Dynamical aspects of some convex acceleration methods as purely iterative algorithm for Newton’s maps

Applied Mathematics and Computation 251 (2015) 507–520

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Dynamical aspects of some convex acceleration methods as purely iterative algorithm for Newton’s maps Gerardo Honorato a,⇑,1, Sergio Plaza b,2 a b

CIMFAV, Facultad de Ingeniería, Universidad de Valparaíso, Av. Pedro Montt 2421, Chile Depto. de Matemáticas, Facultad de Ciencias, Universidad de Santiago de Chile

a r t i c l e

i n f o

Keywords: Whittaker’s iterative method The super-Halley iterative method Dynamics Rational maps Conjugacy classes

a b s t r a c t In this paper we define purely iterative algorithm for Newton’s maps which is a slight modification of the concept of purely iterative algorithm due to Smale. For this, we use a characterization of rational maps which arise from Newton’s method applied to complex polynomials. We prove the Scaling Theorem for purely iterative algorithm for Newton’s map. Then we focus our study in dynamical aspects of three root-finding iterative methods viewed as a purely iterative algorithm for Newton’s map: Whittaker’s iterative method, the super-Halley iterative method and a modification of the latter. We give a characterization of the attracting fixed points which correspond to the roots of a polynomial. Also, numerical examples are included in order to show how to use the characterization of fixed points. Finally, we give a description of the parameter spaces of the methods under study applied to a one-parameter family of generic cubic polynomials. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Newton’s iterative method znþ1 ¼ zn  ff0ðzðznnÞÞ ¼ N f ðzn Þ and other second-order methods have been extensively used and studied. For polynomial equations pðzÞ ¼ 0, the iterative function N p defines a rational mapping on the extended complex plane (Riemann sphere) C ¼ C [ f1g. The simple roots of the equation pðzÞ ¼ 0, or in other words the roots of the equation pðzÞ ¼ 0 which are not roots of the derivative p0 ðzÞ, are super-attracting fixed points of N p , that is, let f be a simple root of pðzÞ, then N p ðfÞ ¼ f and N 0p ðfÞ ¼ 0. For a review of the dynamics of Newton’s method, see [19]. In [28], Smale defined purely iterative algorithms. We modify slightly this definition and define purely iterative algorithms for Newton’s maps (Definition 1 below), which provide a broad class of root finding algorithms, some of them widely known. The main advantage of this definition is to transfer known results of Newton’s method, to purely iterative algorithms for Newton’s maps, and then to others iterative methods. In this paper we accomplish this by showing the Scaling Theorem as follows: the theorem is known for Newton’s method (see [26]) and then we prove that the theorem holds for purely iterative algorithms for Newton’s maps. As a consequence we are able to conclude Scaling Theorem for a wide class of methods such as, Halley, super Halley, Chebyshev, Whittaker and others (see Section 4). One of the main applications concerning the Scaling Theorem, is to reduce the dimension of the parameter space (which can easily be of high dimension). The theory of complex dynamics shows that the one parameter quadratic family z2 þ c is ⇑ Corresponding author. 1 2

E-mail address: [email protected] (G. Honorato). Research supported by CONICYT PAI/ACADEMIA 79112030, Proyecto Anillo ACT 1112 and Dysyrf Anillo 1103, Chile. Sergio Plaza passed away in 2011.

http://dx.doi.org/10.1016/j.amc.2014.11.083 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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representative for the dynamic of the whole family of complex quadratic polynomials. Another famous example is Newton’s method applied to cubic polynomials. In that case the (complex) dimension decreases from 7 to complex dimension 1. In order to approximate solutions of nonlinear equations f ðzÞ ¼ 0, in [15], Ezquerro and Hernández use the super-Halley iterative method, which is convex acceleration of Newton’s iterative method

SHf ðzÞ ¼ z 

! 2 0 00 f ðzÞ 2ðf ðzÞÞ  f ðzÞf ðzÞ ; 0 2f ðzÞ ðf 0 ðzÞÞ2  f ðzÞf 00 ðzÞ

ð1Þ

and, in [32], Whittaker introduces the so-called Whittaker’s iterative method

W f ðzÞ ¼ z 

! 00 f ðzÞ f ðzÞf ðzÞ : 2  0 2 0 2f ðzÞ ðf ðzÞÞ

ð2Þ

For details see [1,29]. The paper is organized as follows. In Section 3 we use a characterization of rational maps which arise as Newton’s method in order to define and characterize purely iterative algorithm for Newton’s maps. Then we give a list of methods in order to illustrate the definition of purely iterative algorithm for Newton’s maps (when k ¼ 1, see Definition 1). We use this notion in Section 4 to prove the Scaling Theorem for purely iterative algorithm for Newton’s maps. In particular, this is sufficient to conclude Scaling Theorem for the examples of the previous Section 3: Whittaker’s method, super-Halley among others. In Section 5, we give a characterization of the attracting fixed points of the iterative methods under study which correspond to the roots of a polynomial. In Section 6 numerical experiments are performed in order to show how to use the characterization of fixed points and also the efficiency of the methods. Section 7 gives a brief description of the parameter spaces of Whittaker’s and the super-Halley methods when we apply them to the family of cubic polynomials pk ðzÞ ¼ z3 þ ðk  1Þz  k. Recent results concerning the parameter space are given in [18,9]. 2. Basic notions and standard definitions Let RðzÞ ¼ QPðzÞ be a rational map of the extended complex plane into itself where P and Q are polynomials with no common ðzÞ factors. We say that f is a fixed point of R if RðfÞ ¼ f. The multiplier of R at a fixed point f is the complex number kðfÞ ¼ R0 ðfÞ. Also, we say that a fixed point f is attracting, repelling, indifferent if its multiplier jkðfÞj is, respectively, less than, greater than, equal to 1. A super-attracting fixed point is a fixed point f of R for which R0 ðfÞ ¼ 0. For z 2 C, we define its orbit as the set orbðzÞ ¼ fz; RðzÞ; R2 ðzÞ; . . . ; Rk ðzÞ; . . .g, where Rk means the k-fold iterate of R. The convergence region, also called the basin of attraction, of an (super) attracting fixed point f of R is the set BðfÞ ¼ fz 2 C : Rn ðzÞ ! f as n ! 1g. In other words, the basin of attraction of an attracting fixed point is the set consisting of the points whose iterates tend to the fixed point f as n tends to 1. Let z0 be a fixed point of Rn which is not a fixed point of Rj , for any j with 0 < j < n. We say that orbðz0 Þ ¼ fz0 ; Rðz0 Þ; . . . ; Rn1 ðz0 Þg is a cycle of length n or simply an n-cycle. Note that orbðzj Þ ¼ orbðz0 Þ for any zj 2 orbðz0 Þ, 0

and R acts as a permutation on orbðz0 Þ. The multiplier of an n-cycle is the complex number kðorbðz0 ÞÞ ¼ ðRn Þ ðz0 Þ. Note that, Q Qn1 0 0 0 k by the chain rule, ðRn Þ ðzj Þ ¼ n1 k¼0 R ðR ðzj ÞÞ ¼ k¼0 R ðzk Þ where the second product is a re-arrangement of the first. Thus, at 0

each point zj of the cycle, the derivative ðRn Þ has the same value. An n-cycle fz0 ; z1 ; . . . ; zn1 g is said to be attracting, repelling, indifferent if its multiplier kðorbðz0 ÞÞ is, respectively, less than, greater than, equal to 1. An n-cycle X ¼ fz0 ; z1 ; . . . ; zn1 g is said to be super-attracting if kðorbðz0 ÞÞ ¼ 0. j If z0 is in an attracting n-cycle of R, the convergence region of the orbit orbðz0 Þ is the set Bðorbðz0 ÞÞ ¼ [n1 j¼0 R ðBðz0 ÞÞ, where n Bðz0 Þ is the convergence region of z0 as an attracting fixed point of R . The Julia set of a rational map R, denoted J ðRÞ, is the closure of the set of repelling periodic points. Its complement is the Fatou set F ðRÞ. If z0 is an attracting fixed point of R, then the convergence region Bðz0 Þ is contained in the Fatou set and J ðRÞ ¼ @Bðz0 Þ. Therefore, the chaotic dynamics of R is contained in its Julia set (see [6,25]). 3. Purely iterative algorithms for Newton’s maps First, we recall the definition of purely iterative algorithms due to Smale, see [28]. Let P d be the space of all polynomials of degree less than or equal to d. Define

j : C  P d ! Jk by 0

½k

jðz; f Þ ¼ ðz; f ðzÞ; f ðzÞ; . . . ; f ðzÞÞ and the rational map F : J k ! C by

Fðz; n0 ; . . . ; nk Þ ¼ z 

Pðz; n0 ; . . . ; nk Þ Q ðz; n0 ; . . . ; nk Þ

ð3Þ

G. Honorato, S. Plaza / Applied Mathematics and Computation 251 (2015) 507–520

509

½k

where P and Q are polynomials in k þ 2 variables with no common factors and f denotes the kth derivative of f. A purely b f : C ! C that depends on f 2 P d and takes the form iterative algorithm is a rational endomorphism T

Tb f ðzÞ ¼ Fðjðz; f ðzÞÞÞ;

ð4Þ

for a rational map as in (3). Consider a modification of the preceding definition. Let Ratd be the space consisting of the rational maps of degree less than or equal to d. We define a subset V  Ratd as

V¼

 R 2 Ratd : finite fixed points a of R are simples;

 1 0 : ð1Þ ¼ d=d  1 2 N; Rð1Þ ¼ 1 and R 1  R0 ðaÞ

Now define

^j : C  V ! J k by

^jðz; RÞ ¼ ðz; RðzÞ; R0 ðzÞ; . . . ; R½k ðzÞÞ: Here ^j is the string of derivatives of R up to order k and J k . Define the rational map G : J k ! C by

Gðz; n0 ; . . . ; nk Þ ¼ z 

Pðz; n0 ; . . . ; nk Þ Q ðz; n0 ; . . . ; nk Þ

ð5Þ

where P and Q are polynomials in k þ 2 variables with no common factors. We define a rational endomorphism T f : C ! C, depending on R 2 Ratd , by

T f ðzÞ ¼ Gð^jðz; RÞÞ where R 2 V. Theorem 1. For every G : J k ! C defined as before, there exists a complex polynomial f of degree d such that for every R 2 V; R ¼ N f , where N f is the Newton method. Also, there exists a linear space H of dimension d þ 1 such that V is contained in Ratd \ H. First we recall a theorem of characterization of Newton’s method. This result has been found independently by Saunders (see Smale [28]), Head [20] and more recently by Ruckert [27], Crane [10], and for König’s method applied to simple polynomials by Buff and Henriksen in [7]. Theorem 2. A rational map R : C ! C of degree d P 2 is the Newton method of a polynomial of degree at least two if and only if Rð1Þ ¼ 1 and for all other fixed points a1 ; . . . ; ad 2 C there exist numbers m1; . . . ; md 2 N such that R0 ðaj Þ ¼ ðmj  1Þ=mj < 1. Q Then, R is the Newton map of the polynomial f ðzÞ ¼ dj¼1 aðz  aj Þmj , where a – 0 is a complex number. Proof of theorem 1. Let G : J k ! C. For every R 2 V, it follows immediately that R satisfies the conditions of Theorem 2. Consequently there exists a complex polynomial f, such that R ¼ N f . This defines a correspondence between elements of V and Newton’s method. For the second part, consider a degree d complex polynomial, namely

gðzÞ ¼ a0 þ a1 z þ    þ ad zd where ad – 0. Now, by applying Newton’s method we obtain N g ðzÞ ¼ z  gðzÞ=g 0 ðzÞ, that is,

Ng ðzÞ ¼

a0 þ a2 z2 þ 2a3 z3 þ    þ ðd  2Þad1 zd1 þ ðd  1Þad zd a1 þ 2a2 z þ 3a3 z2 þ    þ ðd  1Þad1 zd2 þ dad zd1

Thus, if we denote by ½x0 :    : xd : y0 :    : yd  the set of points on the projective space 2d þ 1-dimensional, then Newton’s method are contained on a algebraic subvariety H defined by the following degree 1 homogeneous polynomials: x1 ¼ 0; y1 ¼ 2x2 ; 2y2 ¼ 3x3 ; 3y3 ¼ 4x4 , . . .ðd  2Þyd2 ¼ ðd  1Þxd1 , ðd  1Þyd1 ¼ dxd , yd ¼ 0. So, we have d þ 1 equations on a 2d þ 1 dimensional space. This defines a linear subvariety H of dimension d þ 1. Since the correspondence between elements of V and Newton’s method, it follows that V is contained in Ratd \ H. Theorem 1 motivates the following definition. Definition 1. We define the rational endomorphism T f : C ! C, depending on f 2 P d , by

T f ðzÞ ¼ Gð^jðz; N f ðzÞÞÞ

ð6Þ

where N f is Newton’s map applied to f and G is defined by the formula (5). A rational endomorphism T f as above will be called a purely iterative algorithm for Newton’s maps.

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Remark 1. Note that the degree of the polynomials P and Q in (5) does not depend on the degree of f. Now we will show examples of purely iterative algorithm for Newton’s maps. Note that they are generated by changing the polynomials P and Q in the formula (5) and (6). First, we claim that Whittaker’s iterative method and the super-Halley iterative method are purely iterative algorithms for Newton’s maps.  Whittaker’s iterative method, also known as convex acceleration of Whittaker’s method, is an iterative map of order of convergence two given by

! 00   f ðzÞ f ðzÞf ðzÞ 1 ¼ z  ðz  Nf ðzÞÞ 2  N0f ðzÞ : W f ðzÞ ¼ z  0 2 0 2 2 2f ðzÞ ðf ðzÞÞ Thus, according with (5) and (6), Whittaker’s method is a purely iterative algorithm for Newton’s maps when considering k ¼ 1, and the polynomials

Pðz; n0 ; n1 Þ ¼ ðz  n0 Þð1 

n1 Þ 2

and

Qðz; n0 ; n1 Þ  1:  The super-Halley method, also known as convex acceleration of Newton’s method, is an iterative map of order of convergence three given by

  00 ðzÞ ! 2  f ðzÞf 2 2  N0f ðzÞ ðf 0 ðzÞÞ f ðzÞ 1   ¼ z  z  Nf ðzÞ : SHf ðzÞ ¼ z  0 00 2 1  N0f ðzÞ 2f ðzÞ ðzÞ 1  f ðzÞf 2 0 ðf ðzÞÞ

Considering the polynomials

  n Pðz; n0 ; n1 Þ ¼ ðz  n0 Þ 1  1 2 and

Qðz; n0 ; n1 Þ ¼ 1  n1 ; it follows from (5) and (6), that super-Halley’s method is a purely iterative algorithm for Newton’s maps.  The following method that we denote by SH2f , and should be new, is a modification of the super-Halley method. This is given by the formula

SH2f ðzÞ ¼ z 

! 3  N0f ðzÞ 1 : z  Nf ðzÞ 2 1  N0f ðzÞ

ð7Þ

Consider the polynomials

Pðz; n0 ; n1 Þ ¼

1 ðz  n0 Þð3  n1 Þ 2

and

Qðz; n0 ; n1 Þ ¼ 1  n1 : Again, it follows from (5) and (6) that SH2f is a purely iterative algorithm for Newton’s maps.  The family of iterative methods given by

Mf ;h ðzÞ ¼ z  ðz  Nf ðzÞÞ 1 þ

N 0f ðzÞ 2ð1  hN0f ðzÞÞ

! þ

cN 0f ðzÞ

;

ð8Þ

where h and c are complex parameters conveniently chosen, form a family of purely iterative algorithms for Newtons’s maps. Indeed, this follows by considering the polynomials

Pðz; n0 ; n1 Þ ¼ ðz  n0 Þ½2ð1  hn1 Þ þ n1 þ 2cn1 ð1  hn1 Þ and

Qðz; n0 ; n1 Þ ¼ 2ð1  hn1 Þ;

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in (5) and (6). This family includes families of iterative functions such as Chebyshev’s iterative function, Halley’s method, super-Halley, c-iterative function and Chebyshev–Halley family, among others. See for instance [1,4,2,8,17,29,16,21].  For the so-called Multipoint root-finding algorithms we are unable to determine if they are or not examples of purely iterative algorithms for Newtons’s maps, although they can be purely iterative algorithms (in the sense of Smale’definition, (4)). For example Ostrowski’s Method is given by the formula

  f ðzÞ f ðzÞ f z  f 0 ðzÞ f ðzÞ  Of ðzÞ ¼ z  0   0 f ðzÞ f ðzÞ f ðzÞ  f z 

f ðzÞ f 0 ðzÞ

:

ð9Þ

Consider the polynomials

Pðz; n0 Þ ¼ ðz  n0 Þ½f ðzÞ  3f ðn0 Þ and

Q ðz; n0 Þ ¼ f ðzÞ  2f ðn0 Þ: It follows that polynomials P and Q depends on f, and that contradicts Remark 1. For details on Multipoint root-finding algorithms see [1,29,14]. 4. Conjugacy classes As we have seen in Section 3, the iterative functions W f , SHf and many other methods may be written as purely iterative algorithms for Newton’s maps. In particular, this allows us to describe the conjugacy classes of both Whittaker’s and the super-Halley iterative methods when applied to polynomials. Definition 2. Let f and g be maps from the Riemann sphere into itself. An analytic conjugacy between f and g is an analytic diffeomorphism h from the Riemann sphere onto itself such that h f ¼ g h (conjugacy equation). Conjugacy plays a central role in the understanding of the behavior of classes of maps under iteration in the following sense. Suppose that we wish to describe both the quantitative and the qualitative behaviors of the map z # M f ðzÞ, where Mf ðzÞ is an iterative function resulting from an iterative method znþ1 ¼ Uðzn Þ as described in the introduction. Let f be an arbitrary analytic function. Since conjugacy preserves fixed points, cycles and their character (whether (super) attracting, or repelling, or indifferent), and their basins of attraction, it is a worthwhile idea to try to construct a parameterized family or families consisting of polynomials pa as simple as possible so that, for a suitable choice of the complex parameter a, there may exist a conjugacy between M p ðzÞ and Mpa ðzÞ. In order to describe the conjugacy classes of Whittaker’s and the super-Halley iterative methods, we record the next useful result (see [26, Section 5, Theorem 1]). Theorem 3 (The Scaling Theorem for Newton’s method). Let f ðzÞ be an analytic function on the Riemann sphere, and let AðzÞ ¼ az þ b, with a – 0, be an affine map. If gðzÞ ¼ f AðzÞ, then A N g A1 ðzÞ ¼ N f ðzÞ, that is, N f is analytically conjugated to N g by A.

Proof. We have g ¼ f A, where A : C!C is the affine map AðzÞ ¼ az þ b, 0 0 0 0 f ðzÞ ¼ ðg A1 Þ ðzÞ ¼ g 0 ðA1 ðzÞÞðA1 Þ ðzÞ ¼ 1a g 0 ðA1 ðzÞÞ, that is, g 0 ðA1 ðzÞÞ ¼ af ðzÞ. Therefore,

A Ng A1 ðzÞ ¼ z  a

gðA1 ðzÞÞ g 0 ðA1 ðzÞÞ

¼za

f ðzÞ

af 0 ðzÞ

with

a – 0.

Then

¼ Nf ðzÞ

which was to be proved. h We next prove an extension of the Scaling Theorem for purely iterative algorithms for Newton’s maps. Theorem 4 (Scaling Theorem). Assume that k ¼ 1, that is, we have

Fðz; n0 ; n1 Þ ¼ z 

Pðz; n0 ; n1 Þ Q ðz; n0 ; n1 Þ

where Pðz; n0 ; n1 Þ ¼ ðz  n0 Þða1 þ b1 n1 Þ and Q ðz; n0 ; n1 Þ ¼ ða2 þ b2 n1 Þ. Let f and g be polynomials, and let A be the affine map AðzÞ ¼ az þ b, with a – 0. Consider the purely iterative algorithms

T f ðzÞ ¼ Fð^jðz; Nf ðzÞÞÞ and T g ðzÞ ¼ Fð^jðz; Ng ðzÞÞÞ: If gðzÞ ¼ f AðzÞ, then T f A ¼ A T g , that is, T f is analytically conjugated to T g by A.

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Proof. Assume that there exists a constant k 2 C such that gðzÞ ¼ kf AðzÞ. According to Theorem 3, we have AðN g ðzÞÞ ¼ aN g ðzÞ þ b ¼ N f ðAðzÞÞ. Hence

N0g ðzÞ ¼ N0f ðaz þ bÞ ¼ N0f ðAðzÞÞ: This yields

T f ðAðzÞÞ ¼ Fð^jðAðzÞ; Nf ðAðzÞÞÞÞ ¼ AðzÞ  ¼ az þ b  ¼a z

PðAðzÞ; Nf ðzÞ; N0f ðAðzÞÞÞ Q ðAðzÞ; Nf ðzÞ; N0f ðAðzÞÞÞ

ðaz þ b  AðNg ðzÞÞÞða1 þ b1 N0g ðzÞÞ ða2 þ

b2 N0g ðzÞÞ

! ðz  Ng ðzÞÞða1 þ b1 N0g ðzÞÞ ða2 þ b2 N0g ðzÞÞ

¼ AðzÞ 

¼ az þ b 

ðAðzÞ  Nf ðAðzÞÞÞða1 þ b1 N0f ðAðzÞÞÞ ða2 þ b2 N0f ðAðzÞÞÞ

aðz  Ng ðzÞÞða1 þ b1 N0g ðzÞÞ ða2 þ b2 N0g ðzÞÞ

þ b ¼ AðT g ðzÞÞ

which completes the proof. h The preceding theorem works for various root-finding iterative methods, including those of the preceding section. In particular, for Whittaker’s and the super-Halley iterative methods we have the following. Theorem 5 (The Scaling Theorem). Let f ðzÞ be an analytic function on the Riemann sphere, and let AðzÞ ¼ az þ b, with a – 0, be an affine map. If gðzÞ ¼ f AðzÞ, then A W g A1 ðzÞ ¼ W f ðzÞ and A SHg A1 ðzÞ ¼ SHf ðzÞ. In other words, W f is analytically conjugated to W g by A, and SHf is analytically conjugated to SHg by A. Now if g~ðzÞ ¼ cf ðzÞ, where cis a constant, a straightforward computation shows that W g~ ðzÞ ¼ W f ðzÞ and SHg~ ðzÞ ¼ SHf ðzÞ, or in other words the identity map is a conjugacy between both the maps W g~ ðzÞ and W f ðzÞ, and the maps SHg~ ðzÞ and SHf ðzÞ. From this fact and Theorem 5 we conclude that we may study the iteration of only one of these purely iterative algorithms, up to a change of coordinates. For example, every quadratic polynomial pðzÞ ¼ az2 þ bz þ c reduces to, via an affine change of coordinates, a polynomial belonging to the one-parameter family pl ðzÞ ¼ z2  l. This is nothing but an appropriate change of coordinates that places W p ðzÞ (resp. SHp ðzÞ) inside the conjugacy class of W pl ðzÞ (resp. SHpl ðzÞ), for suitable l. As is well known, every generic cubic polynomial pðzÞ (that is, all of its roots are simple) reduces, via an affine change of coordinates, to a polynomial belonging to the one-parameter family pk ðzÞ ¼ z3 þ ðk  1Þz  k for a suitable choice of the parameter k. Thus, we reduce the study of the iterations of W p ðzÞ (resp. SHp ðzÞ) applied to a cubic polynomial p to the study of the iterations of W pk ðzÞ (resp. SHpk ðzÞ) for a suitable choice of k. For a study of the k-parameter space for Newton’s method applied to the one-parameter family pk , see [11]. Next we recall two well known iterative methods, namely König’s and Schröder’s methods. Let f be an analytic function, and let r ¼ 2; 3; . . .. König’s iterative method K r of convergence order r applied to f is given by

 ðr2Þ 1 f

K r;f ðzÞ ¼ z  ðr  1Þ  ðr1Þ ; 1 f

and Schröder’s iterative method Sr of convergence order

Sr;f ðzÞ ¼ z þ

r1 X ð1Þk k¼1

k!

hf ;k ðzÞ 0

ðf ðzÞÞ

2k1

r applied to f is given by

k

ðf ðzÞÞ

0

0

00

where h1 ðzÞ ¼ 1 and, for k ¼ 1; 2; . . . ; the functions hk satisfy recurrence relations hkþ1 ðzÞ ¼ hk ðzÞf ðzÞ  ð2k  1Þhk ðzÞf ðzÞ: Note that S2;f ¼ K 2;f ¼ N f , or in other words both König’s and Schröder’s iterative methods of order of convergence two coin00 2 2f ðzÞf 0 ðzÞ ðzÞ cide with Newton’s method. On the other hand, K 3;f ¼ z  2f 0 ðzÞ is Halley’s method, and S3 ðzÞ ¼ z  f 01ðzÞ f ðzÞ  2ff 0 ðzÞ 2 3 f ðzÞ f ðzÞf 00 ðzÞ is Chebyshev’s method. Remark that some progress has been made in the study of the k-parameter space for König’s and Schröder’s methods applied to the one-parameter family pk (see [12,13,30,31]). 5. Characterization of attracting fixed points In general, in order to describe the regions of convergence of an iterative method for approximating roots of a nonlinear equation f ðzÞ ¼ 0, we must first determine whether the fixed points corresponding to the roots we are trying to approximate are attracting, or repelling, or indifferent. In terms of convergence, the basins of attraction associated with the roots of the polynomial are robust sites where the convergence of the method is optimal. One of the main problems is the presence of basins of attraction associated with attracting periodic points that are not associated with the roots of the polynomial, in which case the convergence is guaranteed, although to the periodic orbit and not to the roots of the polynomial (the extraneous periodic cycles). One strategy is to attempt to avoid these sites of convergence. To achieve this, at least we must know

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513

that if basins associated with the roots of the polynomial are topologically bi-holomorphic to the disk. Then the multiplier at repelling fixed points will give us information about the width of the basins of attraction, as in Theorem 7 and 8. In this section we study the most general aspects of Whittaker’s, the super-Halley method and a modification of superHalley method (7), by studying the character of its fixed points. We first recall basic facts about Newton’s method. Let

Nf ðzÞ ¼ z 

f ðzÞ 0 f ðzÞ

be Newton’s method applied to the polynomial f. Theorem 6. Let f : C ! C be a polynomial of degree d P 2. We denote its zeros by ai , and denote the multiplicity of ai by ni . Then: (a) The roots ai are (super) attracting fixed points of N f , and their multipliers are ðni  1Þ=ni . When ni ¼ 1, the local degree of N f at ai is 2. (b) The map N f : C ! C has a repelling fixed point at 1 with multiplier 1 þ 1=ðd  1Þ. (c) If f has k distinct roots, then the degree of N f is at most k. For Whittaker’s method we have the following. Theorem 7. Let f : C ! C be a polynomial of degree d P 2. We denote its zeros by ai , and denote the multiplicity of ai by ni . Then: (a) The roots ai are (super) attracting fixed points of W f , and their multipliers are 1  ðni þ 1Þ=2n2i . When ni ¼ 1, the local degree of W f at ai is 2. 2 (b) The map W f : C ! C has a repelling fixed point at 1 with multiplier 1 þ ðd þ 1Þ=ð2d  d  1Þ. 00 (c) The extraneous fixed points (that is, fixed points of W f that are not roots of f) are exactly the zeros of ð1=f Þ . If f has k distinct roots, then the degree of W f is at most 3k  2.

Proof (a) It is not difficult to see that we may rewrite Whittaker’s method as

W f ðzÞ ¼ z 

  1 z  Nf ðzÞ 2  N0f ðzÞ : 2

If f has a zero a of multiplicity n, then a is a (super) attractive fixed point of Newton’s method with multiplier ðn  1Þ=n by Theorem 6. Thus

  n1 þ Oðz  aÞ2 : Nf ðzÞ ¼ a þ ðz  aÞ n It follows that

z  Nf ðzÞ ¼

1 ðz  aÞ þ Oðz  aÞ2 n

and

2  N0f ðzÞ ¼

  nþ1 þ Oðz  aÞ: n

Consequently,

W f ðzÞ ¼ a þ ðz  aÞ 

   ðz  aÞðn þ 1Þ nþ1 2 ðz  aÞ þ Oðz  aÞ2 : þ Oðz  a Þ ¼ a þ 1  2n2 2n2

(b) When jzj tends to 1, we may write Newton’s method applied to the polynomial f as

Nf ðzÞ ¼

      d1 d1 1 z þ Oð1Þ and N0f ðzÞ ¼ þO : d d z

Thus

W f ðzÞ ¼

  dþ1 z þ Oð1Þ: 1 2 2d 2

2

2

Hence, 1 is a fixed point of W f with multiplier 2d =ð2d  d  1Þ ¼ 1 þ ðd þ 1Þ=ð2d  d  1Þ.

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(c) Again, the formula for W f is

  1 W f ðzÞ ¼ z  ðz  Nf ðzÞÞ 2  N 0f ðzÞ : 2 00

The extraneous fixed points of W f are the zeros of 2  N 0f ðzÞ in C, which are the same as those of g ¼ ð1=f Þ . Let ai ; with i ¼ 1; . . . ; k; be the zeros of f , and let ni be their multiplicities. Since f is a polynomial of degree d, we have k X ni ¼ d: i¼1 00

For any rational map, the number of zeros in C is equal to the number of poles in C. The poles of g ¼ ð1=f Þ are the points ai , with multiplicity ni þ 2, and g has a zero of order d þ 2 at 1. Thus, g has k X ðni þ 2Þ ¼ d þ 2k: i¼1

poles counting multiplicities. It follows that g has

ðd þ 2kÞ  ðd þ 2Þ ¼ 2k  2: zeros in C counting multiplicities. Consequently, W f has at most 2k  2 extraneous fixed points. For any rational map, the number of fixed points counting multiplicities is equal to the degree plus one. The fixed points of W f are simple (there are no indifferent fixed points). There are k (super) attracting fixed points (which correspond to the roots), one repelling fixed point at infinity, and at most 2k  2 extraneous fixed points in C. Therefore, the degree of W f is at most

ðk þ 1 þ 2k  2Þ  1 ¼ 3k  2: This concludes the proof of ðcÞ.

h

For the super-Halley method we have the following. Theorem 8. Let f : C ! C be a polynomial of degree d P 2. We denote its zeros by ai , and denote the multiplicity of ai by ni . Then: (a)The roots ai are (super) attracting fixed points of SHf and their multipliers are ðni  1Þ=2ni . When ni ¼ 1, the local degree of SHf at ai is 3. (b) The map SHf : C ! C has a repelling fixed point at 1 with multiplier 1 þ ðd þ 1Þ=ðd  1Þ. 00 (c) The extraneous fixed points of SHf are exactly the zeros of ð1=f Þ . If f has k distinct roots, then the degree of SHf is at most 3k  2. Proof (a) It is not difficult to check that the super-Halley method can be written as

SHf ðzÞ ¼ z 

z  W f ðzÞ 1  N0f ðzÞ

where N 0f is the derivative of Newton’s method, and W f is Whittaker’s method. By an argument similar to that used to prove Theorem 7 and using the fact that

z  W f ðzÞ ¼ ðz  aÞ

  nþ1 þ Oðz  aÞ2 2 2n

we have

   ðz  aÞ nþ1 þ Oðz  aÞ2 ðn þ 1Þ n1 2n2  SHf ðzÞ ¼ z  þ Oðz  aÞ2 ¼ a þ ðz  aÞ þ Oðz  aÞ2 : ¼ a þ ðz  aÞ  ðz  aÞn 1 2 2n 2n a Þ þ Oðz  n (b) When jzj tends to 1, we can write Newton’s method applied to the polynomial f as

Nf ðzÞ ¼



   d1 d1 z þ Oð1Þ and N0f ðzÞ ¼ þ Oð1Þ: d d

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Thus

  dþ1 z þ Oð1Þ: SHf ðzÞ ¼ 1  2d Hence, 1 is a fixed point of SHf with multiplier 2d=ðd  1Þ ¼ 1 þ ðd þ 1Þ=ðd  1Þ. (c) We may rewrite the formula for SHf as

SHf ðzÞ ¼ z 

z  W f ðzÞ 1  N0f ðzÞ

where N 0f is the derivative of Newton’s method, and W f is Whittaker’s method. The fixed points of SHf are the fixed points of W f ðzÞ, that is, the zeros of Whittaker’s method which have already been studied in Theorem 7. Thus, repeating the argument of Theorem 7, part ðcÞ we conclude that the degree of SHf is at most 3k  2. h Remark 2. It is known that super-Halley method applied to a polynomial f may have superattracting fixed points not associated with roots of f, see [23]. Theorem 9. Let f : C ! C be a polynomial of degree d P 2. We denote its zeros by ai , and denote the multiplicity of ai by ni . (a) The roots ai are attracting fixed points of SH2f and their multipliers are 1=2ni . (b) The map SH2f : C ! C has a repelling fixed point at 1 with multiplier 1 þ ð2d  1Þ.

Proof (a) Again, it is not difficult to check that the iterative method SH2 can be written as

3  N0f ðzÞ 1 SH2f ðzÞ ¼ z  z  Nf ðzÞ 2 1  N0f ðzÞ

! ¼z

1 ðz  Nf ðzÞÞ þ ðz  W f ðzÞÞ : 2 1  N0f ðzÞ

ð10Þ

where N 0f is the derivative of Newton’s method, and W f is Whittaker’s method. By an argument similar to that used to prove Theorem 7 and using the fact that

z  W f ðzÞ ¼ ðz  aÞ

  nþ1 þ Oðz  aÞ2 2n2

we have

SH2f ðzÞ ¼ z 

   ðz  aÞ 2nþ1 þ Oðz  aÞ2 ð2n þ 1Þ 1 1 2n2 ðz  aÞ þ Oðz  aÞ2 : þ Oðz  aÞ2 ¼ a  ¼ a þ ðz  aÞ  ðz  aÞ 2n 2n a Þ þ Oðz  n

(b) When jzj tends to 1, we may write Newton’s method applied to the polynomial f as

Nf ðzÞ ¼

      d1 d1 1 z þ Oð1Þ and N0f ðzÞ ¼ þO ; d d z

and

W f ðzÞ ¼ Thus

  dþ1 z þ Oð1Þ: 1 2 2d

  1 z þ Oð1Þ: SH2f ðzÞ ¼  2d

Hence, 1 is a fixed point of SH2f with multiplier 2d ¼ 1 þ ð2d  1Þ. h 6. Numerical experiments The multiplicity n of a root of a polynomial f can be estimated using a ratio formula. In this section we show numerical experiments in order to find a root of a polynomial, with multiplicity greater than one. Also we approximate the multiplier of

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the fixed point, which in particular allows us to approximate the multiplicity of the root. In this sense, the modification of the super-Halley method SH2f (as in (7)) is more efficient than Whittaker’s and super-Halley’s Method.  Let xm denote the iterations of Whittaker’s method starting with a point x0 . For Whittaker’s method the ratio formula is given by

Rmþ1 ¼

xmþ1  xm ¼ W 0f ðnm Þ; xm  xm1

where nm is given by the mean value theorem. In a neighborhood of the root a we have

lim Rm ¼ W 0f ðaÞ ¼ 1 

m!1

ðn þ 1Þ ; 2n2

which is the multiplier of a viewed as a fixed point of W f . Observe that when n > 2; W f is a linear method with rate of convergence 1  ðnþ1Þ . In case of a root of multiplicity n ¼ 1, the order of convergence is exactly 2. 2n2  Let ym denote the iterations of the super-Halley’s method, starting at y0 . In the case of super-Halley’s method the ratio is

~ mþ1 ¼ ymþ1  ym ¼ SH0 ðmm Þ; R f ym  ym1 where

mm is given by the mean value theorem. In a neighborhood of the root a we have

~ m ¼ SH0 ðaÞ ¼ lim R f

m!1

ðn  1Þ ; 2n

which is the multiplier of a as a fixed point of SHf . Observe that when n > 2; SHf is a linear method with rate of convergence ðn1Þ . In case of a root of multiplicity n ¼ 1, the order of convergence is exactly 3. 2n  Let zm denote the iterations of the modification of super-Halley’s method given in (7), starting at z0 . For this iterative method the ratio formula is

b mþ1 ¼ zmþ1  zm ¼ SH20 ðl Þ; R f m zm  zm1 where

lm is given by the mean value theorem. In a neighborhood of the root a we have

b m ¼ SH20 ðaÞ ¼  lim R f

m!1

1 2n

Again, by taking the norm of this number, we have the multiplier of a as a fixed point of SH2f , as computed above in Proposition 9. This method is different from W f and SHf . Observe that when n > 2; SH2f is a linear method with rate of convergence ðn1Þ . However, also in case of a root of multiplicity n ¼ 1, the order of convergence is linear. In spite of this fact, 2n the following examples shows that this method is more efficient than Whittaker and super-Halley’s method.

Example 1. Now we consider the polynomial f ðzÞ ¼ z3  3z2 þ 2:25z, with a root of multiplicity n ¼ 2 at a ¼ 3=2. Hence, for Whittaker’s method we have

lim Rm ¼ W 0f ð3=2Þ ¼ 1 

m!1

ð2 þ 1Þ 2  22

¼

5 ¼ 0:625: 8

For the super-Halley’s method

~ m ¼ SH0 ð3=2Þ ¼ lim R f

m!1

ð2  1Þ 1 ¼ ¼ 0:25: 22 4

For the iterative method SH2 given in (7),

Table 1 Iteration of Whittaker’s method, the super Halley’s method and the iterative method SH2 starting at 1. n

xm

ym

zm

f ðxm Þ

f ðym Þ

f ðzm Þ

Rm

~m R

^m R

1 2 3 4 5 6 7 8

1.33333 1.40325 1.44177 1.46437 1.47801 1.48636 1.49151 1.49471

1.33333 1.45529 1.48864 1.49715 1.49928 1.49982 1.49995 1.49998

1.52919 1.49249 1.50186 1.49953 1.50011 1.49997 1.49999 1.50000

0.037037 0.013132 0.004887 0.001858 0.000714 0.000276 0.000107 0.000041

0.037037 0.002908 0.000191 0.000012 0.00000761 0.00000047 0.00000003 0.00000000

0.001303 0.000084 0.000005 0.000000 0.000000 0.000000 0.000000 0.000000

0.209777 0.550797 0.586789 0.603405 0.612254 0.617299 0.620284 0.622090

0.407116 0.273423 0.255049 0.251207 0.250298 0.250074 0.250020 0.249992

0.0855152 0.2551802 0.2485729 0.2503482 0.2499133 0.2500291 0.2500068 0.2500823

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b m ¼ SH20 ð3=2Þ ¼  lim R f

m!1

1 1 ¼  ¼ 0:25: 22 4

In Table 1 we show the iteration of Whittaker’s method starting at x0 ¼ 1, the super Halley’s method starting at y0 ¼ 1 and the iterative method SH2 starting at z0 ¼ 1:1. Also we show approximations to the respective multiplier. Example 2. Consider the polynomial f ðzÞ ¼ ðz  1Þ9 ðz þ 1Þ, with a root of multiplicity n ¼ 9 at a ¼ 1. Hence, for Whittaker’s method we have

lim Rm ¼ W 0f ð1Þ ¼ 1 

m!1

ð9 þ 1Þ 2  92

¼

76  0:938271 81

For the super-Halley’s method

~ m ¼ SH0 ð1Þ ¼ lim R f

m!1

ð9  1Þ 4 ¼  0:444444 29 9

For the iterative method SH2 given in (7),

b m ¼ SH20 ð1Þ ¼  lim R f

m!1

1 1 ¼  0:055555 29 18

In Table 2 we show the iteration of Whittaker’s method starting at x0 ¼ 0:5, the superHalley’s method starting at y0 ¼ 0:5 and the iterative method SH2 starting at z0 ¼ 0:5. Also we show approximations to the respective multiplier. 7. Parameter space In this section we study the super-Halley and Whittaker’s methods applied to the one-parameter family of cubic polynomials pk ðzÞ ¼ z3 þ ðk  1Þz  k, where k 2 C. For Whittaker’s method W k we see that there are open regions in the parameter space (k-space) such that, for k in these regions, there are open sets of initial values in the complex plane so that, for z0 in these regions, W k ðz0 Þ applied to pk fails to converge to the roots of pk . As is a well known, there are polynomials f for which Newton’s method N f ðzÞ ¼ z  ff0ðzÞ has attracting cycles. For example, ðzÞ if we consider the one-parameter family of cubic polynomials pk ðzÞ ¼ z3 þ ðk  1Þz  k where k 2 C, we let N k denote Newton’s method applied to pk . Then there are open regions in the parameter space (k-space) such that, for kin these regions, there are open sets of points in the complex plane such that N k fails to converge to the roots of f k due to the existence of attracting cycles. A first study of Newton’s method for the family of polynomials f k was carried out by Curry et al. in [11] (also see [5]). Similar phenomena for the same family of cubic polynomials are observed for both Schröder’s and König’s iteration methods (see [30,31].) In fact, these authors show the patterns of non-convergent iterated sequences in the complex domain. For example, for Whittaker’s method, Fig. 1a shows parameter regions (shaded black) for which the iterates of the critical point W k accumulate on an attracting cycle. Fig. 1b shows a zoom of one of these regions (shaded black). For k ¼ 0:5 in the shaded black region of Fig. 1a, the black shaded regions of Fig. 2a are those regions where W k fails to converge to any root of pk due to the existence of attracting cycles. Fig. 2b shows the basin of attraction of the roots when we apply Whittaker’s method to the polynomial pðzÞ ¼ z3  1. In Fig. 3a we see that there are no black shaded regions in the parameter space for which the iterates of the critical points of the super-Halley method SHk accumulate on attracting cycles. Fig. 3b shows the basins of attraction of the super-Halley method applied to the polynomial pðzÞ ¼ z3  1. From the above we see that for Whittaker’s iterative method applied to the polynomial pk0 , for k0 in the shaded black region of Fig. 1a, the black shaded regions of Fig. 2a take up great part of the complex plane. Consequently, the basins of attraction of the roots of pk0 are ‘‘small’’ and the iterates W kpk ðz0 Þ do not converge to any root of pk0 for initial points z0 0 belonging to a ‘‘very extensive’’ set. Fig. 2b shows that there exist vast regions of initial points z0 (black shaded regions) n in the complex plane where the iterates W p ðz0 Þ do not converge to any root of the polynomial pk ðzÞ ¼ z3  1.

Table 2 Iteration of Whittaker’s method, the super Halley’s method and the iterative method SH2 starting at 0:5. n

xm

ym

zm

f ðxm Þ

f ðym Þ

f ðzm Þ

Rm

~m R

^m R

1 2 3 4 5 6 7 8

0.53234 0.56247 0.59055 0.61674 0.64119 0.66402 0.68534 0.70527

0.76665 0.89454 0.95282 0.97897 0.99064 0.99583 0.99815 0.99917

1.00445 0.99975 1.00001 0.99999 1.00000 0.99999 1.00000 1.00000

0.001639 0.000918 0.000514 0.000288 0.000161 0.000090 0.000050 0.000028

0.000003 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0.931432 0.932138 0.932748 0.933279 0.933743 0.934152 0.934515 0.934837

0.479640 0.455656 0.448732 0.446227 0.445213 0.444781 0.444593 0.444510

0.009325 0.055775 0.055542 0.055578 0.055559 0.055555 0.055555 0.055555

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Fig. 1a. k-plane for Whittaker’s method.

Fig. 1b. Zoom of a part of the black region in the k-plane.

Fig. 2a. Whittaker’s method W k with k ¼ 0:5 þ 0i in the black shaded region of Fig. 1a.

As distinguished from what happens in the case of Whittaker’s method, in the case of the super-Halley method, black shaded regions do not occur either in the parameter space or when we apply it to approximate the roots of the polynomial pðzÞ ¼ z3  1. The preceding could lead us to conclude that the super-Halley method has a nicer behavior than Whittaker’s method in both the parameter space and the complex plane. 8. Summary In this paper we give the definition of purely iterative algorithm for Newton’s maps which is a slight modification of the concept of purely iterative algorithm due to Smale. This definition (which to our knowledge is new) provided a broad class of

G. Honorato, S. Plaza / Applied Mathematics and Computation 251 (2015) 507–520

519

Fig. 2b. Whittaker’s method applied to z3  1.

Fig. 3a. k-plane for the super-Halley method.

Fig. 3b. The super-Halley method applied to z3  1.

root finding algorithms and is accomplished using a characterization of rational maps which arise from Newton’s method applied to complex polynomials. As an application of this definition we prove a general Scaling Theorem. Then we give a characterization of the fixed points that correspond to the roots of a polynomial in the complex plane when we use Whittaker’s, the super-Halley iterative methods and a modification of the super-Halley method. This is a local description of the behavior of the iterates when we use these methods. In particular, this gives an idea of the shape of the basins of attraction of the fixed points which correspond to the roots of the polynomial. The fixed point z ¼ 1 is a repeller. Numerically, this implies that the iterations done with these methods applied to a polynomial always stay in the finite part of the extended complex plane. This does not mean that we have ensured the convergence of the iterations to a fixed point that corresponds to a root of the polynomial. As is well known, there may exist either (super) attracting fixed points which do not correspond

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to any of the roots of a polynomial or (super) attracting cycles. There may also exist indifferent cycles and/or indifferent fixed points. For another application of this class of results see [3,22,24]. Then we show numerical experiments in order to find a root of multiplicity greater than one, and to approximate the multiplier of the fixed point, which in particular allows us approximate the multiplicity of the root. In this sense, the modification of the super-Halley method SH2f as in (7) is more efficient than Whittaker’s and super-Halley’s Method. Finally, we give a brief description of the parameter spaces for Whittaker’s and the super-Halley methods applied to the family of cubic polynomials pk ðzÞ ¼ z3 þ ðk  1Þz  k. References [1] S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Sci. Ser. A Math. Sci. (N.S.) (10) (2004) 3– 37. [2] S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157 (1) (2003) 197–205. [3] S. Amat, S. Busquier, A.A. Magreñán, Reducing chaos and bifurcations in Newton-type methods. Abstr. Appl. Anal. 2013, Art. ID 726701, 10 pp. [4] I.K. Argyros, D. Chen, Results on the Chebyshev method in Banach spaces, Proyecciones 12 (2) (1993) 119–128. [5] D.C. Arney, B.T. Robinson, Exhibiting chaos and fractals with a microcomputer, Comput. Math. Appl. 19 (3) (1990) 1–11. [6] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. AMS 11 (1) (1984) 85–141 (new series). [7] X. Buff, Ch. Henriksen, König’s root-finding algorithms, Nonlinearity 16 (2003) 989–1015. [8] A. Cordero, J.R. Torregrosa, P. Vindel, Dynamics of a family of Chebyshev–Halley type methods, Appl. Math. Comput. 219 (16) (2013) 8568–8583. [9] A. Cordero, J. García-Maimó, J.R. Torregrosa, M. Vassileva, P. Vindel, Chaos in King’s iterative family, Appl. Math. Lett. 26 (8) (2013) 842–848. [10] E. Crane, Mean value conjectures for rational maps, Complex Var. Elliptic Equ. 51 (1) (2006) 41–50. [11] J.H. Curry, L. Garnett, D. Sullivan, On the iteration of a rational function: computer experiment with Newton’s method, Commun. Math. Phys. 91 (1983) 267–277. [12] V. Drakopoulos, N. Argyropoulos, A. Böhm, Generalized computation of Schröder iteration functions to motivated families of Julia and Mandelbrot-like sets, SIAM J. Numer. Anal. 36 (1999) 417–435. [13] V. Drakopoulos, On the additional fixed points of Schröder iteration functions associated with a one-parameter family of cubic polynomial, Comput. Graphics 22 (5) (1998) 629–634. [14] J. Dzunic, B. Neta, M.S. Petkovic, L.D. Petkovic, Multipoint Methods for Solving Nonlinear Equations, Elsevier, Amsterdam, 2013. [15] J.A. Ezquerro, M.A. Hernández, On a convex acceleration of Newton’s methods, J. Opt. Theory Appl. 100 (1999) 311–326. [16] J.M. Gutiérrez, M.A. Hernández, A family of Chebyshev–Halley type methods in Banach spaces, Bull. Aust. Math. Soc. 55 (1) (1997) 113–130. [17] J.M. Gutiérrez, M.A. Hernández, An acceleration of Newton’s method: super-Halley method, Appl. Math. Comput. 117 (2–3) (2001) 223–239. [18] J.M. Gutiérrez, A.A. Magreñán, J.L. Varona, The Gauss-Seidelization of iterative methods for solving nonlinear equations in the complex plane, Appl. Math. Comput. 218 (6) (2011) 2467–2479. [19] F. v. Haeseler, H.-O. Peitgen, Newton’s method and dynamical systems, in: Heinz-Otto Peitgen (Ed.), Kluwer Academic Publishers, 1989, pp. 3–58. [20] J. Head, The combinatorics of Newton method for cubic polynomials (Ph.D. thesis), Cornell Univ., Ithaca, N.Y., 1987. [21] M.A. Hernández, M.A. Salanova, A family of Chebyshev–Halley type methods, Int. J. Comput. Math. 47 (1–2) (1993) 59–63. [22] G. Honorato, S. Plaza, N. Romero, Dynamics of a higher-order family of iterative methods, J. Complexity 27 (2) (2011) 221–229. [23] K. Kneisl, Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method, Chaos 11 (2) (2001) 359–370. [24] A.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput. 233 (2014) 29–38. [25] J. Milnor, Dynamics in one complex variable: introductory lectures, third ed., Princeton U. Press, New Jersey, 2006. [26] S. Plaza, Conjugacy classes of some numerical methods, Proyecciones (2001) 1–17. [27] J. Rückert, D. Schleicher, On Newton’s method for entire functions, J. Lond. Math. Soc. 2 (75) (2007) 659–676. no. 3. [28] S. Smale, On the efficiency of algorithms of analysis, Bull. A.M.S. 13 (2) (1985) 87–121. [29] J.L. Varona, Graphic and numerical comparison between iterative methods, Math. Intell. 24 (1) (2002) 37–46. [30] E. Vrscay, Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: a computer assisted study, Math. Comput. 46 (173) (1986) 151–169. [31] E. Vrscay, W. Gilbert, Extraneous fixed points, basin boundary and chaotic dynamics for Schröder and König rational iteration functions, Numer. Math. 52 (1988) 1–16. [32] E.T. Whittaker, A formula for the solution of algebraic and transcendental equations, Proc. Edinburgh Math. Soc. 36 (1918) 103–106.