Fractal Characteristics of Newton's Method on

Report no. 96/14

Fractal Characteristics of Newton's Method on Polynomials M. Drexler I. J. Sobey Oxford University Numerical Analysis Group C. Bracher Technical University at Munich Department of Theoretical Physics In this report, we present a simple geometric generation principle for the fractal that is obtained when applying Newton's method to nd the roots of a general complex polynomial with real coecients. For the case of symmetric polynomials z ? 1, the generation mechanism is derived from rst principles. We discuss the case of a general cubic and are able to give a description of the arising fractal structure depending on the coecients of the cubic. Special cases are analysed and their characteristics, including scale factors and an approximate fractal dimension, are derived. The theoretical results are con rmed via computational experiments. An application of the theory in turbulence modelling is presented. Key words and phrases: Newton's Method, Fractals, Iterative Mappings, Polynomials Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building Parks Road Oxford, England OX1 3QD E-mail: [email protected]

November, 1996

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Contents

1 Introduction 2 The Symmetric Newton Fractal of Order 

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De nitions and Preliminaries : : : : : : : : : : : : Classical Analysis : : : : : : : : : : : : : : : : : : The Inverse Newton Mapping and its Properties : Fractal Map and Properties for the General Cubic 3.4.1 Fractal properties for the General Cubic : 3.4.2 Appearance of the Fractal : : : : : : : : : 3.5 Analysis of Special Cases : : : : : : : : : : : : : : 3.5.1 Julia Set Degeneracy : : : : : : : : : : : : 3.5.2 Root Degeneracy : : : : : : : : : : : : : :

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2.1 2.2 2.3 2.4

De nitions : : : : : : : : : : : : : : : : : : : : : General Properties : : : : : : : : : : : : : : : : Structural Results from Classical Root Analysis Generation via a Rotational Basis : : : : : : : :

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3 The Newton Fractal of a General Cubic 3.1 3.2 3.3 3.4

34 34 34 36 37 37 39 43 44 45

4 Numerical Experiments and Applications

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5 Concluding Remarks

58

4.1 Box-counting the Fractal Dimension : : : : : : : : : : : : : : : : : 50 4.2 Analysis of Local Solvers for the Turbulent k ?  Equations : : : : 54

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List of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

The Newton fractal  = 3 in the interval [?2; 2]  [?2; 2] : : : : Location of the Newton polynomial's largest root via two circles Sector Partition of the complex plane : : : : : : : : : : : : : : Generation of the Julia point structure via axis mapping : : : : Moduli of the rotational basis vectors : : : : : : : : : : : : : : Basins of attraction for z5 = 1. : : : : : : : : : : : : : : : : : : Basins of attraction for cubic, d = drs ? 0:7; c = 1. : : : : : : : Basins of attraction for cubic, d = drs + 0:7; c = 1. : : : : : : : Basins of attraction for cubic, d = 0:7; c = 1. : : : : : : : : : : A hypothetical fractal loop enclosing a root : : : : : : : : : : : One-dimensional restrictions of cubics : : : : : : : : : : : : : : Dierent fractal shapes for general cubics : : : : : : : : : : : : Asymptotic angles for cubic fractals : : : : : : : : : : : : : : : Besicovich Fractal for the root-degenerate cubic : : : : : : : : : Grid arrangement for Box Counting : : : : : : : : : : : : : : : Box-counting plot for the symmetric third-order Newton fractal Box-counting plot for the Besicovich fractal : : : : : : : : : : :

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9 18 21 23 31 32 32 33 33 40 41 41 43 47 51 52 53

Conditions for p roots of the symmetric Newton polynomial on the unit disc : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Box-counting results for the symmetric third-order Newton fractal Box-counting results for the Besicovich fractal : : : : : : : : : : : Convergence for various starting guesses on a k ?  type cubic : :

16 51 53 56

List of Tables 1 2 3 4

4

Description of the Colour Plates Fig. 6

Basins of attraction for z 5 = 1, using the orthodox Newton method. Array of 300  300 equidistant points cast over [?1:5; 1:5]  [?1:5; 1:5]. An oset of k
50 has been added to the actual iteration number according to the converged solution. Colouring according to converged solution  blue for root (1; 0), iteration range 1 . . .45,  green for root e  , iteration range 50 . . .95,  orange for root e  , iteration range 100 . . .145,  yellow for root e  , iteration range 150 . . .195,  red-brown for root e  , iteration range 200 . . .245. 2 5

4 5

6 5

8 5

Fig. 7

Basins ofattractionfor z 3 + dz ? c = 0, using the orthodox Newton method. Coecients d = ? p34 + 0:7 ; c = 1. Array of 240  240 equidistant points cast over [?2; 2]  [?2; 2]. An oset of k
85 has been added to the actual iteration number according to the converged solution. Colouring according to converged solution  blue for positive root, iteration range 1 . . .80,  yellow for large negative root, iteration range 85 . . .160,  green/red for small negative root, iteration range 170 . . .250. 3

Fig. 8

Basins ofattractionfor z 3 + dz ? c = 0, using the orthodox Newton method. Coecients d = ? p34 ? 0:7 ; c = 1. Array of 240  240 equidistant points cast over [?2; 2]  [?2; 2]. An oset of k
85 has been added to the actual iteration number according to the converged solution. Colouring according to converged solution  blue for positive real root, iteration range 1 . . .80,  green/red for negative root with =(z) > 0, iteration range 85 . . .160,  yellow for negative root with =(z) < 0, iteration range 170 . . .250. 3

Fig. 9

Basins of attraction for z 3 + dz ? c = 0, using the orthodox Newton method. Coecients d = 0:7; c = 1. Array of 240  240 equidistant points cast over [?2; 2]  [?2; 2]. An oset of k
85 has been added to the actual iteration number according to the converged solution. Colouring according to converged solution  blue for positive real root, iteration range 1 . . .80,  green/red for negative root with =(z) > 0, iteration range 85 . . .160,  yellow for negative root with =(z) < 0, iteration range 170 . . .250.

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1 Introduction Despite being a widely used algorithm to solve non-linear systems, little is known about the global behaviour of Newton's method. Practitioners usually rely on the local Newton-Kantorovich convergence result [21] and employ Newton's method as a local solver depending on some suitably chosen residual. If the residual does not decrease as speci ed, a globally more stable method is used to nd a better starting estimate for the solution. Some strategies of this kind can be found in [16], [19]. Another approach is to stabilise Newton's method by operations on the variable shifts ([5], [8], [12]). Such a method might be stable for some applications, but good convergence is not guaranteed due to the existence of local minima that can slow convergence down. One aspect of a globally applied Newton method is that the basins of attraction for dierent roots of a non-linear problem might have fractal boundaries. In addition to possible singularities of the problem, this introduces orbits into the convergence history that can slow convergence down considerably. As the fractal structure has not been completely understood, this is used as an indicator that Newton's method has globally unpredictable behaviour. However, it has been shown in [6] for the complex cubic that the convergence behaviour of Newton's method can be explained once the underlying fractal problem has been understood. Historically, the problem of applying Newton's method to complex polynomials has been rst addressed by Schroeder in 1871. Given a quadratic z2 ? 1, he asked to which of the two roots an arbitrary starting point in the complex plane would converge. The solution to this question on the boundary of the basins of attraction was the imaginary axis, found immediately by himself. In 1879, Cayley asked the same question for a symmetric cubic z3 ? 1, not being able to give an answer. The question became known as Cayley's problem . In a substantial paper, Gaston Julia ([11]) used this problem as an example of describing sets that later came to bear his name. The Julia set may be described as the union of all points that are eventually mapped onto a singular point. An exact de nition is given later. Julia also derived some properties of the set solving Cayley's problem, namely a re ective symmetry with respect to the real axis, and invariance under rotations by multiples of 23 and the inclusion of z = 0 and z = 1 as images of each other. Also, the fragmented character of the set was mentioned in the sense that it could not immediately described by Jordan curves. However, a comprehensive solution to Cayley's problem was not given. Almost seven decades later, interest in iterated polynomial mappings and their properties re-emerged, but much attention was devoted to the Mandelbrot set ([2], [13], [17]) and Newton's method was only treated in a historical context and as an accessible example to introduce the concept of Julia sets [18]. For the numerical use of Newton's method, some eorts were made to bound the number of iterations needed for convergence from any starting point ([14], [20])

6 and to show convergence in a statistical sense. However, a comprehensive solution to Cayley's problem still was not available. The aforementioned research focused on the general properties of Julia sets rather than on speci c properties of Newton's method. By changing this approach and examining the mechanism by which Newton's method generates a fractal structure, a rst comprehensive solution to Cayley's problem was given in [6], deriving all characteristic scale factors and symmetries, and uniting these results to give a fractal dimension of the structure. In addition, an approximation to the fractal structure was given that consists only of Jordan curves and thus can be computed without the extensive use of computer graphics. This work will continue on the lines set out in [6]. It will generalise the results of the symmetric cubic z3 ? 1 to the general symmetric polynomial z ? 1. The fractal structure will be explained from rst principles and a quantitative description will be stated as far as analytic solutions permit this. Two dierent approaches will be taken for the generation of the fractal structure, both leading to the same result. One approach relies on the classical root analysis, whereas the other uses a rotational representation of the solutions to polynomials similar to the one presented in [6]. The rotational approach to us seems more intuitive and generalisable to other problems. We will also state a way to approximate the fractal structure by Jordan curves. The quantitative results stated include both local and global symmetries, scale factors and a general way to estimate the fractal dimension of the structure. In the second part, we will restrict the analysis to general polynomial problems of degree 3. A general 'fractal map' is established, showing how the appearance of the fractal varies with the parameters of the cubic. Again, characteristic properties are stated quantitatively. Special cases are examined and are shown to be transitionary states between the three general fractal types that are possible for general cubics. The symmetric cubic is one of these cases. Another case is shown to yield a fractal of Besicovich type, that has so far not been associated with Newton's method. The fractal dimension for this fractal is computed using the previously established theory for the generation of polynomial Newton fractals. The third part will verify the theoretical results on the fractal dimension using an experimental box-counting approach and show how the results on general cubic fractals can be applied directly to a problem in turbulence modelling. It will be shown that for this problem, no stabilisations of Newton's method are necessary if the starting point is chosen in accordance with the fractal results. These results, however, contradict the physical intuition that is commonly used to nd a starting approximation. The superior numerical behaviour of such a counter-intuitive guess is demonstrated in experiments. The work concludes with a brief summary of the results. We will also discuss the numerical relevance of the theoretical results on the fractal structure.

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2 The Symmetric Newton Fractal of Order  As mentioned above, it is well established that the boundaries of the basins of attraction for dierent roots of certain polynomials are fractals. We will rst give the necessary de nitions for a technical discussion. For a certain class of polynomials, properties for the associated fractals will be stated in a quantitative fashion. For a detailed derivation and an example of the signi cance of these properties in the case of a cubic polynomial, we refer to [6]. We will then present a general construction principle for the fractal structure that has not appeared in the literature and can be used to explain most of the fractal features.

2.1 De nitions

This section will be discussing a special class of polynomials de ned in the following fashion. De nition 2.1 The symmetric polynomial of order  is de ned by

z = 1; z 2 C;  2 N;  > 2: (2.1) We will examine the basins of attraction for the roots of these symmetric polynomials when Newton's method is used as a numerical procedure for root- nding. Throughout this report, we will be concerned with Newton's method de ned in the standard way. De nition 2.2 The orthodox Newton method on f (z) is de ned by the iteration (k)   z(k+1) = z(k) ? ff ((zz(k))) = g z(k) (2.2) z with z (k) denoting the k th iterate, and fz denoting dzdf . It is noted from the de nition that the roots zk of f (z) with f (zk ) = 0 are stable xed points of the Newton iteration. The view we shall take of the Newton method is slightly dierent to that implied by (2.2). Rather than nding the next iterate given a starting point, we shall ask which points z(k) = z are mapped into a given point z(k+1) = z0 by (2.2). We therefore de ne De nition 2.3 The complex Newton polynomial of order  is de ned by ( ? 1)z ? z0z?1 + 1 = 0; z; z0 2 C (2.3) Despite the existence of a 'common-sense concept' of a fractal, it can be dicult to give a general de nition for such a structure. Following Mandelbrot [13], a strict de nition of a fractal is

8

De nition 2.4 A fractal is a set for which the Hausdor-Besicovitch dimension

strictly exceeds the topological dimension. This de nition, however precise, is hard to apply in practice if the HausdorBesicovitch dimension is dicult to determine or even unknown - which in fact is the case for many well-known fractals. Therefore, alternative and more intuitive de nitions of a fractal are in use. For the purposes of this study, we use 'fractal' according to the de nition by Falconer [7]. De nition 2.5 We refer to a set S as a fractal with the following in mind.  S has a ne structure, i.e. detail on arbitrary small scales.  S is too irregular to be described in traditional geometrical language, both locally and globally.  Often S has some form of self-similarity, perhaps approximate or statistical.  Usually, the 'fractal dimension' of S (de ned in some way) is greater than its topological dimension.  In most cases of interest S is de ned in a very simple way, perhaps recursively.

A suitable working de nition for the purposes of this study is De nition 2.6 The Newton fractal of order  is de ned by the union of all points that are mapped into the singular origin by the Newton mapping (2.3). It is obvious from this de nition that with z belonging to the Newton fractal, z0 also belongs to the fractal. A picture of the Newton fractal in the vicinity of the origin can be seen in Fig. 1. It is worth noting that the fractal is the union of points that lie on the boundary of the basins of attraction, and does not consist of the basins themselves. Fig. 5 also provides a good illustration for most of the points in de nition 2.5. As this study is concerned with the iteration of a function of a complex variable, we will refer to the underlying framework of Julia set theory. Following Falconer [7], we de ne the necessary sets as follows. De nition 2.7 The Julia set J (g) of a complex-variable function g is the closure of the set of repelling periodic points of g . De nition 2.8 The Fatou set F (g) of a complex-variable function g is the complement of the Julia set J (g). F (g) is also known as the stable set of g . Using this, we could equivalently de ne the Newton fractal as the union of all Julia points of the Newton mapping. An important notion for identifying Julia points on the fractal structure will be their order.

9 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Figure 1: The Newton fractal  = 3 in the interval [?2; 2]  [?2; 2]

De nition 2.9 The order of a speci c Julia point on a Newton fractal is de ned

as the number of Newton iterations it takes to reach the origin from that point.

An excellent overview of the Julia set theory concerned with polynomials of a complex variable can be found in Falconer's book [7]. In this work, we want to highlight one particular theorem that is very instructive for understanding the character of fractals associated with polynomials and Newton's method. We rst have to de ne the important concept of a basin of attraction. De nition 2.10 The basin of attraction A(w) of an attractive xed point w of a function g is de ned by A(w) = fz 2 C : g k (z ) ! w as k ! 1 g. With this de nition, we are able to state the theorem. Theorem 2.11 Let w be an attractive xed point of g. Then, denoting the boundary of the basin of attraction A(w) by @ A(w), we get: J (g ) = @ A(w). The same is true if w = 1. For a proof and further background on Julia set theory, see [7]. One implication of the theorem is that any point of the Julia set must lie on the boundary of all basins of attraction for all attractive xed points of g. Thus, an approximate close to a Julia point with only a small perturbation might converge to any of the roots. Further aspects of this theorem will be discussed in later sections. A colour plot of the basins of attraction for a polynomial of degree 5 illustrating the fractal character and theorem 2.11 is given in Fig. 6.

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2.2 General Properties

After having prepared the ground, we can now present a more technical analysis of the general inverse mapping (2.3) and the emerging Newton fractal. The necessary results are derived brie y, referring to [6] for a more detailed discussion and the example  = 3.

Classical Results It follows directly from (2.1) that the roots associated with

the vth-order symmetric polynomial are

(2.4) zk = e  (k?1)i; k = 1; 2; . . . ; : The primary Julia point that is caused by a singularity of the derivative in (2.2) is always 0 = 0: (2.5) 2

Parent Structure It can be easily obtained from (2.3) that the rst-order Julia points which arise from applying the inverse Newton iteration to the origin once are 1;k = p 1
e  (2k?1)i; k = 1; 2; . . . ; : (2.6) ?1 Therefore, their distance from the origin for the  th order Newton fractal is (2.7) r = p 1 ;  ?1 approaching 1 for large  . Comparing (2.4) and (2.6), it can be seen that the rst-order Julia points always lie between two neighbouring roots. As we shall see later, these points de ne the parent structure that contains the whole information needed to describe the fractal. The concept of the attractive circle that was a very useful approximation for  = 3 cannot be extended for general  , as the parent structure contains points with a greater radius than r. However, all the concepts associated with the attractive circle generalise. Global Symmetry By inspection, it can be established that (2.3) is invariant

under the coordinate transformation

m  i ; z0 7! z0e  i m 2 N: z 7! ze m (2.8) This is equivalent to subsequent rotations by 2 with the origin as a centre. Furthermore, we can see that (2.3) is invariant to 2

2

z 7! z; z0 7! (z0);

(2.9)

11 with z denoting the conjugate of z. The combination of rotational and re ective symmetry dictates that the branches of the fractal lie on straight lines. Therefore, the Newton fractal of order  has a  -fold global rotational symmetry and hence consists of  branches. It also has a re ective symmetry regarding the real axis. The  branches lie along straight lines.

Global Scale Factor For large z; z0 (jzj; jz0j  1), the governing equation (2.3) can be expanded as

  z0 =  ? 1 z + O z1? :

(2.10)

As both z and z0 are contained in the fractal, the fractal is invariant under the scaling z 7! ;1 z; ;1 =  ? 1 : (2.11) It has to be noted that this exact invariance is only achieved in the limit jzj ! 1. As an approximation, it may however be used much earlier. We therefore state that the Newton fractal of order  has a global scale factor of ;1 .

Local Symmetry Following the result on global symmetry, we de ne the global symmetry axes Ls to be the lines through the origin and the rst-order

Julia points as de ned in (2.6). We parametrise the axes by

z0 = eis ; s =  + 2 s; 0   < 1: Approximating the solution to (2.3) for large z0 by ! ? ?? 1 z = ?pz + O z0 ; 0 2

(

1 1)2

1

we obtain for the sth global symmetry axis and   1 zsm( )  ?q1
e ?i s
e ?i m ; m = 0; 1; . . . ; ( ? 2): j j 1

1

2

1

(2.12) (2.13)

(2.14)

Again, the zsm asymptotically approach the origin on straight lines for large  . The succession of Julia points on these lines will be discussed in relation with local scale factors. The angle !sm , from which the zsm approach the origin, can immediately be written as !sm =  ( ? 1) [1 + 2(m + s)] : (2.15)

12 By inspection, (m + s) ranges from 0 to  ( ? 1) ? 1, and therefore the angles are  k; k = 0; 1; . . . ;  ( ? 1) ? 1: (2.16) !k =  ( ? 1) +  (2? 1) It can be seen that the angles !k = 0 and !k =  never occur. The interpretation of this formula is that  ( ? 1) locally identical branches of the fractal approach the origin with an angle of   =  (2? (2.17) 1) between neighbouring branches. An example of this discussion for  = 3 is given in section 3.4.1 of [6]. We see that the local symmetry is an integer multiple of the global symmetry. This property is characteristic for all Newton fractals. We note that, as each Julia point is a locally conformal image of the origin, the local symmetry holds throughout the fractal. We can summarize that the Newton fractal of order  is locally invariant to rotations by  and exhibits  ( ? 1)-fold local rotational symmetry at each point.

Local Scale Factors We note that (2.13) yields small z for large z0 and therefore is suitable for approximating the local behaviour at the origin. Considering the geometric progression z0; ;1 z0; (;1 )2 z0; . . .

(2.18)

that describes Julia points for a suitable starting point jz0j  1, the progression mapped close to the origin by (2.13) is 1 1 ; q1 q ;...: (2.19) ; p  ? z0  ?  z 2 ;1 0  ? (;1 ) z0 1

1

1

This again is a geometric progression; the scale factor can be determined as s q ?  ? 1 ? ?1  = = (  (2.20) ;1 ) :  We see that  is rapidly approaching 1 as  grows  1  ?  1 ?  ( 1? 1) : (2.21)  = 1 ?  The geometrical interpretation of this result is that the Julia points on a chain of order n (obtained from the negative real axis after n inverse Newton mappings) 1

1

1

1

13 approach the Julia point of order (n ? 1) in a geometrical progression with factor  . This holds particularly for the straight lines approaching the origin that were described in the preceding paragraph. As  is growing, the Julia points on that branch are more equally spaced and, as there is an in nite number of them on each branch, they appear more densely crowded. Following an argument in [6] and using the previously established properties, we can state that the parent structure of the fractal will be a blob bordered by two fractal chains and a number of chains running inside, the total number of chains per blob being ( ? 1) due to the results on symmetry. Aside from each chain having a scale factor  along itself, there exist scale factors that determine the local scaling when a point changes from one branch to the other in a Newton step, the so-called cross-chain scale factors ;k . It is not possible to state general expressions for all cross-chain scale factors involved. According to [6], we can give an expression for the scale factors associated with xed points of cycle  1 ;2;k = ; k = 1; 2; . . . ;  ? 1: (2.22) 2( ? 1) sin k Although not being able to state them explicitly, we can give an argument for the number of dierent cross-chain scale factors that should occur. It can be noted that the blobs of the  th order Newton-fractal consist of ( ? 1) branches which are arranged symmetrical around the centre line of the blob. If  is even, the centre line coincides with one of the branches. Furthermore, the blobs on each branch (which are relevant for the next step of the inverse mapping) are composed of ( ? 1) sub-branches. This leads to a total number of ( ? 1)2 subbranches. We now take the symmetry with respect to the centre line into account, in particular the fact that for even  , one (the centre) branch is symmetrical in itself. This reduces the number of dierent sub-branches to h i (2.23) b = b 21 ( ? 1)2 + 1 c; with each of which a cross-chain scale factor should be associated. b
c denotes the integer truncation operator (e.g. b2:8c = 2). Again, due to the local conformity of the inverse Newton mapping, we can generalise these results for the origin for all points of the fractal. The Julia points of order n approach the point of order (n ? 1) on a branch with local scale factor  as a geometrical progression. There are b 21 ( ? 1)c dierent cross-chain scale factors ;2;k for cycle  and b dierent cross-chain scale factors in general.

Fractal Dimension Following an argument presented in [6], we state an ap-

proximation to the fractal dimension of the  th -order Newton fractal involving all the dierent scale factors. If the fractal dimension is d, it must satisfy the

14 following equation b X k=1

(;k )d + d = 1:

(2.24)

This equation was derived on the assumption of geometric progressions throughout the fractal chains and therefore only is an approximation to the real fractal structure. However, we consider it a rather useful and elegant one. Comparisons of the obtained fractal dimensions with box-counting experiments will be presented in the section on numerical experiments. The equation corresponds to the one stated in [2] for hyperbolic systems of iterated functions and derived independently there. It gives the Hausdor-Besicovich dimension for the attractor of disconnected or just-touching systems, a class to which the Newton fractals belong.

2.3 Structural Results from Classical Root Analysis

Being a polynomial of degree  with complex coecients, the Newton polynomial can be subjected to an analysis concerning the location of its roots. This will give results on the fractal structure and can explain the appearance of the symmetric Newton fractal of order  . We start by establishing bounds on the moduli of the roots and then determine approximations for their arguments. As most of the theorems used to determine the results are well established in complex analysis, we refer to the literature for proofs and the general statement of the theorems. Without loss of generality, it is assumed that the roots can be ordered according to their modulus as

jz1j  jz2j  . . .  jz j:

(2.25)

When quoting a theorem from the literature, we will assume that the polynomial is written in the form  X p(z) = ak zk (2.26) k=0

with coecients ak 2 C, unless otherwise stated.

Bounds for the root of largest modulus Dierentiating (2.3) with respect to z, we obtain

fz (z) =  ( ? 1)z?2 (z ? z0) :

(2.27)

This derivative has roots at

f1 = z0g _ fk = 0g;

k = 2; . . . ;  ? 1:

(2.28)

15 Knowing the location of the roots of the derivative fz (z), the Gauss-Lucas theorem (Theorem 6.5a, [10]) states that they must lie in the convex hull of the set of zeros of the polynomial f (z). Therefore, for the given geometry of derivative roots, at least one zero of the Newton polynomial must have modulus larger than jz0j (and be located in roughly the same direction in the complex plane as z0). Furthermore, for a polynomial of the form (2.26), the zero of largest modulus can be bound by (Theorem (27,3), [15])

  jz1j  p 2?1 ; a  ?k k  = 1max k a C (; k ) 1

(2.29)

with C (; k) = k!(?! k)! denoting the binomial coecient. Substituting the coecients ak of the Newton polynomial, it is obtained for  ! j z 0j p 1 : (2.30)  = max  ? 1 ;  ?1 The rst bound in the maximum operator can be discarded in favour of the sharper bound obtained from the above argument using the Gauss-Lucas theorem. We can therefore state the following Lemma 2.12 The zero of largest modulus z1 of the Newton polynomial (2.3) can be bounded from below by ! 1 jz1j  max jz0j; p  ? 1 : (2.31) For a more convenient upper bound, we examine p (2.32) g(z) =  2 ? 1 ? 1 to nd it has a root at  = 1 and a local minimum with g(z) < 0 where p  gz (z) = 12  2 ln 2 ? 1 = 0: (2.33) Zero is approached from below as  ! 1. Hence, we can state for the region of interest  > 2 p 2 ? 1 < 1 (2.34) and therefore establish

16

Lemma 2.13 The zero of largest modulus z1 of the Newton polynomial (2.3) can

be bounded from above by

jz1j  p 21? 1 max jz?0j1 ; p 1? 1 For large  and jz0j, this bound approaches jz1j   ? 1 jz0j:

!

(2.35) (2.36)

Bounds for the remaining ( ? 1) roots Following a result by Cohn ([15], p.130), a polynomial of the form (2.26) has exactly p zeros on the unit disc, if  X japj > jak j(1 ? kp): (2.37) k=0

The various possibilities for p substituting the coecients of the Newton polynomial (2.3) are given in Table 1.

japj P jak j(1 ? kp) condition for jzpj  1 ?1  jz0j + 1 jz0j < ? 2 ?1  jz0j  jz0j > 1 1 1, there exists a real root z1 with jz1j > jz0j and an argument identical to that of z0. We therefore have Lemma 2.17 For z0 being a Julia point on one of the fractal's global symmetry (2

(2

axes,

arg(z1) = arg(z0):

(2.44)

18 For later discussion, it is also noted that with the coecients in (2.42) being real, all roots must be in conjugate complex pairs. Recalling the de nition of Newton's method (2.2), the Newton polynomial can also be stated as p(z) ?  z?0 1 pz (z) = 0; p(z) = z +  ?1 1 : (2.45) In this form, a result by Walsh and Marden (Corollary (18,1), [15]) can be employed directly, stating that the roots of (2.45) must be located within the disc that contains the roots for p(z) and its translation by z?1 . We therefore obtain Lemma 2.18 The roots of the symmetric Newton polynomial (2.3) lie in the union of the discs with radius  and center ck , where  = p 1 ; c1 = 0; c2 =  ? 1 z0: (2.46) ?1 0

y

y

c2 z0

z0 x

c2 x

Figure 2: Location of the Newton polynomial's largest root via two circles Knowing from lemma 2.12 that the root of largest modulus must lie in the circle with centre c2, we can use a geometrical argument on the intersection of the two circles with radius  and displacement jc2j and obtain Lemma 2.19 The argument of z1, the root of largest modulus of the symmetric Newton polynomial can be bounded from above by p arg(z1)  arg(z0)  arccos j2c2j ; jc2j  2 (2.47a) q 2 2 p jc2j ?  arg(z1)  arg(z0)  arccos jc j ; jc2j  2: (2.47b) 2

19 It can be seen that for large jz0j, the argument of z1 approaches that of z0. To further improve the bound on the argument for small z0, we state the following Lemma 2.20 The lines through the origin and the roots of the symmetric polynomial (2.1)

z = e k i ;  > 0; k = 0; 1; . . . ;  ? 1 2

(2.48)

contain no Julia points.

We note that this lemma can be used for an elegant proof of theorem 2.11, taking into account that every point of the fractal is a locally conformal image of the origin and the origin connects to each root in a straight line that contains no Julia points. Proof Without loss of generality, we restrict the proof to the case k = 0, the

positive real axis. The proof will then hold for all the other lines due to the rotational symmetry of the structure. We consider the Newton iteration (2.2) on the real axis   a =  ?  ? 1 = ( ? 1) + 1 : (2.49)

 ?1

 ?1

As a > 0 for  > 0, the positive real axis is mapped onto itself under the Newton iteration. Furthermore,

d2 f =  ( ? 1) ?2 (2.50) dz 2 for the symmetric polynomial of order  , every point on the positive real axis converges into the root  = 1 due to convexity. As the positive real axis is mapped onto itself and contains only convergent points, it must be fully contained in the Fatou set and therefore cannot contain any Julia points. 2

This result will be needed again when establishing bounds on the remaining roots of the Newton polynomial as the lines through the roots of the symmetric polynomial constitute a separatrix of the fractal. For the root of largest modulus, it helps establishing Lemma 2.21 The argument of the largest root of the symmetric Newton polynomial cannot exceed the upper bound (2.51) arg(z1)  arg(z0)   :

As the modulus of z1 is bounded from below by lemma 2.12, the above bounds on the argument con rm the 'prolongation in the direction of z0'.

20

Bounds on the argument of the ( ? 1) remaining roots To estimate

the location of the remaining roots of the symmetric polynomial, we have to consider the location of the coecients of (2.3) in the complex plane. We notice that a and a0 both are positive real, and only a?1 has a variable location, solely determined by z0. Using this information, we can state the important result (Theorem (41,3), [15]) Lemma 2.22 If = arg(z0), then the sector in the positive real half-plane de ned by arg(z) < 1 ( ? 2 ) (2.52) contains  roots of the symmetric Newton polynomial with   = 0, if 0;

 ?1

(2.79)

these solutions lie inside the parent structure mentioned in the previous subsection.  z0 small In this case, the term containing z0 in (2.75) can be neglected and the equation is restated as z = ?  ?1 1 : (2.80) This equation has the  rst-order Julia points 1;k as its solution. To obtain results for the rotational basis, the system Vr =  is solved, with  being the (complex) solution gained from the above approximations. Again, we will equate the 2-vector rm with the complex number rm 2 C. We consider two cases:

29

 z0 large

From the above, de ning

 = e ?i ; 2

it follows that

(2.81)

1

 ?2 #  1   (2.82) =  ? 1 z0; ?pz ; ?pz ; . . . ; ?pz 0 0 0 is obtained. Solving (2.61) for r, using lemma 2.27 and 2.28, this yields 2 3 2 z 3 z  ? 1 66 77 66 z?1 77 ?  ? ?1 z ! 1 6 7 p r = 666 ?1 + !? ?.. 1
 ? z 777 jz j1 6666 ?. 1 7777 : (2.83) .. 5 64 75 . 4 z !?  ? ?1 1 z  ?1 + !?  ? ?1
  ?pz  ?1

T

"

1

1

1

0

0

0

(

0

1)

1

1

(

1)2

(

1)

0

0

1

0

0

0

In the above calculation, the form of rb arising from the multiplication with the inverse of V was considered: X ?1 rb =  ? 1 z0 +  ?p1 z !?bk k 0 k=1 (2.84) ? b  ? 1 = ? 1 z0 +  ?pz
! !?b?1?1 : 1

(

1

1)

0

The numerator of this expression is only zero for b = 0. It is further noted by comparing the exponential exponents in the denominator, 2i = 2mi , ?b +  = m: ? 2bi + (2.85)  ?1  ?1 This cannot be true for b; m 2 N0 and therefore the denominator cannot be zero.  z0 small De ning a principal rotation i

=e ; we can write the solution  obtained above as k = p 1 2k+1; k = 0; 1; . . . ;  ? 1: ?1 For the bth element of r, obtained from r = V?1

(2.86) (2.87) (2.88)

30 using lemma 2.27, it is therefore obtained  ?b 2  ? 1 ! ?1 X rb = p 1 !?kb 2k+1 = p
(!?b 2) ? 1   ? 1 k=0  ?1 (2.89) = p  ?1 1b: The evaluation of the fraction obtained from the geometric progression follows an argument identical to that presented in the proof of lemma 2.27. It is noted that for the denominator to become zero, !?b 2 = 1; (2.90) and therefore by comparing exponential exponents (2.91) ? b + 1 = m; m 2 N0; has to hold. As 0  b <  , this can only be satis ed for m = 0 and b = 1, thereby giving rise to the above delta function. Another interesting result is gained from evaluating one of the Vieta root conditions on the solutions to (2.3). The condition states for the sum of all solutions  X (2.92) k =  ? 1 z0: k=1 Writing down the left-hand side in terms of a rotational basis, sorting by the rk , and using (2.65) for the geometric sums, we obtain ! X ?1 X  ?1 X bk k = !
rb =  r 0 : (2.93) k=1

b=0 k=0

From these results, we are able to summarize. Lemma 2.30 The rotational basis has the following appearance:  For large z0,

rm =  ?1 1 z0; all basis vectors being equal in the limit z0 ! 1.

(2.94)

 For small z0,

2 3 cos  1 m rm = p  ? 1 4  5 ; sin 

only the basis vector r1 being non-zero in the limit z0 ! 0.

(2.95)

31 In addition, for all z0, the stationary vector is determined by r0 =  ?1 1 z0: (2.96) We further postulate, supported by the results from classical root analysis that r1 will stay the dominant vector and is asymptotically approached by the other vectors as z0 increases in modulus. This explanation is consistent with all fractal properties observed. However, we have not been able so far to state a proof for this proposal from rst principles. |rk| n/(n-1)·|z 0 |

χ

k=1 1

k=... k=0

k=2

|z 0 |

Figure 5: Moduli of the rotational basis vectors

32 ABOVE

250

230 210 205 204 203 -

250 230 210 205 204

202 200 190 160 -

203 202 200 190

155 154 -

160 155

153 152 -

154 153

150 130 -

152 150

110 105 104 103 -

130 110 105 104

102 100 90 60 55 -

103 102 100 90 60

54 53 -

55 54

52 50 -

53 52

30 10 -

50 30

5432-

10 5 4 3

BELOW

1.0

0.5

0.0

-0.5

-1.0

2

-1.0

-0.5

0.0

0.5

1.0

Figure 6: Basins of attraction for z5 = 1. ABOVE

240

200 190 180 -

240 200 190

179 178 177 -

180 179 178

176 175 174 -

177 176 175

173 172 165 -

174 173 172

115 105 95 -

165 115 105

94 93 92 -

95 94 93

91 90 89 -

92 91 90

88 87 -

89 88

80 30 20 -

87 80 30

10 98-

20 10 9

765-

8 7 6

432-

5 4 3

BELOW

1

0

-1

2

-1

0

1

Figure 7: Basins of attraction for cubic, d = drs ? 0:7; c = 1.

33 ABOVE

240

200 190 180 -

240 200 190

179 178 177 -

180 179 178

176 175 174 -

177 176 175

173 172 165 -

174 173 172

115 105 95 -

165 115 105

94 93 92 -

95 94 93

91 90 89 -

92 91 90

88 87 -

89 88

80 30 20 -

87 80 30

10 98-

20 10 9

765-

8 7 6

432-

5 4 3

BELOW

1

0

-1

2

-1

0

1

Figure 8: Basins of attraction for cubic, d = drs + 0:7; c = 1. ABOVE

240

200 190 180 -

240 200 190

179 178 177 -

180 179 178

176 175 174 -

177 176 175

173 172 165 -

174 173 172

115 105 95 -

165 115 105

94 93 92 -

95 94 93

91 90 89 -

92 91 90

88 87 -

89 88

80 30 20 -

87 80 30

10 98-

20 10 9

765-

8 7 6

432-

5 4 3

BELOW

1

0

-1

2

-1

0

1

Figure 9: Basins of attraction for cubic, d = 0:7; c = 1.

34

3 The Newton Fractal of a General Cubic Having described the general symmetric Newton fractal, we now restrict the discussion to polynomials of degree 3, where it is possible to state the structure of a general solution using Cardan's formula. In this section, we will consider the case of a general cubic and describe the fractal structures that arise when Newton's method is used to numerically determine its roots. It will be shown that any cubic in combination with Newton's methods yields a fractal structure, the shape of which will be described depending on the coecients of the cubic. We will state classical results and analyse the inverse Newton mapping. Finally, special cases including the symmetric fractal will be identi ed as transition states of the general cubic fractal.

3.1 De nitions and Preliminaries

Throughout this section, we consider without loss of generality a cubic of the form z3 + dz ? c = 0; d; c 2 R; (3.1) to which Newton's method is applied to nd the roots. This gives rise to the Newton polynomial 2z3 ? 3z0z2 ? dz0 + c = 0; (3.2) using the notation introduced in the de nitions of the previous section, namely z denoting the current iterate z(k) and z0 the future iterate z(k+1).

3.2 Classical Analysis

Following the literature, it is rstly stated that given a general cubic of the form a3u3 + a2u2 + a1u + a0 = 0; ai 2 R; u 2 C (3.3) the representation (3.1) is obtained via the substitution u 7! z ? 3aa2 (3.4) 3 and division by a3, yielding for the real coecients in (3.1) 2 2a32 ? a0 : (3.5) d = aa1 ? 3aa22 ; c = a32aa21 ? 27 a33 a3 3 3 3 Allowing complex coecients, we can also restate (3.1) with a quadratic term, substituting z 7! v ?  v3 + hv2 ? e = 0; (3.6)

35 where

2 = ? 3d ; h = ?3; e = c ? 22:

(3.7)

The roots of the cubic (3.1) that are found using Newton's method can be stated analytically using Cardan's formula 1 = s1 + s2; (3.8a) p 2;3 = ? 21 (s1 + s2)  i 23 (s1 ? s2) ; (3.8b) with v v s 3 2 s 3 2 u u u u c c d c d +c : t t (3.8c) s1 = 2 + 27 + 4 s2 = 2 ? 27 4 It is easy to see from this representation that there is a critical value s2 drs = ?3 c4 (3.9) 3

3

3

with the three roots being real for d < drs, and the two roots z2;3 being conjugate complex for d > drs. Analysing the derivative of (3.1), we obtain for its roots s (3.10) 0;k =  ? 3d ; k = 1; 2

this specifying the primary Julia points 0;k of the fractal. Again, there is a critical value djs = 0; (3.11) with the primary Julia points being real for d < djs and conjugate complex on the imaginary axis for d > djs . From Vieta's condition on the roots of the cubic and its derivative, we can also state z1 + z2 + z3 = 0; ^ z1z2z3 = c 2 R; (3.12) the roots of the cubic (3.1) lie in dierent half-planes of the complex plane and their arguments add up to a multiple of 2. Furthermore, 0;1 + 0;2 = 0 ^ 0;10;2 = ? 3d 2 R; (3.13) the primary Julia points are conjugate complex and are located with equal distance to the origin on one of the axes in the complex plane.

36

3.3 The Inverse Newton Mapping and its Properties

Solving the Newton polynomial (3.2) for z, in a fashion similar to that suggested in [6], we obtain for the solution " # 2 1 p ( z 0) z1 = 2 ' + p' + z0 ; (3.14a)   p q 2 1 p 2 2 (3.14b) z2 = ? 4p' [z0 ? '] ? i 3 ' ? (z0) ;   p q 2 1 p 2 2 z3 = ? 4p' [z0 ? '] + i 3 ' ? (z0) ; (3.14c) with the nonlinearity r i h 3 (3.15) ' = (z0) ? 2 (c ? dz0) + 2 (c ? dz0) c ? dz0 ? (z0)3 : 3

3

3

3

3

3

3

3

It is noted that all dierences between this solution and the symmetric solution presented in [6] are incorporated in the nonlinearity. Using theorem 2.26, we know that (3.14) can also be stated using a rotational basis, x1 = b1 + b23 + t (3.16a) x2 = R120b1 + R240 b23 + t (3.16b) x3 = R240b1 + R120 b23 + t (3.16c) with xk = [xk ; yk ]T , and R120 and R240 denoting the rotation matrices about an angle of 23 and 43 , respectively. By identifying the rotational basis as introduced in the previous section in the following fashion, consistent with [6] and obeying the complex notation introduced in (2.57), 2 (3.17) b1 = 12 p'; b23 = 2(zp0)' ; t = 21 z0; it can be immediately veri ed that for the L2 norm, kb1k
kb23k = ktk2: (3.18) Therefore, t can never be the dominant vector and the basis rotations in (3.16) have an in uence on the location of the three solutions. We see from (3.16) and (3.17) that the generation process of the fractal is not dierent in principle from the previously discussed symmetric case. The only dierences rest with the nonlinearity ' that yields more distorted rotational vectors and the existence of two disjoint primary Julia points 0;k. The general mechanism of the inverse mapping, however, is unchanged. A point z0 will have three images under the inverse mapping: 3

3

37

 A prolongation 1 with 1 = 23 z0 in the limit for large jz0j.  One solution 2 in the parent structure on the same fractal chain as the

prolonged solution.  One solution 3 in the parent structure on the fractal chain that does not contain the prolonged solution. The k do of course not correspond in general to the zk with the same index k, but the index might be permuted. Due to the asymmetry of the rotational basis even for large jz0j, the rotated solutions 2;3 do not both converge into the same point as in the symmetric case. Their behaviour will be quanti ed in the next subsection.

3.4 Fractal Map and Properties for the General Cubic

In this subsection, we will rst analyse the properties of the fractal that arises from applying Newton's method to the general cubic (3.1). From these, we can state a 'fractal map' that relates the shape of the fractal structure to the parameters of the cubic.

3.4.1 Fractal properties for the General Cubic

Inspecting (3.1), it is immediately obvious that there is no global rotational invariant associated with the fractal structure similar to that of the symmetric case. Therefore, the only global symmetry that can be associated with the fractal is a re ective symmetry with respect to the real axis. To determine scale factors, we assume jz0j  fjcj; jdjg and consider two cases, according to the expected magnitude of the solution z.

Global Scale Factor Assuming jzj  fjcj; jdjg, we can simplify the Newton polynomial (3.2) to obtain

2z3 ? 3z2z0  0 , z  23 z0:

(3.19)

To verify the validity of this rst approximation, we perturb the solution z = 23 z0 +  (3.20) and substitute this perturbation into the Newton polynomial to obtain 9 z2 + 6z 2 + 23 ? dz + c = 0: (3.21) 0 0 2 0

38 Dropping higher-order terms, we obtain a decaying solution for ,   92zd : (3.22) 0 Therefore, a global scale factor (3.23) 3;1 = 23 is obtained. As in the symmetric case, the fractal structure grows like a geometric progression as jz0j gets larger. Equation (3.19) also states that for large jzj, the fractal structure will asymptotically approach a straight line.

Local Scale Factor In this case, we assume jzj to be small, thus simplifying the Newton polynomial (2.3) to obtain

?3z2z0 ? dz0  0 , z2  ? 3d :

(3.24)

We note that this constitutes the equation de nining the primary Julia points 0;k. To determine the scale factor, we perturb (3.24) z2 = ? 3d + ; (3.25) yielding in a rst-order approximation, assuming jj  d3 : s s   d d 3 z = i 3 ?   i 3 1 ? 2d  + . . . : (3.26) Dropping higher-order terms again, we expand !   9 d 3 2 ? d : (3.27) 2z  i 3 Substituting these results for z2 and z3 into the Newton polynomial and considering only zero- and rst-order terms in , we obtain q3 c  2i d3 q : (3.28) =? z0  i d3 3 2

Using (3.26), z can be expanded at the primary Julia points by 0 s 1 s d 3 1 1 d z  ? 3  z @ 2 ? d c ? 3 A : 0

(3.29)

39 As the sequence of large z approaches 1 like a geometric progression with scale factor 3;1, this gives rise to a similar sequence of small z (when substituting the sequence of large z as the z0, see [6]) approaching the primary Julia point in a geometric progression with scale factor 3 =  1 = 32 : (3.30) 3;1

As each Julia point of the fractal is just an image of one of the primary Julia points, this local scale factor prevails throughout the fractal. Throughout this calculation, we had to assume d 6= 0, and it is noted that this general scale factor is dramatically dierent from the symmetric case. The subsection on special cases for the cubic will relate the previous calculation to the case d = 0. In addition to an expression for the local scale factor, (3.29) also yields the result that for jz0j ! 1, the solutions converge to the two primary Julia points 0;k, respectively. This con rms the property of the inverse Newton mapping that one image of z0 will always change the fractal branch. Similar to the symmetric case, there are two cross-chain scale factors that can be determined via the evaluation of xed points of period 2 and 3. As their determination is rather tedious and they are not needed for further arguments, we omit their calculation here. It is pointed out, however, that they can be used to calculate an approximate fractal dimension and it is referred to [6] for an example of their determination in the symmetric case. We remark that throughout this chapter, we are concerned with the Julia points that are eventually mapped onto in nity. There however is another type of divergence associated with Newton's method, stemming from periodic cycles. These can lead to the inclusion of Mandelbrot-type fractals in the structure an explanation is given in [3] via the Douady-Hubbard theory. As this local phenomenon is known and explained, we shall not be concerned with it here and concentrate on the global description of the fractal structure.

3.4.2 Appearance of the Fractal

After having discussed most of the quantitative properties of the fractal associated with the general cubic, we will now give a global description of the fractal's topology. It will turn out that the shape of the fractal depends on the choice of the parameters c; d of the cubic, and at the end of the discussion, we therefore give a map that decribes the fractal structure according to the choice of parameter. From the de nition of the fractal, it is obvious that it must contain the primary Julia points. Also, it must 'contain' in nity - meaning that its branches must extend towards in nity. This is con rmed by the far- eld approximation (3.19) stating furthermore that each branch approaches in nity in a straight line. It is furthermore noted that for c; d 2 R, the fractal must be symmetric with respect to the real axis. This follows from the fact that with fz; z0g satisfying the

40 Newton polynomial (3.2), the conjugate complex pair fz; z0g also constitutes a solution. According to theorem 2.11, the fractal structure must separate the roots, because a Julia point (with all basins of attraction meeting) occurs whenever two basins of attraction meet. To conclude the preliminaries from which the map of the fractal is derived, we have to state the following Theorem 3.1 The fractal cannot contain a closed loop around any root in the complex plane.

Proof The de nition of the fractal asserts that if a point is contained in the Fatou

set, its image after one Newton iteration also belongs to the Fatou set. In the same fashion, a point contained in the Julia set is mapped onto another point of the Julia set within one Newton iteration - the fractal is self-contained and invariant. Due to the local conformity properties of the Newton mapping, we can generalise this result to paths in the Fatou set. A path that contains only points in the Fatou set will stay within the Fatou set. Without loss of generality, we now assume that there exists a fractal loop containing a Julia point p of order p around a root and that p is the Julia point with minimal distance to the root. A path is chosen that connects p with the root in a straight line. It is noted that this path contains only Fatou points in its interior. We consider the image of that picture after subjecting it to p Newton iterations. By the properties of the iteration, p is mapped into in nity. As the root is a xed point for the Newton mapping, it will remain unchanged. The fractal as a topological structure also is xed with respect to the Newton mapping. As the Newton mapping is locally conformal, the connected path will now connect in nity and the root. However, by the assumption that the root is enclosed by a fractal loop (which is dense by the fractal properties and theorem 2.11), it will contain another Julia point in its interior. This results in a contradiction, therefore proving that a closed loop around a root cannot exist. A geometrical illustration of the proof is given in Fig. 10. 2 y

y Newton

χp

x

χp →∞

x

Figure 10: A hypothetical fractal loop enclosing a root It remains to be shown that the fractal cannot constitute a loop around a root that connects at in nity with a near-zero angle on the Riemann sphere. For this purpose, we recall that the primary Julia points are mapped into in nity by the Newton mapping (2.2) and therefore are locally conformal images of in nity

41 under the inverse Newton mapping. In fact, by the same argument, every point of the fractal is a locally conformal image of in nity. Therefore, if the fractal branches joined at in nity with a near-zero-angle, they would have to branch out from every point of the fractal with the same angle, locally. Topologically, a dense structure with this property could not extend to in nity - therefore a non-vanishing angle has to exist at in nity by contradiction. From these principles and the results from classical analysis, the appearance of the fractal can be determined. It is stressed that the fractal appearance is solely governed by the location of the roots of the cubic (3.1) and its derivative. These in turn can be classi ed by conditions on the cubic coecients. f(x)

f(x)

f(x)

x

x

a)

x

b)

c)

Figure 11: One-dimensional restrictions of cubics α2

y

α1

α2 a) root

y

y α1

x

x

α1

α1

α1

α2

x α2

b)

α1

c)

primary Julia point

fractal

Figure 12: Dierent fractal shapes for general cubics

q d < drs (Real-Real condition) For d < ?3 c4 , the cubic (3.1) has three real roots, and its derivative 2 real roots separating the cubic roots. Therefore, the primary Julia points are located on the real axis, being of equal modulus and opposite sign. The fractal consists of two separate branches, each passing through a primary Julia point and approaching in nity. Each branch is located in a half-plane and is symmetric with respect to the real axis. Each branch approaches in nity with 3

2

42 an asymptotic angle 1; 2. In general, 1 6= 2, but as d ! ?1, the angles become equal for reasons of symmetry with the case d > 0. A schematic sketch of the one-dimensional cubic and the fractal is shown in Fig. 11a and 12a.

drs < d < 0 (Conjugate-Real condition) The cubic now has two conjugate complex roots and one real root, whereas the roots of the derivative are still located on the real axis. A schematic sketch of the real restriction of such a cubic is shown in Fig 11b. The two fractal branches intertwine on the negative real axis, reaching in nity jointly in the negative half-plane. They split in the vicinity of the positive primary Julia point, from where they independently and symmetrically approach in nity as straight lines with angle 1. This fractal is sketched in Fig. 12b. d > 0 (Conjugate-Imaginary condition) The roots of the cubic split into one conjugate complex pair and a real root. Now the derivative also has a conjugate complex pair of roots that reside on the imaginary axis with equal modulus and opposite sign. The corresponding real restriction of the cubic is shown in Fig. 11c. The fractal again consists of two independent branches, each containing a primary Julia point and approaching in nity as a straight line in its half-plane. Each branch approaches ?1 with an angle 1 and +1 with an angle 2. For d ! 1, 1 = 2 for symmetry reasons with the case d < 0. This holds because the root pattern of both (3.1) and its derivative for d ! ?1 can be obtained from that for d ! +1 by the substitution z 7! iz, a rotation by 2 . The fractal is sketched schematically in Fig. 12c.

Asymptotics The asymptotics of the fractal branches are very dicult to ex-

amine and we could not arrive at an analytic expression governing the asymptotic angle with which in nity is approached. We give numerical results for the angle with d as a parameter. For this study, c = 1 was xed without loss of generality as this implies only a scaling of the results. To obtain the asymptotic angle of the fractal branch, the primary Julia points were calculated and starting from these, a recursion was set up that involved only the prolongated solution of (3.16) in the desired direction. Numerically, the following asymptotic cases were determined for large d:

d ! ?1 : 1 = 2 = 54:460 d ! +1 : 1 = 2 = 35:540 : The transition between these two limits is depicted in Fig. 13.

(3.31)

43 70

alpha_1

60

50

40

30 -5

-2.5

0

2.5

5

7.5

10

5

7.5

10

d

60

alpha_2

50 40 30 20 10 0 -5

-2.5

0

2.5

d

Figure 13: Asymptotic angles for cubic fractals

3.5 Analysis of Special Cases

After having discussed the general cubic problem, we will now consider the special cases that can occur. From the previous remarks, it is obvious that only two distinguished cases exist. The rst case has a degenerate Julia set, where the two primary Julia points coincide. With d = 0, this case is equivalent to Cayley's problem and yields the symmetric Newton fractal of order 3. The second special case is when the roots of the original polynomial degenerate, with two roots coinciding on the real axis. As it should be evident from the previous discussion, this case also yields a fractal. It turns out to be of Besicovitch type, and we will provide a detailed analysis, including an estimate for the fractal dimension. We can see from the previous discussion of the general case, that the dynamics for d = 0 arises from the primary Julia points moving closer on the real axis for d ! 0 ?  and separating onto the imaginary axis for d > 0. Therefore, d = 0 marks the highly symmetrical situation where the two fractal branches

44 just touch each other on the negative real axis before nally splitting with d further increasing. The dynamics for d = drs , the root degeneracy, arises from the behaviour of the cubic roots. Due to the generation principles of the fractal, it always has to separate the roots on the real axis. Therefore, the fractal branch between the two real roots is squeezed thinner and thinner as d ! drs ?  until it nally 'evaporates', leaving only one fractal branch in the positive half-plane separating the positive root from the now degenerate root in the negative half-plane. As d is then increased beyond drs , the separating fractal branch reappears, but now rotated by 2 and on the negative real axis to separate the now conjugate complex roots in the negative half-plane. The intertwined branches on the negative real axis grow thicker as d is increased further, until they nally separate for d > 0.

3.5.1 Julia Set Degeneracy

According to (3.10), the rst derivative of the cubic has two coinciding roots for d = 0. This marks the transition point from two real roots for d < 0 to two imaginary roots for d > 0, with the separation of the fractal branches as discussed in the previous subsection. The original problem for d = 0 rewrites as

z3 ? c = 0;

(3.32)

with a corresponding Newton polynomial 2z3 ? 3z0z2 + c = 0: Via the scaling substitution

(3.33)

p

p

z 7! cz; ) z0 7! cz0; 3

3

(3.34)

this can be restated as Newton's method applied to

z3 = 1;

(3.35)

constituting the symmetric problem of order 3 or, in historical terms, Cayley's problem . This has been extensively and quantitatively discussed in the section on symmetric Newton fractals and as a speci c example in [6], so that we can refer to these sources for the complete coverage of the problem. In this context, we want to point out the dierence in the local scale factor that is caused by the Julia set degeneracy. As derived previously, the local scale factor for the general cubic is 3 = 32 : (3.36)

45 This derivation was, however, valid only for d 6= 0. For the symmetric case, we obtain from (3.24), applying the perturbation

z2 = :

(3.37)

Substituting this into the Newton polynomial, and dropping high-order terms in , it follows that   ? 3cz (3.38) 0 and therefore rc p 1 z =   pz
3 : (3.39) 0 With a global scale factor of 3;1 = 23 , according to the same argument as in the nonsymmetric case, this gives rise to a local scale factor of s 3;s = 23 : (3.40) This dierence in the local behaviour of the symmetric problem is caused by the degenerate primary Julia set f0;kg.

3.5.2 Root Degeneracy

A root degeneracy of (3.1) occurs when d = drs , marking the transition from three real roots of the problem into one real root and a conjugate complex pair. In the complex plane, it can be envisaged by two of the real roots moving closer together on the (negative) real axis, meeting for d = drs, and then separating into the complex plane. The value of the roots for this degenerate case is, according to (3.8) rc 1 = 2 2 (3.41a) rc (3.41b) 2;3 = ? 2 : It is therefore possible to express (3.1) in terms of its roots, yielding 3

3

f (z) = (z ? 1) (z ? 2)2 = (z + 22 ) (z ? 2)2 :

(3.42)

Taking the derivative of the left-hand side, we obtain

fz (z) = 3 (z ? 2) (z + 2) :

(3.43)

46 Therefore, cancelling the common factor, the Newton iteration can be written as z0 = z ? (z ?3(z2)(+z+)22) ; (3.44) 2 thus giving rise to the Newton polynomial 2z2 ? (3z0 ? 22 )z + (222 ? 32z0) = 0: (3.45) Hence, in this degenerate case, the Newton polynomial is only of second order! We note that, according to the fractal generation principles with one solution to the Newton polynomial always being prolonging, this is the simplest Newton polynomial that generates a fractal. At this point, we state a short remark. Every quadratic polynomial with real coecients a2 z 2 + a1 z + a 0 = 0 (3.46) can be normalised into

z2 + ac = 0

(3.47)

0

with

z 7! z ? a21 ; c = a0 ? a41 : 2

(3.48)

The associated Newton polynomial is, according to (2.3) z2 ? 2z0z + ac = 0 (3.49) 0 with solutions for the inverse Newton mapping s (3.50) z1;2 = z0  (z0)2 ? ac : 0 Newton's method applied to the equation (3.47) gives rise to Schroeder's problem, which, as is well known, exhibits no fractal behaviour. Therefore, (3.45) really states the simplest Newton fractal and no quadratic problem can yield a similar fractal. In a way, however, the fractal associated with the degenerate cubic is a generalisation of Schroeder's problem, as will become clear at the end of this discussion. Back to the case of the degenerate cubic, we are able to state the solution to (3.45), the inverse Newton mapping   q z1;2 = 14 3z0 ? 22  9(z0)2 + 122 z0 ? 1222 : (3.51)

47 The fractal that arises from subjecting the primary Julia point 0 = ?2 to this mapping is depicted in Fig. 14. It is of Besicovich type [13], a curve being formed by a dense sequence of points. The curve is continuous, yet has no point at which it is dierentiable, as it consists of kinks with angle  at each point. As, by the property of the Newton mapping, in nity and the primary Julia point are locally conformal images of each other,  can be determined numerically, using the results on the global angle of the fractal for the general cubic. For this case,

 = 21  109:8

(3.52)

Analysing (3.51), we obain that each Julia point has two images under the inverse mapping. One image is prolongated on the same fractal branch, the other is inverted in modulus and located on the branch in the other half-plane. It is again reasonable to speak of a parent structure connecting the Julia points 0 and 1 where all the non-prolongated images are located. As there are only two dierent roots, points to the left of the Besicovich line depicted in Fig. 14 converge to the negative root 2 and points to the right of the line to the positive root 1. It is of course also possible to treat the degenerate problem as a cubic - then the negative Julia point 2 that coincides with the root is a xed-point solution to the cubic inverse mapping. 2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0 0.0

0.5

1.0

1.5

2.0

Figure 14: Besicovich Fractal for the root-degenerate cubic Comparing (3.50) and (3.51), we note that the generation process for Schroeder's problem is similar to that of the degenerate cubic. The only dierence is that for Schroeder's problem  = 180, therefore the Julia set is con ned to the imaginary axis and appears as a straight line with no fractal character. As (3.45) is still inherently a cubic problem, the global scale factor of (3.53) 3;1 = 23

48 prevails, as can be seen from (3.51) for large z0. The local scale factor for nondegenerate primary Julia points is, as previously established 3 = 32 : (3.54) Both of these scale factors describe the fractal on one chain of Julia points, either converging into the origin (or any other lower-order Julia point) or diverging towards in nity. As there are two chains of the fractal, one above the real axis and one below the real axis, we have to nd the scale factor 3 associated with a mapping between these two. That such a mapping exists, can be seen from the root condition on (3.45) and the global scaling property for third-order problems demanding for one solution z2  23 z0 1 2 (3.55) z1z2 = 22 ? 32 z0 , z1  f z : 0 To nd the scale factor, we look for the xed points of cycle 2 of the inverse Newton mapping, following the ideas presented in [6]. The condition for these is that, under the inverse Newton mapping (z0 7! z 7! z0) constitutes an orbit. Therefore, the Newton polynomial must hold in the following two forms 2z2 ? (3z0 ? 22)z + (222 ? 32 z0) = 0 (3.56a) 2 2 2(z0) ? (3z ? 22 )z0 + (22 ? 32 z) = 0: (3.56b) Subtracting these and simplifying, we arrive at (3.57) z + z0 = ? 25 2: The xed points are now obtained from substituting this relation back into the Newton polynomial and solving for z. This yields for the xed points  p   3 2 1;2 = 4 ?5  i 5 15 : (3.58) It is of course possible to con rm these xed points numerically by evaluating the inverse mapping. Similarly as in [6], the scale factor is determined via a linear approximation of the inverse Newton mapping and obeys dz : (3.59) 3 = dz 0 To obtain the total dierential, we dierentiate (3.45) implicitly dz ? (3z ? 2 ) dz ? 3z ? 3 = 0: 4z dz 0 2 2 dz0 0

(3.60)

49 Therefore, the governing equation for the scale factor is 3( z +  2 ) 3 = (3.61) 4z ? 3z0 + 22 : To evaluate it, we have to substitute the xed points for z and z0. We immediately see from (3.58) that 2 cancels and therefore has no in uence on the scale factor. Equation (3.61) is symmetrical with regard to the imaginary part of z and z0, thus making no dierence between the possible two assignments between fz; z0g and f1; 2g. Evaluating the expression, we obtain (3.62) 3 = p2 : 19 With these scale factors at hand, an estimate of the fractal dimension of this Besicovitch fractal can be given. Employing (2.24), the fractal dimension d must satisfy d3 + d3 = 1;

(3.63)

this being solved numerically to yield

d  1:213:

(3.64)

This fractal dimension is consistent with the estimate on the Hausdor dimension of iterated rational functions given by Douady [4].

50

4 Numerical Experiments and Applications In this nal section, we state two practical results. Firstly, we empirically determine the fractal dimension of the two special cases of cubic fractals by boxcounting. The results strongly con rm the theoretically established values for the fractal dimension. Secondly, we apply the results on the fractal appearance for a general cubic to the turbulent k ?  equations when solved pointwise by Newton's methods. This is a common strategy in codes for problems in Computational Fluid Dynamics. We show that the fractal properties can have strong in uence on convergence, particularly for commonly used approximations of turbulence. We point out a computationally cheap and eective way to improve convergence.

4.1 Box-counting the Fractal Dimension

In order to get an estimate on the accuracy of the theoretical fractal dimensions of the special cases for  = 3, we conduct box-counting experiments. The boxcounting dimension of a fractal is, according to [2] de ned in the following fashion. De nition 4.1 If the plane R2 is covered by square boxes of side length 21a , and Na(A) boxes intersect an attractor A, the fractal dimension D of A is " # ln ( N a (A)) D = alim (4.1) !1 ln (2a) : To establish the box-counting dimension, we note from theorem 2.11 that the fractal is a connected structure separating the basins of attraction. Therefore, the following algorithm is used. For each box of the grid that is cast over the fractal, its centre point is compared with the four corner points (un lled circles for the shaded box in Fig. 15). Each point is assigned a 'colour' according to the basin of attraction it is located in. If the colours of these ve points dier, the fractal intersects the box and therefore it is counted as being part of the fractal. If all colours are equal, the neighbouring centre points ( lled circles in Fig. 15) are compared. If any of their colours dier from the colour obtained for the box so far, ten additional points are cast on the appropriate side of the box and examined for their colour. As soon as any of them converges to a dierent root, the box is also counted as part of the fractal. Otherwise, it is regarded as not containing parts of the fractal. The algorithm as described is easy to implement and can compute boxcounting results over a great variety of length scales without actually having to compute the Julia set. It is possible that some boxes are falsely considered not being part of the fractal (given the appearance of the Newton fractals, this is however fairly unlikely). Therefore, the algorithm provides a secure lower bound

51

Figure 15: Grid arrangement for Box Counting for the fractal dimension. It is noted from the discussion in [7] that the boxcounting dimension is not always an accurate measure of the fractal dimension de ned in more theoretical ways. However, in the absence of other measures, we will use it as a benchmark for the theoretical results established earlier.

Symmetric Newton Fractal  = 3 Applying the box-counting algorithm to

two areas containing the symmetric fractal, the following counts were obtained. [?0:8; 0]  [?0:3; 0] a box counts Na(A) total boxes 2 8 8 3 16 21 4 34 65 5 87 260 6 249 1040 7 638 4017 8 1653 15 785 9 4627 63 140 10 12 601 252 560 11 34 203 1 007 985 12 93 266 4 027 433 13 253 276 16 109 732

[?5; 5]  [?5; 5] Na(A) total boxes 224 1600 600 6400 1636 25 600 4469 102 400 12 174 409 600 32 932 1 638 400 89 586 6 553 600 243 660 26 214 400 665 179 104 857 600

Table 2: Box-counting results for the symmetric third-order Newton fractal According to Def. 4.1, the logarithm of Na(A) is plotted against the logarithm of the box size. The slope of the line tted to this data determines the fractal dimension. For the results in Table 2, the plot in Fig. 16 is obtained. The data is well tted by a straight line. Running a regression analysis, we obtain a slope of D = 1:431 for the smaller area, and D = 1:443 for the larger area. The standard errors are  = 4:5
10?3 and  = 8:7
10?4 , respectively. We

52 8

log(N)

6 4 2 0 0

1

2 log(2ª)

(-0.8..0)x(-0.3..0)

3

4

(-5..5)x(-5..5)

Figure 16: Box-counting plot for the symmetric third-order Newton fractal can therefore infer the estimate for the fractal dimension

D  1:44:

(4.2)

This is considerably lower than the theoretical dimension of d  1:80, a deviation that is explained from the fact that the Newton fractal does not consist of perfect geometric progression as assumed for the theoretical estimate. The Julia points of lower order are much wider spaced than the geometric progression suggests, so that appropriate correction terms would have to be used that would then lower the theoretical estimate. For further details, we refer to the discussion in [6].

Besicovich Fractal of Root-degenerate Cubic As for the symmetric Newton fractal, we apply the box-counting algorithm to two areas of the complex plane that contain the fractal and obtain the results in Table 3. To establish the fractal dimension, a logarithmic plot is obtained and the slope of the best- tting line determined. The plot is depicted in Fig. 17. Again, in both cases the data is well tted by a straight line, particularly for small boxes. This suggests the validity of the box-counting dimension in a limit sense even for box sizes smaller than the ones considered here. A regression analysis yields a slope of D = 1:028 for the smaller area and D = 1:026 for the larger one. The standard error is  = 4:0
10?3 in both cases, giving rise to an estimate of D  1:027

(4.3)

for the fractal dimension of the Besicovich fractal. Again, this is considerably lower than the theoretical estimate d  1:213. The reason is similar to that for the symmetric Newton fractal. In both cases, we assumed perfect geometric progressions of the Julia points for the theoretical estimate. In reality, this

53 [0:75; 1:5]  [0; 1:8] [0; 2]  [?2; 2] a box counts Na(A) total boxes Na(A) total boxes 2 10 24 28 128 3 23 90 56 512 4 44 348 108 2048 5 87 1392 218 8192 6 188 5568 478 32 768 7 390 22 176 984 131 072 8 768 88 512 1946 524 288 9 1572 354 048 3968 2 097 152 10 3201 1 416 192 8076 8 388 608 11 6451 5 663 232 16 262 33 554 432 12 13 069 22 64 856 33 022 134 217 728 13 26 412 90 599 424 Table 3: Box-counting results for the Besicovich fractal 5

log(N)

4 3 2 1 0 0

1

2 log(2ª)

(0.75..1.5)x(0..1.8)

3

4

(0..2)x(-2..2)

Figure 17: Box-counting plot for the Besicovich fractal assumption only holds for high-order Julia points and is violated considerably in the beginning of a chain of Julia points. We postulate that by introducing proper correction terms for this phenomenon, the theoretical estimate will get smaller and into better agreement with the box-counting results.

54

4.2 Analysis of Local Solvers for the Turbulent k ?  Equations

One of the most commonly used models in turbulence computations is the k ?  model introduced by Jones and Launder (for a derivation, see e.g. [8]). It amends the Navier-Stokes equations for uid ow with two transport equations, introducing the quantities k and  that describe the turbulent ow features. The employed approximation of the Reynolds stress tensor assumes isotropy, a restriction that limits the practical accuracy of the model. However, it is widely used due to its relative computational ease. Despite the possibility of using a stabilised form of Newton's method as a global solver for the k ? equations [5], most available software uses time-stepping techniques and solves the equations pointwise. It is common to locally decouple the turbulence equations from the Navier-Stokes equations, establishing a nested iteration that only solves one system of equations at a time. In this context, we are interested in the nonlinearity introduced by the k ?  system. In symbolic notation, this system can be written as

c1k2 + c2k + c3 + c42 = 0 d1k2 + d2k + d3k + d42 = 0;

(4.4a) (4.4b)

and is solved for fk; g. Usually, Newton's method is used for this purpose. The coecients are real and xed for each point, but as they are determined by the ow variables, they vary considerably across the geometry. To analyse the properties of the system (4.4), we eliminate one of the variables to obtain  b + b  + b 2 + b 3 = 0; (4.5) 3 c21 0 1 2 with the coecients b0 = c1c3d23 b1 = c3d1(c3d1 ? c2d3) + c1d3(2c3 d2 + c4d3) b2 = c3d1(2c4d1 ? c2d2 ? 2c1d4) + c1d2(c3d2 + 2c4d3) (4.6) ?c2d3(c4d1 + c1d4) b3 = (c1d4 ? c4d1)2 + (c4d2 ? c2d4)(c1d2 ? c2d1): The spurious, turbulence-free solution fk; g = f0; 0g can be eliminated from (4.5), yielding the cubic

b0 + b1 + b22 + b33 = 0:

(4.7)

As the coecients are real, this corresponds to the general cubic discussed in the previous section. As mentioned above, the coecients of this cubic contain information about the ow variables and are therefore likely to vary considerably

55 throughout the geometry. In particular, all dierent fractal forms discussed for the general cubic are likely to appear, if  would be a complex variable. However, the Newton search is con ned to the real axis only and we are therefore interested in the one-dimensional Julia set of the fractals that resides on the real axis. This will determine the convergence behaviour of Newton's method when started to nd a real root of (4.5). In general, the more Julia points are found on the real axis, the longer the convergence history will be, as the vicinity of any Julia point will be subject to large Newton shifts eventually. Following the discussion in the previous section, we can distinguish three cases, depending on the roots i of (4.7) and the roots i of the derivative with respect to .  fig 2 R: The real axis contains two Julia points with same modulus and opposite sign.  1 2 R; 2;3 2 C; i 2 C: The real axis contains no Julia points.  1 2 R; 2;3 2 C; i 2 R: The real axis contains an in nite number of Julia points left of 1 > 0. Of this list, only the last case will pose problems for a considerable range of starting values, where it will be quite likely that Newton's method gets close to a Julia point and therefore takes a long time converging with a linear rate (see the discussion in [6] for a more detailed explanation). Furthermore, the Julia points are in a positive region where physical intuition would suggest suitable guesses for the solution - starting from little turbulence, close to the origin. It is therefore very advisable that a local Newton solver in a k ?  code avoids this critical region. This can be done by choosing a starting value that is bounded from below by   q 1 (0) 2 (4.8)  > 1 = 6b ?2b2 + 4b2 ? 12b1b3 : 3 Relying only on the known coecients of (4.4), this estimate can be obtained at little computational cost. As the bound gives the coordinate of the largest Julia point, it is not advisable to start too close to it. Any positive correction of the bound, however, does not impair convergence dramatically as starting values with (0) > 1 are located in the convex region of (4.7) and therefore converge directly to the root. Numerical results for an exemplary cubic of the form x3 ? x ? 1 = 0 (4.9) with a real root at x = 1:3247179 con rm this theoretical bound. We considered three intervals in which we started Newton's method for a sequence of equally spaced initial guesses and counted the iterations until convergence with a residual

56 of 10?13 . Table 4 states the results. In addition to the average iteration count x until convergence, we state the empirical variance for n runs n X s2 = n ?1 1 (xi ? x)2 (4.10) i=1 as a measure on the predictability of the convergence history. The number of initial guesses n = 577 was held constant throughout the experiment. Timing was done on an Intel 80486 based system, for 50 runs to convergence on each guess and then scaled down for one run, yielding the gure stated. As a representative of a commonly used stabilisation technique, dynamic shift scaling for the downward shifts was implemented with the limiting value as stated. For a detailed discussion of the stability issues of that method in a fractal context, see [6]. Interval iterations x variance s2 time [sec] 0.001, 0.577 24.54 227.593 0.30 109.96 0.037 1.46 160.96 0.037 1.84 0.600, 1.176 6.73 3.069 0.11 6.67 2.546 0.12 6.67 2.546 0.12 5.000, 5.576 8.00 0.000 0.12 8.00 0.000 0.13

method orthodox scaling 0.999 scaling 0.99 orthodox scaling 0.999 scaling 0.99 orthodox scaling 0.99

Table 4: Convergence for various starting guesses on a k ?  type cubic The rst interval is entirely located left of the primary Julia point 0 = p13 . The orthodox method exhibits a large variance - indicator for the presence of many Julia points - and a large number of iterations to converge on average for a starting guess. A closer survey of the convergence path shows that large shifts occur and the iterates are often negative. The chosen shift limits for the downward shift scaling prevent negative iterates, and increase predictability of convergence. This however happens at the expense of dramatically slow convergence. The improved predictability of the stabilised method is coherent with the results on fractal depletion presented in [6]. The second interval is located just above the bound given in (4.8), and displays improved convergence in every aspect. The runtime gain shows almost a factor of 3, as does the gain in iterations. As the interval is rather close to the root, iteration counts vary due to normal Newton convergence, thus the value of the variance. It is noted that even for the orthodox method, the iterates stay positive, and shift scaling can be employed at almost no additional cost.

57 The third interval shows that it is safe to use the physically counterintuitive approach of choosing large starting values and exploiting the convexity of the function. Despite an initial error that is much larger than for the rst interval, convergence compares to the case of starting very close to the root in the second interval. With the interval being further from the root, all guesses converge in the same number of iterations, thus the vanishing variance.

58

5 Concluding Remarks In this work, we have examined the properties of Newton's method employed to nd the roots of complex polynomials numerically. For the symmetric case z ? 1, a generation mechanism for the resulting fractal structure has been established from rst principles, yielding quantitative results for the properties of the structure. The results generalise from the case  = 3, where a connection between the fractal generation mechanism and numerical convergence from any starting point was established in [6]. As the generation principle is conserved, the numerical convergence pattern is conserved as well. We can state that converging from any starting point for any complex polynomial of order  , the residual convergence path of Newton's method obeys the following four distinct patterns.  A stationary residual with magnitude de ned by the primary Julia point 0 and the rst-order Julia point 1, when the iterates move within the parent structure and constantly change the fractal branch.  A sharp increase in residual, when the iterates leave the parent structure with a change of branch. This has to be followed by  Linear convergence with an approximate rate ? 1 as the iterates move laterally along one branch. This is the only type of convergence possible outside the parent structure for large iterates.  Quadratic convergence once the iterates get suciently close to the root for classical Newton-Kantorovich stability to hold. For symmetric polynomials, the rst statement translates into 'stationary residual with magnitude 1'. For general polynomials, the factor for linear convergence holds as their global behaviour is determined by the highest order  of the variable (see the general cubic for an example). We can therefore conclude that for general complex polynomials, the global numerical convergence path of Newton's method is explicable once the underlying fractal structure of the problem is known. It is likely that these principles generalise to other functions, but we have no quantitative results there. Discussing the general cubic problem, a comprehensive description of the possible fractal structures for third-order polynomials has been stated. As this description gives bounds on the location of the numerically critical Julia points, it can be used to speed up numerical computations involving the solution of cubics. A straightforward example for the k ? equations in turbulence modelling has been presented. Of course, the results can be applied to any other twodimensional polynomial system of overall degree 3 (i.e. that can be transformed into a complex cubic).

59 It is expected that a general description of the fractal structures is also possible for systems of degree four due to the existence of analytic formuli for the solution. The derivation of this description would be technically equivalent to the cubic case and was therefore omitted in this work. However, it should be straightforward to implement it using the concepts presented here. For systems with degree larger than four, no analytic solution exists in general. The discussion will therefore most likely be restricted to the symmetric case, where this work has given a comprehensive list of properties. The fractal dimension can be estimated numerically, determining the necessary scale factors via computation. In determining the fractal properties of symmetric polynomials in association with Newton's method, a gap in the subject of fractal geometry has been closed. Furthermore, by analysing the general cubic, applications of the fractal theory to practical systems of equations have been made possible. It was established that the results concerning the fractal properties can be applied directly to understand and improve numerical convergence. We feel that the results obtained for the still rather simple case of polynomials might generalise for other, more dicult classes of problems and help to improve the adaptions of Newton's method for engineering and physics applications.

Acknowledgements This work was nished while M. Drexler was visiting the program of Scienti c Computing / Computational Mathematics at Stanford University. He would like to express his gratitude to Prof. G.H. Golub for the kind invitation and the academic support at Stanford. He also wants to thank M. Gander (SC/CM, Stanford) for helpful discussions. The nancial support of the German Academic Exchange Service (DAAD) through the programme HSPII/AUFE is acknowledged gratefully. C. Bracher acknowledges the nancial support of the 'Studienstiftung des deutschen Volkes'.

60

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