julia sets of generalized newton's method

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Fractals 2007.15:323-336. Downloaded from www.worldscientific.com by DALIAN UNIVERSITY OF TECHNOLOGY on 01/23/16. For personal use only.

Fractals, Vol. 15, No. 4 (2007) 323–336 c World Scientific Publishing Company

JULIA SETS OF GENERALIZED NEWTON’S METHOD XINGYUAN WANG∗ and TINGTING WANG School of Electronics and Information Engineering Dalian University of Technology Dalian, Liaoning 116024, China ∗[email protected] Received November 20, 2006 Accepted May 5, 2007

Abstract The Julia sets theory of generalized Newton’s method is analyzed and the Julia sets of generalized Newton’s method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton’s method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets. Keywords: Generalized Newton’s Method; Julia Sets; Fixed Points; Basins of Attraction.

1. INTRODUCTION

and English. Among which “The Fractal Geometry of Nature” gives the best explanation of fractal geometry.1,2 Now the fractal theory is advancing ahead with great speed and new achievements emerge endlessly.2–4 Newton’s method is a fascinating problem in the study of the fractals. In order to reveal the profound meanings of the fractal images, Peitgen, Wegner, Walter and Chen successively

The word “Fractal,” from the Latin “fractus,” meaning “broken,” was introduced in 1975 by mathematician Mandelbrot to describe irregular and intricate natural phenomena such as coastlines, plant branching, and mountains that could not be described by Euclidean geometry. Since 1975, he has published three books successively in French 323

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constructed and studied the Julia sets of the standard Newton’s method,5–10 and Gilbert analyzed the dynamic behavior of the Newton’s method for multiple roots.11,12 In this paper, the works of Peitgen and Gilbert are generalized, and the Julia sets of the function


f (z) = (z − ξ1 )α (z − ξ2 )β (z − ξ3 )γ × (ξ1 , ξ2 , ξ3 ∈ C; α, β, γ > 0 and ∈ R) (1) are constructed with methods such as the standard Newton’s method, the relaxed Newton’s method, the Newton’s method for multiple roots, the Collatz method, the Schr¨ oder method, the K¨ onig method and the Steffensen method. And then the variation regularity of the basins of attraction is studied in detail.

2. THEORY AND METHOD 2.1. The Generalized Newton’s Method (1) The standard Newton’s method Newton’s method is usually used to solve equations with real coefficients. However, it could also be used on the complex plane to solve equations with complex coefficients. Let f : C → C be a polynomial with complex coefficients, define the rational function N : C → C as follows N (z) = z − f (z)/f (z).

(2)

N (z) is called the standard Newton’s transformation for f (z). Similar to the real coefficient condition, if f (z) = 0, then ξ ∗ is the root of the equation f (z) = 0 if and only if N (ξ ∗ ) = ξ ∗ , that is, ξ ∗ is the fixed point of N (z). The Newton’s method will be at least quadratically convergent at a simple root and linearly convergent at a multiple root. In most of the time, the initial point would converge to a fixed point that is a root, but sometimes the orbit of the initial point could converge to an attracting periodic cycle.13 Now various modifications are made to the standard Newton’s method such as the relaxed Newton’s method, the Newton’s method for multiple roots, the Collatz method and so on. It has been proved that these methods could be at least quadratically convergent even at a multiple root.14 (2) The relaxed Newton’s method The standard Newton’s method is linearly convergent if ξ ∗ is a root of order k of f (z) = 0.

Apply Newton’s method to

k f (z) to obtain

Nk (z) = z − kf (z)/f (z).

(3)

It has been proved that the relaxed Newton’s method is at least quadratically convergent at the multiple root of order exactly k.14 But the problem is that usually we do not know the exact order of the root before computation. (3) The Newton’s method for a multiple root Consider the function u(z) = f (z)/f (z). It can be proved that if ξ ∗ is a root of order k of f (z) = 0, then ξ ∗ is a simple root of u(z) = 0.14 Apply the Newton’s method to u(z) = 0 to obtain M (z) = z − u(z)/u (z) = z − f (z)f (z)/{[f (z)]2 − f (z)f (z)}. (4) The iteration Eq. (4) is at least quadratically convergent at every root of f (z) = 0, but it is more complicated to calculate because it involves second derivatives. (4) The Collatz method Take the average of Eqs. (3) and (4), we obtain the iteration formula Nk (z) + M (z) Ck (z) = 2 f (z){(k + 1)[f (z)]2 − kf (z)f (z)} . =z− 2f (z){[f (z)]2 − f (z)f (z)} (5) The Collatz method is cubically convergent at a root of exactly k.15 (5) The Schr¨ oder method The Schr¨ oder method will converge to any given oder iteration order at a simple root.16,17 The Schr¨ method of the third order is defined by f (z)f (z) 2 f (z) − f (z) 2f (z)3 f (z){2[f (z)]2 + f (z)f (z)} =z− . 2f (z)3

S(z) = z −

(6)

(6) The K¨ onig method Assume that h(z) is finite and nonzero at a simple root of f (z), apply Newton’s method to f (z)h(z), then the iteration will always be at least of second onig iteration order at that simple root.16 In the K¨ method of the rth order, Kr (z), h(z) is chosen so the iteration converges with order r at a simple root of f (z) = 0. K2 is just the standard Newton’s method.

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Julia Sets of Generalized Newton’s Method ∗ K3 must satisfy K3 (ξ ∗ ) = 0 at a simple root ξ of f (z) = 0, and we can take h(z) = 1/ f (z). K3 is defined by

K3 (z) = z −

2f (z)f (z) . 2[f (z)]2 − f (z)f (z)

(7)

K3 is also called the Hally method.18


(7) The Steffensen method The Steffensen method does not require the calculation of derivatives and only uses one initial value. The Steffensen method is defined by St(z) = z −

f (z)2 . f (f (z) + z) − f (z)

(8)

The Steffensen method may be derived from Aiken’s process that is used to speed up the convergence of a sequence.19

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For ξ ∗ , i.e. the root of f (z) = 0, if f (ξ ∗ ) = 0 and k ≥ 2, then |Nk (ξ ∗ )| ≥ 1. Therefore ξ ∗ is a neutral ˆ or a repelling fixed point of Nk (z). And for ∞ ∈ C, if |z| is big enough, then k . Nk (z) ∼ z 1 − m So ∞ is a fixed point of Nk . When k < m, for |z| that is big enough, there is |Nk (z)| ∼ 1 − k +

km(m − 1) k < 1. =1− 2 m m

So ∞ is an attracting fixed point of Nk . (3) The Newton’s method for a multiple root If f (z) = 0, then ξ ∗ is the root of f (z) = 0 for Newton’s method for a multiple root if and only if ξ ∗ is a fixed point of Eq. (4). From Eq. (4) we can get M (z) = (2[f (z)f (z)]2 − f (z)[f (z)]2 f (z)

2.2. The Theory of Julia Sets of Generalized Newton’s Method Let f (z) = a0 + a1 z + · · · + an z m (m ≥ 2), if f (ω) = ω, then ω is called the fixed point of f . The least integer p(p > 1) that satisfies f p (ω) = ω is called the period and ω is called a periodic point of f . Let (f p ) (ω) = λ, if λ = 0, then ω is super-attracting; if 0 ≤ |λ| < 1, then ω is attracting; if λ = 1, then ω is neutral; if |λ| > 1, then ω is repelling. (1) The standard Newton’s method From Eq. (2) we can get N (z) = f (z)f (z)[f (z)]2 . For ξ ∗ , the root of f (z) = 0, if f (ξ ∗ ) = 0, then ξ ∗ is a super-attracting fixed point of N (z). And for ˆ if |z| is big enough, then ∞ ∈ C, 1 . N (z) ∼ z 1 − m Therefore ∞ is fixed point of N . For |z| that is big enough we can also get m(m − 1) 1 < 1. =1− |N (z)| ∼ 2 m m So ∞ is an attracting fixed point of N .

(2) The relaxed Newton’s method If f (z) = 0, then ξ ∗ is the root of f (z) = 0 for the relaxed Newton’s method if and only if ξ ∗ is a fixed point of Eq. (3). From Eq. (3) we can get Nk (z) = 1 − k + kf (z)f (z)/[f (z)]2 .

− [f (z)]2 f (z)f (z))/{[f (z)]2 − f (z)f (z)}2 .

(9) For ξ ∗ , the root of f (z) = 0, if f (ξ ∗ ) = 0, then ξ ∗ is a super-attracting fixed point of M (z). And for ˆ if |z| is big enough, then ∞ ∈ C, m . M (z) ∼ z 1 − m Here M (z) is a finite value, therefore ∞ is not a fixed point of M . From Eq. (4) we can know that the roots of f (z) = 0 are not the only fixed points of M (z). The roots of f (z) = 0 that are not the roots of f (z) = 0, are also the fixed points of M (z), we call them the extraneous fixed points.20,21 If z is such an additional fixed point, then there is f (z) = 0 and f (z) = 0. From Eq. (9) we can get M (z) = 2. Therefore the extraneous fixed points are always repelling, and the Newton’s method for multiple roots can hardly get attracted to these points. (4) The Collatz method If f (z) = 0, then ξ ∗ is the root of f (z) = 0 for Collatz method if and only if ξ ∗ is a fixed point of Eq. (5). From Eq. (5) we can get Ck (z) = 1 − ([(k + 1)(f )3 + 2f f f − k(f )2 f ] × [(f )3 − f f f ] − [(k + 1)f (f )2 − k(f )2 f ][2(f )2 f − f (f )2 − f f f ])/2{f (z)[f (z)2 − f (z)f (z)]}2 .

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Here the f , f , f and f respectively represent for f (z), f (z), f (z) and f (z). Thus for ξ ∗ , the root of f (z) = 0, if f (ξ ∗ ) = 0, then


Ck (ξ ∗ ) =

1−k . 2

If k = 2, then |Ck (ξ ∗ )| < 1, ξ ∗ is an attracting fixed point of Ck (z); if k = 3, then |Ck (ξ ∗ )| = 1, ξ ∗ is a neutral fixed point of Ck (z); if k ≥ 4, then |Ck (ξ ∗ )| > 1, ξ ∗ is a repelling fixed point of Ck (z). ˆ when |z| is big enough there is And for ∞ ∈ C, m+k . Ck (z) ∼ z 1 − 2m So ∞ is a fixed point of Ck . For |z| that is big enough we can also get 2 + 3k − 6km > 1, |Ck (z)| ∼ m so ∞ is a repelling fixed point of Ck . From Eq. (5) we can deduce that the roots of (k + 1)[f (z)]2 − kf (z)f (z) = 0 that are not the roots of f (z) = 0 are extraneous fixed points of Ck (z).21 (5) The Schr¨ oder method If f (z) = 0, then ξ ∗ is the root of f (z) = 0 for the Schr¨ oder method if and only if ξ ∗ is a fixed point of Eq. (6). From Eq. (6) we can get S (z) = {2f (z)[f (z)]2 f (z) + 3[f (z)]2 f (z) − [f (z)]2 f (z)f (z)}/{2[f (z)]4 }. For ξ ∗ , the root of f (z) = 0, if f (ξ ∗ ) = 0, then ξ ∗ is a super-attracting fixed point of S(z). And for ˆ if |z| is big enough, then ∞ ∈ C, 3m − 1 . S(z) ∼ z 1 − 2m2 Therefore ∞ is a fixed point of S. For |z| that is big enough we can also get 1 1 2− . |S (z)| ∼ 1 − m 2m |S (z)|

< 1, ∞ is an attracting If m = 2, then fixed point of S; if m ≥ 3, then |S (z)| > 1, ∞ is a repelling fixed point of S. From Eq. (6) we can deduce that the roots of 2[f (z)]2 + f (z)f (z) = 0 that are not the roots of f (z) = 0 are extraneous fixed points of S(z).20

(6) The K¨ onig method If f (z) = 0, then ξ ∗ is the root of f (z) = 0 for the K¨ onig method if and only if ξ ∗ is a fixed point of Eq. (7). From Eq. (7) we can get K3 (z) = {3[f (z)f (z)]2 − 2[f (z)]2 f (z)f (z)}/{2[f (z)]2 −f (z)f (z)}2 .

(10)

For ξ ∗ , the root of f (z) = 0, if f (ξ ∗ ) = 0, then ξ ∗ is a super-attracting fixed point of K3 . And for ˆ if |z| is big enough, then ∞ ∈ C, 2 . K3 (z) ∼ z 1 − m+1 Therefore ∞ is a fixed point of K3 . For |z| that is big enough we can also get m−1 < 1. |K3 (z)| ∼ m+1 So ∞ is an attracting fixed point of K3 . The roots of f (z) = 0 that are not the roots of f (z) = 0 are extraneous fixed points of K3 (z). For extraneous fixed points there is K3 (z) = 3, so they are all repelling.20 (7) The Steffensen method If ξ ∗ is a root of f (z) = 0, then the denominator in Eq. (8) is zero, and ξ ∗ could not be called the fixed point of the Steffensen method. So in the iteration process of Eq. (8), when f (f (z) + z) − f (z) = 0, we set St(z) = z and halt the iteration. The Steffensen method will be linearly convergent at a multiple root and quadratically convergent at a simple root. But for most of the initial values the Steffensen method does not converge to any root. From Eq. (8) we can get St (z) = 1 − 2f (z)f (z)f (f (z) + z) − f (z)2 f (z) − f (z)2 f (z)f (f (z) + z) −f (z)2 f (f (z) + z))/[f (f (z) + z)− f (z)]2 . ˆ if |z| is big enough, then For ∞ ∈ C, 1 . St(z) ∼ z 1 − (m−1)2 1 z − z m−1 Therefore ∞ is a fixed point of St. For |z| that is big enough and m ≥ 2 we can also get |St (z)| ∼ 1−

2 −1)

(2m − m2 )z −(m−1) − mz −(m z (m−1)(m−2)

= 1. So ∞ is a neutral fixed point of St.

− m3

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Julia Sets of Generalized Newton’s Method

Assume F method, i.e.

is

the

generalized

Newton’s


A(ξ ∗ ) = {z : F n (z) → ξ ∗ , (n → ∞)}. A(ξ ∗ ) is the basin of attraction of ξ ∗ , here ξ ∗ is the root of f (z) = 0. By the iteration of F , all the points in A(ξ ∗ ) will get attracted to ξ ∗ . Since ξ ∗ is the super-attracting fixed point of the methods such as the standard Newton’s method, the Newton’s method for multiple roots, the Schröder method and the K¨ onig method, the A(ξ ∗ ) is an open region including ξ ∗ . Consider the Julia sets of the rational function F (z), all the above-mentioned theories for polynomials are still tenable. The pivotal difference is that for rational function, the Julia sets of F will not always be bounded (but definitely closed). And JF is the boundary of A(ω) for every attracting fixed point ω. So JF is an important factor in the analyzing of the basins of attraction of Newton’s method. For the polynomial f (z), the order of which is bigger than 2, assume the roots of f are ω1 , ω2 , . . . , ωn and there is f (ωi ) = 0, then the Julia sets of the generalized Newton’s method F will be the boundary of the basins of each root. JF = ∂A(ω1 ) = · · · = ∂A(ωn ). The points on the boundary of any basin of attraction will definitely be on the boundaries of all the other basins of attraction. As the Julia sets, JF , is uncountable, there are a large numbers of such multiple boundary points. It is hard to imagine and also quite fantastic. Theorem 1.22 The Julia sets, JF , is the closure of

the repelling periodic points of polynomial F . It is an uncountable close subset not containing the isolated points. If z ∈ JF , then JF is the closure of ∞ −k (z). The Julia set is the boundary of the k=1 F basin of attraction of each attracting fixed point of F including the infinite point and the effect of F on JF is chaotic.

integer and d ≥ 1. (2) Transform from Eqs. (2) to (8) to zn+1 = F (zn ). (3) Fix the window W (W ⊂ Z), ∀z0 ∈ W , zn (n = 0, 1, . . . , N ) is calculated. (4) If |F (zn ) − zn | ≤ EOF (EOF is the extent of error, here we set EOF = 0.00001), then we can say that zn+1 is the root of f (z) = 0, and z0 will be colored according to the convergence of the orbit {F n (z0 )}. We use white for divergent area and attracting cycle, and other different colors for different roots and extraneous fixed points. The relative data for the drawing of the colors are in Table 1. (5) Repeat (3) and (4) for all the points in W , and we could get the Julia sets for the generalized Newton’s method. From Theorem 1 we could know that the Julia set of rational function is the boundary of the stable region related with all the fixed points, and the above-mentioned Julia set is the boundary of the colored basins in the fractal image constructed with this method. In this paper we use different colors for the simple or multiple roots and also the extraneous fixed points. And the gradual changed color is adopted to express the inner structure of the basins of attraction. The lighter the color is the slower the convergence will be, whereas the darker the color is the faster the convergence will be. If the maximum iteration numbers needed for the points in W to converge to ξi∗ or ξ˜j∗ is count, ∀z0 ∈ W , we denote the iteration numbers needed for z0 to converge to ξi∗ or ξ˜j∗ as num. Assume the corresponding color for ξi∗ or ξ˜j∗ is red, we set the RGB value r of z0 as follows num − 1 . r = 114 + (249 − 114) × count − 1 When num = 1 the RGB value will be the smallest, and the color will be the darkest; when num = count the RGB value will be the largest and the color will be the lightest. Therefore the gradual Table 1

2.3. The Constructing Method of Julia Sets of Generalized Newton’s Method We will construct the Julia sets of the generalized Newton’s method of f (z) using the following steps. (1) First the roots of f (z) = 0 are resolved, i.e. ξi∗ (i = 1, 2, . . . , d), here d is the order of function f . Then the extraneous fixed points are resolved from Eqs. (4) to (7), i.e. ξ˜j∗ (j = 1, 2, . . . , d ), here d is

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The Description of the Color Palette.

Color

RGB Value R

G

B

Red

114 249

43 222

18 213

Green

22 217

83 247

17 215

Yellow

110 255

110 255

0 125

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changed colors are formed according to the value of num within the same basin.

3. EXPERIMENTS AND RESULTS Take Eq. (1) as our research object. It is obviously that the equation has three roots, ξ1 , ξ2 and ξ3 with orders α, β and γ.


3.1. The Julia Sets of the Standard Newton’s Method Set the maximum iteration number N = 20, and use the colors green, red and yellow separately for A(ξ1 ), A(ξ2 ) and A(ξ3 ), we construct the Julia sets of standard Newton’s method as in Fig. 2. The coordinates of Fig. 2 are [−6, 4] × [−5, 3]. In the triangle formed by the three roots ξ1 , ξ2 and ξ3 , set its three borders l1 , l2 and l3 (Fig. 1a). Assume the orders of the roots are equal, that is α = β = γ, if l1 = l2 = l3 , then A(ξ1 ), A(ξ2 ) and A(ξ3 ) possess a rotational symmetry, the boundary of A(ξ1 ), A(ξ2 ) and A(ξ3 ) is the Julia set of the standard Newton’s method (Fig. 2a). If l1 = l2 > l3 (see the black triangle in Fig. 1a), the basins are symmetrical with respect to the midnormal of l3 . The size of A(ξ1 ) and A(ξ2 ) are the same and bigger than that of A(ξ3 ) (Fig. 2b). If l2 > l1 > l3 , the size of A(ξ2 ) is the biggest and the areas of A(ξ1 ) and A(ξ3 ) are separated away by A(ξ2 ) (Fig. 2c). If l1 > l2 > l3 (see the green triangle in Fig. 1a), the size of A(ξ1 ) is the biggest and the areas of A(ξ2 ) and A(ξ3 ) are separated away by A(ξ1 ) (Fig. 2d). The distribution of the basins of attraction also depends on the orders of the roots. Set α = 2, and let the other parameters be the same as Fig. 2c.

(a) Corresponding to the standard Newton’s method.

Fig. 1

Compare Fig. 2e with Fig. 2c, we can see that A(ξ1 ) becomes the biggest basin instead of A(ξ2 ), and the areas of A(ξ2 ) and A(ξ3 ) are separated away by A(ξ1 ). Set β = 0.8, and let the other parameters be the same as Fig. 2c. Compare Fig. 2f with Fig. 2c, we can see that A(ξ2 ) becomes smaller and A(ξ1 ) becomes the biggest basin. This shows that the higher the order of the root is, the bigger the basin is. Besides, we can also find that if the order of the root does not equal to 1, then the inner structure of the basin will become asymmetry instead of the smooth nesting ellipses (see A(ξ1 ) in Fig. 2e and A(ξ2 ) in Fig. 2f). In Fig. 2e, there is α = 2 and the color of A(ξ1 ) is lighter, which shows that the convergence is slower here, and in Fig. 2f, there is β = 0.8, correspondingly the color of A(ξ1 ) is darker, which shows that the convergence is faster here. This is because the standard Newton’s method is linearly convergent at a multiple root. The above analyses show that the structure of basins of attraction in the Julia sets of the standard Newton’s method depends on the distribution and the orders of the roots.

3.2. The Julia Sets of the Relaxed Newton’s Method Set the maximum iteration number N = 15 (for Fig. 3c there is N = 900), and use the colors of red, yellow and green separately for A(ξ1 ), A(ξ2 ) and A(ξ3 ), we construct the Julia sets of the relaxed Newton’s method as in Fig. 3. The coordinates of Fig. 3 are [−3, 7] × [−2, 6]. Set the positions of ξ1 , ξ2 and ξ3 as the three nodes in the black triangle in Fig. 1b, then move ξ3 horizontally which forms the green triangle in Fig. 1b. Set α = β = γ = 1, we obtain the Julia

(b) Corresponding to the the Newton’s method for multiple roots.

The distribution of the roots of f (z) = 0.

(c) Corresponding to the relaxed Newton’s method.

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(a) α = β = γ = 1, ξ3 = −0.634 − 0.634i.

(b) α = β = γ = 1, ξ3 = 0.

(c) α = β = γ = 1, ξ3 = 0.3i.

(d) α = β = γ = 1, ξ3 = −0.3i.

(e) α = 2, β = γ = 1, ξ3 = 0.3i.

(f ) α = γ = 1, β = 0.8, ξ3 = 0.3i.

Fig. 2

329

The Julia sets of the standard Newton’s method, here ξ1 = −1 − 2i, ξ2 = −2 − i.

(a) α = β = γ = 1, ξ3 = 1.8 + 1.268i.

(b) α = β = 1, γ = 2, ξ3 = 2 + 1.268i.

(c) α = β = 1, γ = 2, ξ3 = 2 + 1.268i.

(d) α = β = k = 2, γ = 3, ξ3 = 2 + 1.268i.

(e) α = β = 2, γ = k = 3, ξ3 = 2 + 1.268i.

(f ) α = β = γ = 0.3, ξ3 = 2 + 1.268i.

Fig. 3

The Julia sets of the relaxed Newton’s method, here ξ1 = 1 + 3i, ξ2 = 3 + 3i.

sets of the relaxed Newton’s method as in Fig. 3a. Here l1 > l3 > l2 , A(ξ1 ) is the biggest basin and the areas of A(ξ2 ) and A(ξ3 ) are separated away by A(ξ1 ). When k = 2, ξ ∗ is a neutral fixed point

of Nk (z). If N is small, we will not be able to observe the basins of attraction of the simple roots (Fig. 3b, the precision of convergence for Fig. 3b is |f (zn )| ≤ 0.00001). When N is big enough and the

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precision of convergence is bigger we could observe the basins of the simple roots (Fig. 3c, the precision of convergence for Fig. 3(c) is |f (zn )| ≤ 0.001). When k > 2, ξ ∗ is a repelling fixed point, and the basins of the simple roots will never be seen. When α = β = 2 and γ = 3, set k = 2 and k = 3, we construct the Julia sets of the relaxed Newton’s method (Figs. 3d and 3e). In these two figures, we can see that the size of A(ξ3 ) is the biggest and the area of A(ξ1 ) and A(ξ2 ) are separated away by A(ξ3 ). When γ = k = 3, the inner structure of A(ξ3 ) shows a series of smooth nesting ellipses, or else the inner structure of A(ξ3 ) will become asymmetry. When α = β = γ = k, we can deduce from the formula (3) that Nk (z) = z −

(z − ξ1

)−1

1 . + (z − ξ2 )−1 + (z − ξ3 )−1

The equation shows that Nk (z) has no relation with the parameter k and the orders of the roots. Thus if α = β = γ = k, then the Julia sets of the relaxed Newton’s method is unique (Fig. 3f). The relaxed Newton’s method for non-integral values of k has been studied extensively (see Refs. 23 and 24 for results when 0 < k < 1). By a mass of experiments we discover that when α, β and γ are not equal, the Julia sets of the relaxed Newton’s method depend on the orders of the roots

and the distances between the roots. And the regularity is the same with that of the standard Newton’s method.

3.3. The Julia Sets of the Newton’s Method for Multiple Roots Set the maximum iteration number N = 20, and use the colors of red, green and yellow separately for A(ξ1 ), A(ξ2 ) and A(ξ3 ), we construct the Julia sets of the Newton’s method for multiple roots as in Fig. 4. The coordinates of Fig. 4 are separately [−3, 7]×[−2, 6], [−500, 500]×[−400, 400], [−8, 12]× [−6, 10], [−400, 600] × [−500, 300], [−750, 250] × [−250, 550] and [−500, 500] × [−400, 400]. Set the positions of ξ1 , ξ2 and ξ3 as the three nodes in the black triangle in Fig. 1b and let α = β = γ = 1, then we obtain the Julia sets of the Newton’s method for multiple roots (Figs. 4a and 4b, Fig. 4a gives a partial enlarged detail of Fig. 4b). Here l1 = l2 = l3 , A(ξ1 ), A(ξ2 ) and A(ξ3 ) possess a rotational symmetry. There is an extraneous fixed point ξ˜1 = 2 + 2.4i of M (z) and all of the points in the dark blue area, i.e. the basins of the extraneous fixed point, get attracted to ξ˜1 = 2 + 2.4i. Maintain the values of ξ1 , ξ2 , ξ3 and set α = 1, β = 0.6, γ = 1.2, we get Fig. 4c. In Fig. 4c the size of A(ξ3 ) is the biggest, A(ξ1 ) is smaller, A(ξ2 ) is the smallest,

(a) α = β = γ = 1, ξ3 = 2 + 1.268i.

(b) Zoom of Fig. 4a.

(c) α = 1, β = 0.6, γ = 1.2, ξ3 = 2 + 1.268i.

(d) α = β = γ = 1, ξ3 = 1.7.

(e) α = β = γ = 1, ξ3 = 1.5.

(f ) α = β = γ = 1, ξ3 = 2 + 1.1i.

Fig. 4

The Julia sets of the Newton’s method for multiple roots, here ξ1 = 1 + 3i, ξ2 = 3 + 3i.

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and the area of A(ξ1 ) and A(ξ2 ) are rounded by the area of A(ξ3 ). This shows that in the Newton’s method for multiple roots the higher the order is the bigger the basin will be. Move ξ3 horizontally to the left (shown as the green triangle in Fig. 1c), we get Fig. 4d. Now l1 > l2 > l3 , and the size of A(ξ2 ) is the biggest. This is not coincident with the regularity we get in the above chapters. Keep on moving ξ3 to the left and get Fig. 4e. This time the size of A(ξ1 ) becomes the biggest. This is because there is an extraneous fixed point ξ˜1 = 2+2.8i near the root ξ2 = 3+3i in Fig. 4d which intensifies the attracting power of A(ξ2 ) (the basin of the extraneous fixed point is very small, and we can only see a dark blue point in the figure). Whereas there is no extraneous fixed point in Fig. 4e, and the result is consist with the regularity we obtained above. Then we move the ξ3 in Fig. 1c downwards and get Fig. 4f, here l1 = l2 > l3 . According to our foregoing observations, we will believe that the size of A(ξ1 ) and A(ξ2 ) are the same and bigger than A(ξ3 ). But as there is an extraneous fixed point ξ˜1 = 2 + 2.1i which is near ξ3 , it comes out that the area of A(ξ3 ) is the biggest and the area of (ξ1 ) and A(ξ2 ) are rounded by the area of A(ξ3 ). Theorem 2. If α = β = γ, l1 = l2 = l3 and there is no extraneous fixed point, then the Julia sets constructed by the standard Newton’s method, the relaxed Newton’s method or the Newton’s method for multiple roots have three times rotation symmetry.

Using the inductive method of mathematics, we give the proof of the standard Newton’s method, and it is similar for the other two methods. When α = β = γ, the corresponding Newton’s transform deduced from Eq. (2) is

Proof.

N (z) = z −

1 1 α( z−ξ 1

+

1 z−ξ2

+

. 1 z−ξ3 )

(11)

When l1 = l2 = l3 , we move the origin to the center of the regular triangle with ξ1 , ξ2 and ξ3 as its three nodes by rotation and the transla2π tion of the axis. Then there is ξ2 = ξ1 ei 3 , ξ2 = 4π ξ1 ei 3 , and we can deduce from the Eq. (11) that 2πj |N 1 (z)| = |N 1 (zei 3 )|. Assume that |N k−1 (z)| = 2πj |N k−1 (zei 3 )|. 2πj 2πj 2πj ∵ N k (zei 3 ) = N k−1 [N (zei 3 )] = N k−1 [ei 3 2πj N (z)], let N 1 (z) = z , then N k (zei 3 ) = 2πj N k−1 (z ei 3 ). While N k (z) = N k−1 [N 1 (z)] =

N k−1 (z ), and ∵ |N k−1 (z )| = |N k−1 (z ei i 2πj 3

∴ |N k (z)| = |N k (ze

2πj 3

331

)|,

)| (k = 1, 2, . . . , N ; j = 0, 1, 2).

This shows that the Julia sets constructed by the standard Newton’s method, the relaxed Newton’s method or the Newton’s method for multiple roots which satisfy the above conditions will have three times rotation symmetry (Figs. 2a and 3f). As there is an extraneous fixed point in Fig. 4a, it does not satisfy Theorem 2. The above study shows that when α = β = γ and there is no extraneous fixed point, then the Julia sets of the Newton’s method for multiple roots depend on the orders of the roots and the distances between the roots. And the regularity is the same with that of the standard Newton’s method. For the Newton’s method for multiple roots, the two relatively smaller basins may be rounded by the biggest basin. Gilbert25 gives a study of the Julia sets of the Newton’s method for multiple roots of the equation f (z) = z 2 (z 3 − 1).

3.4. The Julia Sets of the Collatz Method Set N = 20, and use the colors of red, yellow and green separately for A(ξ1 ), A(ξ2 ) and A(ξ3 ), we construct the Julia sets of the Collatz method as in Fig. 5. The coordinates of Fig. 5 are [−3, 7]×[−2, 6]. Set the positions of ξ1 , ξ2 and ξ3 as the three nodes in the black triangle in Fig. 1c and let α = β = γ = k = 2, then we obtain the Julia sets of the Collatz method (Fig. 5a). Now l1 = l2 , the size of A(ξ1 ) and A(ξ2 ) are the same and symmetrical with respect to the line y = 2. Move ξ3 horizontally to the left, we get Fig. 5b. Now l1 > l2 > l3 . In Fig. 5a the red blocks on the boundary of A(ξ2 ) and A(ξ3 ) are connected to each other, while in Fig. 5b the yellow blocks on the boundary of A(ξ1 ) and A(ξ2 ) are disjoined. This shows that as ξ3 moves to the left, the red blocks between A(ξ2 ) and A(ξ3 ) are connected to each other and make the area of A(ξ2 ) and A(ξ3 ) separated away by A(ξ1 ). Set the positions of ξ1 , ξ2 and ξ3 as the three nodes in the black triangle in Fig. 1b, then the area of A(ξ1 ), A(ξ2 ) and A(ξ3 ) possess a rotational symmetry. Let γ = 3, we get Fig. 5c. In this figure we could see that as γ increases, the area of A(ξ3 ) becomes larger and the inner structure of A(ξ3 ) becomes asymmetry while

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(a) α = β = γ = 2, ξ3 = 2.


Fig. 5

(b) α = β = γ = 2, ξ3 = 1.9.

The Julia sets of the Collatz method, here k = 2, ξ1 = 1 + 3i, ξ2 = 3 + 3i.

the inner structure of A(ξ1 ) and A(ξ2 ) still shows a smooth nesting ellipses character. This is coincident with our foregoing conclusions.

3.5. The Julia Sets of the Schr¨ oder Method Set N = 20, and use the colors of green, red and yellow separately for A(ξ1 ), A(ξ2 ) and A(ξ3 ), we construct the Julia sets of the Schr¨ oder method as in Fig. 6. The coordinates of Fig. 6 are [−6, 4]×[−5, 3]. Set the positions of ξ1 , ξ2 and ξ3 as the three nodes in the black triangle in Fig. 1a and let α = β = γ = 1, then we obtain the Julia sets of the Schr¨ oder method (Fig. 6a). In Fig. 6a the Julia sets are symmetrical with respect to the line y = x. Move ξ3 upwards we get Fig. 6b. Now l2 > l1 > l3 , the size of A(ξ2 ) is the biggest. As ξ3 keeps on moving upwards, the area of A(ξ1 ) and A(ξ3 ) are separated away by A(ξ2 ). Set α = 0.8, β = 1, γ = 1.6 and ξ3 = −0.634 − 0.634i, then ξ1 , ξ2 and ξ3 are just the three nodes of a regular triangle, that is l1 = l2 = l3 . Now the higher the order of the root is the bigger the basin is. (In Fig. 6c the area of A(ξ3 ) with the order γ = 1.6 is the biggest and the area of

(a) α = β = γ = 1, ξ3 = 0.

Fig. 6

(c) α = β = 2, γ = 3, ξ3 = 2 + 1.268i.

A(ξ1 ) and A(ξ2 ) are separated away by A(ξ3 ). The area of A(ξ1 ) with the order α = 0.8 is the smallest.) Besides, the basins of high order polynomial are studied with the Schr¨ oder iteration function.21,26 And the structure of the classical Mandelbrot set is found in the Julia sets of the Schr¨ oder function on the parameter plane.

3.6. The Julia Sets of the K¨ onig Method Set N = 20, and use the colors of green, red and yellow separately for A(ξ1 ), A(ξ2 ) and A(ξ3 ), we construct the Julia sets of the K¨ onig method as in Fig. 7. The coordinates of Fig. 7 are [−6, 4]×[−5, 3]. Set the positions of ξ1 , ξ2 and ξ3 as the three nodes in the black triangle in Fig. 1a and let α = β = γ = 1, then we obtain the Julia sets of the K¨ onig method (Fig. 7a). In Fig. 7a the Julia sets are symmetrical with respect to the line y = x. Move ξ3 upwards we get Fig. 7b. Now l2 > l1 > l3 , the size of A(ξ2 ) is the biggest. As ξ3 keeps on moving upwards, the area of A(ξ1 ) and A(ξ3 ) are separated away by A(ξ2 ). Set α = 0.6, β = 1, γ = 2 and ξ3 = −0.634 − 0.634i, then ξ1 , ξ2 and ξ3 are just the

(b) α = β = γ = 1, ξ3 = i.

(c) α = 0.8, β = 1, γ = 1.6, ξ3 = −0.634 − 0.634i.

The Julia sets of the Schr¨ oder method, here ξ1 = −1 − 2i, ξ2 = −2 − i.

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(a) α = β = γ = 1, ξ3 = 0.


Fig. 7

(b) α = β = γ = 1, ξ3 = 6i.

333

(c) α = 0.6, β = 1, γ = 2, ξ3 = −0.634 − 0.634i.

The Julia sets of the K¨ onig method, here ξ1 = −1 − 2i, ξ2 = −2 − i.

three nodes of a regular triangle, that is l1 = l2 = l3 . Now the higher the order of the root is the bigger the basin is. (In Fig. 7c the area of A(ξ3 ) with the order γ = 2 is the biggest and the area of A(ξ1 ) and A(ξ2 ) are separated away by A(ξ3 ). The area of A(ξ1 ) with the order α = 0.6 is the smallest.) Theorem 3. When the three roots ξ1 , ξ2 , ξ3 and their orders possess an axial symmetry, and the extraneous fixed points have the same symmetry as well, then the Julia sets constructed by the methods such as the standard Newton’s method, the relaxed Newton’s method, the Newton’s method for multiple roots, the Collatz method, the Schr¨ oder method and the K¨ onig method possess a symmetry with respect to the axis.

Theorem 3 can be proved similarly as Theorem 2. There are a series of figures of the Julia sets which satisfy Theorem 3 such as Figs. 2b, 3b–3f, 4a, 4b, 4f, 5a, 5c, 6a and 7a, all of the figures possess a symmetry with respect to the axis.

3.7. The Julia Sets of the Steffensen Method Comparing with the above methods, the Steffensen method may be a little different. The Steffensen method does not require the computation of derivatives, so the complexity of computation is smaller. But the Steffensen method converges much slower at a large initial value, therefore there are a great deal of white points in the Julia sets of the Steffensen method which shows that the points here does not converge to any value. Here we use a very small precision for the judgement of the convergence, or else the judgement will fail as there are many false fixed points.

Set N = 20, and use the colors of green, red and yellow separately for A(ξ1 ), A(ξ2 ) and A(ξ3 ). We construct the Julia sets of the Steffensen method as in Fig. 8. The coordinates of Fig. 8 are [−3.5, 1.5] × [−3, 1]. Set the positions of ξ1 , ξ2 and ξ3 as the three nodes in the black triangle in Fig. 1a and let α = β = γ = 1, then we obtain the Julia sets of the Steffensen method (Fig. 8a). Move ξ3 vertically along the y-axis we get Figs. 8b and 8c. As we move ξ3 upwards away from ξ1 and ξ2 , the area of A(ξ3 ) also gets away from the area of A(ξ1 ) and A(ξ2 ), and the structure of the Julia sets on the boundary of the basins becomes simple (Fig. 8b). When we move ξ3 downwards near ξ1 and ξ2 , the area of A(ξ1 ), A(ξ2 ) and A(ξ3 ) interweaves with each other, and the structure of the Julia sets on the boundary of the basins becomes very complex (Fig. 8c). Set α = γ = 1 and β = 1.2, we get Fig. 8d. The Steffensen method is quadratically convergent at a simple root and linearly convergent at a multiple root, so the color of A(ξ2 ) in Fig. 8d is lighter which shows that the convergence is slower here. As β increases, the area of A(ξ2 ) enlarges greatly. If N is big enough we could observe the complete A(ξ2 ). The Julia sets of Steffensen method is not coincident with our above conclusions.

3.8. The Effect of the Scope of the Principal Phase Angle If there is a decimal fraction in the values of α, β and γ, then the Julia sets of the generalized Newton’s method depend on the different scope of the principal phase angle. The phase angle of the Julia sets before this section is [−π, π). Assume there is an instance of Eq. (1), f (z) = (z 2.2 + 2i)z 0.5 .

(12)

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Fig. 8

(a) α = β = γ = 1, ξ3 = 0.

(b) α = β = γ = 1, ξ3 = 0.5i.

(c) α = β = γ = 1, ξ3 = −0.5i.

(d) α = β = 1 γ = 1.2, ξ3 = 0.

The Julia sets of the Steffensen method, here ξ1 = −1 − 2i, ξ2 = −2 − i.

Set N = 20, and use the colors of red, yellow and green separately for different roots, we construct the Julia sets of the Newton’s method for multiple roots as in Fig. 9. The coordinates of Fig. 9 are [−10, 10]× [−8, 8]. We utilize the DeMoivre theory for the computation of z α , z β , z α−1 and z β−1 , for example z α = z α (cos αθ + i sin αθ).

(13)

Here the computation of z α depends on the choice of the phase angle. In this paper we use four different scopes of the principal phase angle, i.e. [0, 2π),[−3π/2, π/2), [−π, π) and [−π/2, 3π/2). If α and β are positive integers, there is no effect on the use of Eq. (13) as there is cos(αθ) = cos(αθ + 2πα) . (14) sin(αθ) = sin(αθ + 2πα) When α is a positive decimal fraction, the different sets of θ will cause a different evolution of the Julia sets. Moreover if the value of αθ exceeds the above four scopes, we can adjust it by increasing or decreasing the integer multiple of αθ, which causes the rupture of the Julia sets. Because of the discontinuity of the phase angle, the rupture takes place at the positive x-axis, positive y-axis, negative x-axis and negative y-axis (Fig. 9).

4. CONCLUSION The research works of Peitgen and Gilbert are generalized. The theory of the Julia sets is analyzed and the characters of the fixed points of the generalized Newton’s method are given out. Then the Julia sets of the generalized Newton’s method are constructed using the iteration method. From this study we discover that: (1) the basins of attraction and its inner structure of the Julia sets of the generalized Newton’s method depend on the positions and the orders of the roots and also the existence of the extraneous fixed point; (2) if the orders of the roots are equal, and the distribution of the roots and the extraneous fixed point possess a rotational symmetry, then the Julia sets possess a rotational symmetry; (3) if the distribution of the roots and the orders of the roots, and the distribution of the extraneous fixed point possess axis symmetry, then the Julia sets have the axis symmetry; (4) take no account of the extraneous fixed points, if the roots are distributed evenly on the same cycle, then the higher the order is, the bigger the basin is; (5) take no account of the extraneous fixed points, if the orders of the roots are equal, then the size of the basins depends on the distances between the roots; (6) if there is an extraneous fixed point, then the basins of the root which is near the extraneous fixed

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Fig. 9

(a) θ ∈ [0, 2π).

(b) θ ∈ [−3π/2, π/2).

(c) θ ∈ [−π, π).

(d) θ ∈ [−π/2, 3π/2).

335

The Julia sets of different scopes of the principal phase angle.

point will be enlarged; (7) the Julia sets constructed by the Steffensen method are not consistent with the above regularity; and (8) if the order of the root is a decimal fraction, then the different sets of the principal phase angle will cause different evolutions of the Julia sets.

ACKNOWLEDGMENTS This research was funded by the Chinese National Natural Science Foundation (No: 60573172) and the Natural Science Foundation of Liaoning province (No. 20040081).

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