ 2 gðt 1 ðzþþ ¼ g 0 T 1 ðzþ 2 f ðzþ 3 af 0 ðzþ ¼ af 0 ðzþ a 2 f 00 ðzþ 2 3 ¼ g 0 ðt 1 ðzþþ g 00 ðt 1 ðzþþ 2 3 f ðzþ af 0 ðzþ þ f ðzþ af 0 ðzþ þ¼a f 0 ðzþ f 00 ðzþ 2 f ðzþ 3 f 0 ðzþ þ ¼ af 0 z 2 3 f ðzþ f 0 ðzþ ¼ af 0 ðyþ: We tus obtain from (26) TR g T 1 (z)=r f (z), tis completing te proof. Te scaling teorem establised above indicates tat up to a suitable cange of coordinates te study of te dynamics of te iteration function (25) for polynomials can be reduced to te study of te dynamics of te same iteration function for simpler polynomials. For example, for any quadratic and any cubic polynomials, we can easily prove te following results by an affine cange of variable and multiplication by a constant tat. Teorem 4.2. Let p(z) = az 2 + bz + c, wit a 0 and qðzþ ¼z 2 l; ð27þ l ¼ b2 4ac 4a. Ten tere is an analytic conjugacy between R p and R q. Teorem 4.3. Let p(z) = (z z 0 )(z z 1 )(z z 2 ), wit 0 6 jz 0 j 6 jz 1 j 6 jz 2 j and let qðzþ ¼z 3 þðk 1Þz k; k 2 C: ð28þ Ten tere is an analytic conjugacy between R p and R q. Tus analyzing te iteration function (25) for any quadratic and cubic reduces to analyzing it for te q s in (27) and (28), respectively. Definition 10 [9]. We say tat a one-point iterative root-finding algoritm p? T p as a universal Julia set (for polynomials of degree d) if tere exists a rational map S suc tat for every degree d polynomial p, J(T p ) is conjugate by a Möbius transformation to J(S). Te following teorem establises a universal Julia set for quadratics for our metod (24). Teorem 4.4. For a rational map R p (z) arising from te metod (24) applied to p(z) = (z a)(z b), a b, R p (z) is conjugate via te Möbius transformation given by MðzÞ ¼ z a z b to Fig. 1. Basin of attraction for SðzÞ ¼z 4 zþ2 2zþ1.

9 6434 C. Cun et al. / Applied Matematics and Computation 218 (2012) SðzÞ ¼z 4 z þ 2 2z þ 1 Proof. Let p(z)=(z a)(z b), a b and Let M be te Möbius transformation given by MðzÞ ¼ z a wit its inverse z b M 1 ðuþ ¼ ub a, wic may be considered as a map from C [f1g. We ten ave u 1 M R p M 1 ðuþ ¼M R p ub a u 1 ¼ u 4 u þ 2 : 2u þ 1 Fig. 1 illustrates te dynamic structure of SðzÞ ¼z 4 zþ2. Te basin of attraction for S(z) clearly reveals te structure of te universal Julia set for S wen p is quadratic. Te points in te blue area converge to te origin, te red area points converge to 2zþ1 te point at infinity. 5. Fixed points and critical points In te following teorem, we establis te dynamical caracterization regarding te fixed points of R p. Teorem 5.1. Assume p is a generic polynomial of degree d P 2 wit simple roots. If z 0 is a simple root of p, ten it is a superattracting fixed point of R p. All oter additional fixed points of R p are roots of p 0 (z) = 0. Proof. Let p(z) be a generic polynomial of degree d P 2 wit simple roots. Suppose tat z 0 is a root of p(z). Ten R p satisfies tat R p ðz 0 Þ¼z 0 ; R ðjþ p ðz 0Þ¼0; j ¼ 1; 2; 3; R ð4þ p ðz 0Þ 0 since it is of order four [1]. Hence R p as a super-attracting fixed point at eac root of p. Since R p (z 0 )=z 0 only wen p(z 0 )=0orp 0 (z 0 ) = 0, all oter additional fixed points of R p are roots of p 0 (z)=0. Note tat if p 0 (z 0 ) = 0, ten we ave p(z 0 ) 0 since z 0 would oterwise be a multiple root. For a generic polynomial p, additional fixed points of R p and teir dynamical beavior can be found and determined by Teorem 5.1. For example, for te cubic polynomial p(z)=z 3 1, R p as te additional fixed point z 0 = 0. Since jr 0 pð0þj ¼ 1 (see (5)), it is an indifferent fixed point, tis altering te basins of attraction of te roots of te cubic. Critical values of a function f are tose values v 2 C for wic f(z)=v as a multiple root. Te multiple root z = c is called te critical point of f. Tis is equivalent to te condition f 0 (c) = 0. Let p(z) be a generic polynomial wit simple roots. Te free critical points of R p are tose critical points tat are not roots of p(z). Te underlying reason for studying te free critical points is due to te following well-known fact. Teorem 5.2 (Fatou-Julia). Let R(z) be a rational map. If z 0 is an attracting periodic point, ten te immediate basin of attraction B (z 0 ) contains at least one critical point. As a consequence of Teorem 5.2, it is important to detect te existence of attracting periodic cycles. If tere exist an attracting periodic cycle, ten tere exists at least one critical point near te cycle, and te iterates of R p starting wit te critical point converge to tat cycle and not to a root. Tus te existence of attracting periodic cycles could interfere wit our R p searc for a root of te equation p(z) = 0. To detect te existence of attracting periodic cycles, te orbits of te free critical points of te R p function sould be observed and teir set of limit points determined. Upon differentiating (25) we ave R 0 p ðzþ¼1 þ½5p02 ðzþ 30p 0 ðzþp 0 ðyþþ9p 02 ðyþš 2 ð16p 02 ðzþþ16pðzþp 00 ðzþþð5p 02 ðzþ 30p 0 ðzþp 0 ðyþþ9p 02 ðyþþ 16pðzÞp 0 ðzþ 10p 0 ðzþp 00 ðzþ 30p 00 ðzþp 0 ðyþ 30p 0 ðzþp 00 ðyþ p02 ðzþþ2pðzþp 00 ðzþ þ 18p 0 ðyþp 00 ðyþ p02 ðzþþ2pðzþp 00 ðzþ : ð29þ 3p 02 ðzþ 3p 02 ðzþ It follows from (29) tat te equation for te critical points of te iterative metod R p is given by 0 ¼ p 0 ðzþð5p 02 ðzþ 30p 0 ðzþp 0 ðyþþ9p 02 ðyþþð21p 02 ðzþ 30p 0 ðzþp 0 ðyþþ9p 02 ðyþþ16pðzþp 00 ðzþþ 16pðzÞ½10p 03 ðzþp 00 ðzþ 30p 02 ðzþp 00 ðzþp 0 ðyþ 2ðp 02 ðzþþ2pðzþp 00 ðzþþð5p 0 ðzþp 00 ðyþ 3p 0 ðyþp 00 ðyþþš: For te cubic p(z)=z 3 1, we ave R p ¼ zð206z12 þ544z 9 6z 6 14z 3 1Þ 449z 12 þ301z 9 6z 6 14z 3 1 R 0 p ðzþ ¼ð92494z15 þ 36994z 12 þ 2062z 9 298z 6 31z 3 1Þðz 3 1Þ 3 ð449z 12 þ 301z 9 6z 6 14z 3 1Þ 2 : wic is of degree d = 13. R0 pðzþ ¼0 gives us So, te critical points of R p are roots of te equation (92494l l l 3 298l 2 31l 1)(l 1) 3 = 0, l = z 3. Te roots are z 3 ¼ 1; 1; 1; 0: ; 0: ; 0: ; 0: : i:

10 C. Cun et al. / Applied Matematics and Computation 218 (2012) R p as d + 1 = 14 fixed points and 2d 2 = 24 critical points in C 1 ¼ C [ f1g, counting multiplicity. Te use of Maple sows tat nine of tese critical points z 0 ave eigenvalue jr 0 (z 0 )j = 1, and te rest ave jr 0 (z 0 )j <1. Given a polynomial p(z), an iteration function T p (z) is said to be generally convergent if for almost all z 2 C its orbit converges to a root of p(z). Te fact tat Newton s metod is not generally convergent for polynomials was investigated by Barna [17]. We also investigate tis aspect for te metod (24) by constructing a specific polynomial p(z) suc tat te rational map R p arising from (24) applied to te polynomial as an attracting periodic orbit of period 2. Our approac is based on an argument of Smale [18,19] and we ave te following result. Proposition 5.1. Te metod (24) is not generally convergent for te polynomial pðzþ ¼z 3 þ az 2 þ bz þ c; a ¼ 0: ; b ¼ 1: ; c ¼ 0: : Proof. Consider te polynomial p(z)=z 3 + az 2 + bz+ c. We find te coefficients a, b and c so tat R p arising from (24) applied to p(z) will ave a super-attracting periodic pointof period 2 at te origin, tat is, R p ð0þ ¼1; R p ð1þ ¼0; R 0 0ð0Þ p ð0þ ¼0; R0 pð1þ 1, wic by te cain rule would give R 2 pð0þ ¼0; R2 p ¼R 0 p ðr pð0þþr 0 p ð0þ ¼R0 p ð1þr0 pð0þ ¼0. Hence tere exists an open neigborood of te origin suc tat te fixed point iteration does not converge to any of te roots of p(z). Terefore te metod (24) is not generally convergent for tis polynomial. Te conditions R p ð0þ ¼1; R p ð1þ ¼0; R 0 pð0þ ¼0 imply tat a, b and c are te solution of te system c 3 ðc 5 2c 3 b 3 þ 2cb 6 4c 4 ab þ 16c 2 ab 4 4ab 7 þ 6c 3 a 2 b 2 22ca 2 b 5 4c 2 a 3 b 3 þ 8a 3 b 6 þ ca 4 b 4 Þ ðb 3 c 2 þ b 6 c 4 b 4 ac þ 2c 3 ab a 2 c 2 b 2 Þ 2 ¼ 0; 16cb 5b 2 30l 1 b þ 9l 2 1 ¼ 1; 16ð1 þ a þ b þ cþð3 þ 2a þ bþ ¼ 1; 5ð3 þ 2a þ bþ 2 30l 2 ð3 þ 2a þ bþþ9l 2 2 ð30þ ð31þ ð32þ l 1 ¼ 4c2 4ac 2 3b 3b þ b; 2 2ð1 þ a þ b þ cþ l 2 ¼ 3 1 þ 2a 1 3ð3 þ 2a þ bþ 2ð1 þ a þ b þ cþ 3ð3 þ 2a þ bþ þ b:r: Solving te system (30) (32) by Maple wit te number of digits set to 20 produces a ¼ 0: ; b ¼ 1: ; c ¼ 0: : Tus te polynomial pðzþ ¼z 3 0: z 2 1: z þ 0: makes te metod (24) fail to converge to a root of p(z) over a set of positive Lebesgue measure. 6. Numerical results Four fourt order optimal metods are considered, wic are King s metod [20] wit b ¼ 1 given by 2 w n ¼ x n f ðx nþ f 0 ðx n Þ ; x nþ1 ¼ w n f ðw nþ f ðx n Þþbfðw n Þ f 0 ðx n Þ f ðx n Þþðb 2Þfðw n Þ ; ð33þ

11 Autor's personal copy 6436 C. Cun et al. / Applied Matematics and Computation 218 (2012) Kung Traub s metod [4] given by f ðxn Þ ; f 0 ðxn Þ f ðwn Þ 1 ¼ 0 ; f ðxn Þ ½1 f ðwn Þ=f ðxn Þ 2 wn ¼ xn xnþ1 ð34þ Kou et al. s metod [21] given by f ðxn Þ ; f 0 ðxn Þ f 2 ðxn Þ þ f 2 ðwn Þ ¼ xn 0 f ðxn Þ½f ðxn Þ f ðwn Þ wn ¼ xn xnþ1 and our metod (24). Fig. 2. King s wit b ¼ 12 (left) and Kung Traub s metod (rigt) for te roots of te quadratic polynomial z2 1. Fig. 3. Kou s metod (left) and te metod (24) (rigt) for te roots of te quadratic polynomial z2 1. ð35þ

12 Autor's personal copy C. Cun et al. / Applied Matematics and Computation 218 (2012) Fig. 4. King s wit b ¼ 12 (left) and Kung Traub s metod (rigt) for te roots of te cubic polynomial z3 1. Fig. 5. Kou s metod (left) and te metod (24) (rigt) for te roots of te cubic polynomial z3 1. Te basins of attraction of te four metods applied to te quadratic polynomial z2 1 are presented and compared in Figs. 2 and 3. Te results for te cubic polynomial z3 1 are given in Figs. 4 and 5. Te basin of attraction for te metod (24) is better tan any of te oter metods. For our metod (24) applied to te quadratic polynomial wit distinct roots, an almost arbitrary point converges to te root closer to te point. 7. Conclusion In tis paper we ave constructed new optimal fourt order root-finding metods for solving nonlinear equations, wic contains well-known Jarratt s metods as special cases. We presented results wic describe te conjugacy classes and dynamics of presented optimal metods for complex polynomials of degree two and tree. We constructed a specific polynomial suc tat te rational map arising from our metod applied to te polynomial as an attracting periodic orbit of period 2. Te basins of attraction of existing optimal metods and our metod are considered to deal wit initial value problems of iteration metods and compared to illustrate teir performance as a criterion for comparison.

13 6438 C. Cun et al. / Applied Matematics and Computation 218 (2012) Acknowledgements Tis researc was supported by Basic Science Researc Program troug te National Researc Foundation of Korea (NRF) funded by te Ministry of Education, Science and Tecnology ( ). Te fourt autor was partially supported by te Serbian Ministry of Education and Science under grant References [1] J.F. Traub, Iterative Metods for te Solution of Equations, Celsea publising company, New York, [2] W. Gander, On Halley s iteration metod, Am. Mat. Mon. 92 (2) (1985) [3] A.M. Ostrowski, Solution of Equations in Euclidean and Banac Space, Academic Press, New York, [4] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mac. 21 (1974) [5] J. Kou, Y. Li, X. Wang, Fourt-order iterative metods free from second derivative, Appl. Mat. Comput. 184 (2007) [6] J.R. Sarma, R.K. Goyal, Fourt-order derivative-free metods for solving non-linear equations, Int. J. Comput. Mat. 83 (1) (2006) [7] W.J. Gilbert, Generalizations of Newton s metod, Fractals 9 (3) (2001) [8] V. Drakopoulos, How is te dynamics of König iteration functions affected by teir additional fixed points?, Fractals 7 (3) (1999) [9] K. Kneisl, Julia sets for te super-newton metod, Caucy s metod and Halley s metod, Caos 11 (2) (2001) [10] S. Plaza, Review of some iterative root-finding metods from a dynamical point of view, Scientia 10 (2004) [11] M. Scott, B. Neta, C. Cun, Basin attractors for various metods, Appl. Mat. Comput. 218 (2011) [12] B. Neta, M. Scott, C. Cun, Basin attractors for various metods for multiple roots, Appl. Mat. Comput. in press. [13] S. Amat, S. Busquier, S. Plaza, Dynamics of te King and Jarratt iterations, Aequationes Mat. 69 (2005) [14] J. Milnor, Dynamics in One Complex Variable, Annals of Matematics Studies, tird ed., vol. 160, Princeton Univ. Press, Princeton, NJ, [15] P. Jarratt, Some fourt-order multipoint iterative metods for solving equations, Mat. Comput. 20 (1966) [16] A.F. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, [17] B. Barna, Über die Divergenzpunkte des Newtonsces Verfarens zur Bestimmumg von Wurzeln algebraiscen Gleicungen. II, Publ. Mat. Debrecen 4 (1956) [18] S. Smale, On te efficiency of algoritms of analysis for solving equations, Bull. Am. Mat. Soc. 13 (1985) [19] B. Kalantari, Polynomial Root-Finding and Polynomiograpy, World Scientific Publising Co., Singapore, [20] R.F. King, A family of fourt order metods for nonlinear equations, SIAM J. Numer. Amal. 10 (1973) [21] J. Kou, Y. Li, X. Wang, A composite fourt-order iterative metod for solving non-linear equations, Appl. Mat. Comput. 184 (2) (2007)