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 conrmed 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: namd@comlab.oxford.ac.uk
November, 1996
2
Contents
1 Introduction
2 The Symmetric Newton Fractal of Order 2.1
2.2
2.3
2.4
Denitions : : : : : : : : : : : : : : : : : : : : :
General Properties : : : : : : : : : : : : : : : :
Structural Results from Classical Root Analysis
Generation via a Rotational Basis : : : : : : : :
3 The Newton Fractal of a General Cubic
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Denitions 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 : : : : : : : : : : : : : :
3.1
3.2
3.3
3.4
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4 Numerical Experiments and Applications
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5 Concluding Remarks
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4.1 Box-counting the Fractal Dimension : : : : : : : : : : : : : : : : : 50
4.2 Analysis of Local Solvers for the Turbulent k Equations : : : : 54
3
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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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
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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.
5
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 specied, 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
denition is given later. Julia also derived some properties of the set solving
Cayley's problem, namely a reective 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 specic 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.
7
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 denitions 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 signicance 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 Denitions
This section will be discussing a special class of polynomials dened in the following fashion.
Denition 2.1 The symmetric polynomial of order is dened 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 dened in
the standard way.
Denition 2.2 The orthodox Newton method on f (z) is dened 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 denition 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 dene
Denition 2.3 The complex Newton polynomial of order is dened 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 denition for such a structure. Following Mandelbrot
[13], a strict denition of a fractal is
8
Denition 2.4 A fractal is a set for which the Hausdor-Besicovitch dimension
strictly exceeds the topological dimension.
This denition, 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
denitions of a fractal are in use. For the purposes of this study, we use 'fractal'
according to the denition by Falconer [7].
Denition 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 (dened in some way) is greater than
its topological dimension.
In most cases of interest S is dened in a very simple way, perhaps recursively.
A suitable working denition for the purposes of this study is
Denition 2.6 The Newton fractal of order is dened by the union of all
points that are mapped into the singular origin by the Newton mapping (2.3).
It is obvious from this denition 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 denition 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 dene the necessary sets as follows.
Denition 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.
Denition 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 dene 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]
Denition 2.9 The order of a specic Julia point on a Newton fractal is dened
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 dene the important concept of a basin of attraction.
Denition 2.10 The basin of attraction A(w) of an attractive xed point w of
a function g is dened by A(w) = fz 2 C : g k (z ) ! w as k ! 1 g.
With this denition, 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.
10
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 briey, 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
zk = e (k 1)i; k = 1; 2; . . . ; :
(2.4)
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 dene 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 reective
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 reective 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
(2.11)
z 7! ;1 z; ;1 = 1 :
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 dene the
global symmetry axes Ls to be the lines through the origin and the rst-order
Julia points as dened 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
;
2
(
1
1)2
0
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 = ( 1) + (2 1) k; k = 0; 1; . . . ; ( 1) 1:
(2.16)
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 1)
(2.17)
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 there-
fore 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
q1
q 1 2 ;...:
(2.19)
pz0 ; z ; (;1 ) z0
;1 0
1
1
1
This again is a geometric progression; the scale factor can be determined as
s
q
= 1 = (;1 ) 1:
(2.20)
We see that is rapidly approaching 1 as grows 1 1 ( 1 1) :
= 1 (2.21)
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 innite 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 12 ( 1)2 + 1 c;
with each of which a cross-chain scale factor should be associated. bc 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
= max 1 ; :
(2.30)
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
g(z) = 2 1 1
(2.32)
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 jz0j1 ; 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
jz0 j
jz0j > 1
1<p< 1 0
+ jz0j
not possible
1
1
1 + jz0j
not possible
p
Table 1: Conditions for p roots of the symmetric Newton polynomial on the unit
disc
From this table, we can conclude
Lemma 2.14 Of the roots of the Newton polynomial (2.3), the unit disc jzj 1
contains
all roots for jz0j < 2 ,
exactly ( 1) roots for jz0j > 1.
Using a generalisation of Cauchy's theorem on a disc containing a given number of roots by Van Vleck (Theorem 6.4n, [10]), we obtain
17
Lemma 2.15 The disc dened by
jzj < q1
jz0j
(2.38)
1
contains at least ( 1) roots of the Newton polynomial (2.3). It contains exactly
( 1) roots for jz0j > 1, according to lemma 2.12.
Regarding lower bounds for the roots, we can expand the polynomial at z = 0,
and obtain (Theorem 6.4b, [10]) for the radius of a circle that contains no roots
a0 m
1
:
(2.39)
= 2 1min
m am Substituting the coecients of the Newton polynomial, we can state
Lemma 2.16 The roots of the Newton polynomial (2.3) are bounded from
below by
1
0
1
1
1
jzk j > 2 min @ p 1 ; q A k = 1; 2; . . . ; (2.40)
jz0j
1
1
Estimates for the Argument of the Largest Root Parametrising the New-
ton polynomial along the fractal's global symmetry axes and rotating the structure accordingly,
k
k
z0 = e i; z = e i; k = 0; 1; . . . ; 1;
(2.41)
a polynomial with real coecients
( 1) 1 + 1 = 0
(2.42)
is obtained. As, parametrising = ; 2 R
1+ 1 (2.43)
changes sign for > 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
1)
(2
1)
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 denition of Newton's method (2.2), the Newton polynomial
can also be stated as
p(z) z0 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
jc j arg(z1) arg(z0) arccos 2jc 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
(2.49)
a = 1 = ( 1) + 1 :
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
arg(z1) arg(z0) :
(2.51)
As the modulus of z1 is bounded from below by lemma 2.12, the above bounds
on the argument conrm 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 dened by
arg(z) < 1 ( 2 )
(2.52)
contains roots of the symmetric Newton polynomial with
= 0, if <(z0) < 0.
2, if <(z0) > 0.
Proof The coecients of the symmetric Newton polynomial are, ordered by their
indices,
a = ( 1); a 1 = z0; a0 = 1:
(2.53)
Therefore, the double sector dened by j arg(z0) j contains all coecients. If
<(z0) < 0, all coecients lie in the same sector, whereas for <(z0) > 0, the sequence
of coecients changes sector twice. According to Theorem (41,3), [15] this denes the
maximal number of roots in the stated positive sector. 2
Considering the rotational symmetry of the fractal, we now divide the complex plane into sectors with angle 2 centered at the origin by drawing lines
connecting the origin with the rst-order Julia points 1;k . Recalling the local
conformity of the inverse Newton mapping, we know that for small z0, the Newton polynomial has a simple root close to each of the points 1;k . Furthermore,
lemma 2.20 states that for points on the fractal, the bisectors of the sectors
dened by the 1;k cannot be crossed. Therefore, for small z0, the roots are
conned to the global branches of the fractal. However, as the roots of a
polynomial vary smoothly with changes in the coecients according to Rouche's
theorem (Theorem 4.10c, [10]) and the fractal is a dense structure by theorem
2.11, they will be conned to the vicinity of the global symmetry axes for all z0
on the fractal. From lemma 2.17, we can conlude that there must be an approximate conjugacy of the roots with respect to the symmetry axis closest to z0. We
can therefore conclude
Lemma 2.23 The roots of the Newton polynomial (2.3) are located close to
the global symmetry axes of the fractal, if z0 is a Julia point. The ( 1) axes not
pointing in the general direction of z0 each have a simple root in their vicinity.
The roots obey an approximate conjugacy with respect to the symmetry axis close
to z0, depending on the deviation angle of z0 from that axis.
21
Lemma 2.22 can be used to establish further bounds on how close the roots
lie to the symmetry axes regarding their position to z0. As these are only of
technical interest, we will not pursue this further.
Structural Conclusion We have now established the necessary framework for
a description of the fractal structure from rst principles.
To do so, we partition the complex plane into sectors centered at the origin
that open with an angle of 2 and are bisected by the rays Ls connecting the
origin with the rst-order Julia points 1;k. It is immediately clear from (2.4) that
the roots of the polynomial (2.1) dene the borderlines between these sectors.
Fig. 3 shows such a partition for the case = 5.
y
x
root
first-order Julia point
Figure 3: Sector Partition of the complex plane
Furthermore, as stated earlier, we denote that part of the fractal that connects
the origin and the rst-order Julia points 1;k as the parent structure . We denote
the structure connecting two Julia points of ascending order on the bisecting axis
Ls of a sector as a blob structure . We are now able to state
Theorem 2.24 For any Julia point k , there exist images under the inverse
Newton mapping dened by (2.3) with the following properties:
One solution is located in the same sector as k , prolongated along the
bisecting axis of that sector by one blob structure.
( 1) solutions are located on the parent structure in the ( 1) sectors
that do not contain k , each sector containing one solution. The parent
structure is contained within the unit disc.
Proof We rst comment on the preliminary denitions and statements. It is
immediately obvious from (2.3) that the rst-order Julia points exist as stated. By
the local conformity property of the inverse Newton mapping, both the origin and
the 1;k must be locally of similar structure. Due to theorem 2.11, there has to exist
22
a dense fractal structure connecting these two points, the parent structure. Due to
lemma 2.17, the rst-order Julia points are the beginning of a chain of Julia points
on the bisecting axis in each sector. As the fractal structure approaches the origin in
straight lines at equal angles (see the earlier results on local symmetry), it approaches
each of these points in the same fashion, thereby dening "knots" on the bisecting axis
Ls . The structure connecting two of these knots must by theorem 2.11 be dense, hence
follows the existence of the blob structure.
The rst assertion of theorem 2.24 is proven by combining lemma 2.12 and 2.13 for
the modulus, and lemma 2.17, 2.19, 2.21 for the argument of the largest root. Lemma
2.18 gives a geometrically suggestive result on the prolongation along the bisecting
axis, which coincides with the general direction of k for Julia points.
The second assertion is proven by lemma 2.14 and 2.15 for the modulus, and lemma
2.23 for the argument of the solutions. Lemma 2.14 denes the upper bound on the
size of the parent structure. The images of k can be bounded from below using lemma
2.16. 2
Theorem 2.24 has stated properties of the pointwise mapping of Julia points.
Combined with the other results on the inverse Newton mapping, we can use it
to state a theorem giving a description of the fractal structure via Jordan curves.
Theorem 2.25 The fractal structure can be approximated by successive inverse
Newton mappings of the bisecting axes of the sectors dening the fractal, each
containing an innite number of Julia points. In the limit of innitely many
inverse mappings of that structure, the Newton fractal of order is obtained.
An example of the theorem for the case = 3 is given in [6].
Proof The existence of innitely many Julia points on each bisecting axis can be
concluded from lemma 2.17, with the primary Julia points 1;k starting the recursion
on the axis. The same lemma ascertains that one image of any point on the axis will
lie on the same axis (and its extension through the origin), this structure constituting
an invariant of the mapping. Due to local conformity of the inverse Newton mapping,
the image of the axis (which is obviously a Jordan curve) will also be a Jordan curve.
As the origin 0 is mapped onto innity and 1;k onto the origin by one Newton step,
the part of the axis connecting (0; 0) and 1;k extending to innity must have ( 1)
images connecting the 1;k and the origin 0 . These lie, according to theorem 2.24, on
the parent structure of the fractal.
By the denition of the inverse Newton mapping, every curve that connects two
Julia points of order (k 1) and k will be mapped onto a line that connects Julia points
of order k and (k +1) by that mapping and the order of any Julia point on that line will
also be increased by one. On the other hand, the order of all Julia points on a curve
will be decreased by one when the curve is subjected to one Newton step. However,
all Julia points on the fractal will eventually be mapped onto one of the bisecting axes
Ls - namely onto 1;k - when subjected to consecutive Newton steps. As every inverse
of the Ls also maps the rst-order Julia points contained on them and every Julia
point is an image of 1k , the whole fractal can eventually be obtained by subjecting
the bisecting axes Ls to the inverse Newton mapping. 2
23
(0)
(1)
(2)
(3)
1
2-3
3-1
2
2-1
3-2
3
2-2
3-3
Figure 4: Generation of the Julia point structure via axis mapping
Fig. 4 illustrates the generation of the fractal structure via consecutive axis
mappings for the case = 3. We consider the thick line on the left of the tree
to be the initial axis. It has, according to theorem 2.24, a prolongated solution
and two images in the other sectors. Geometrically, these images connect the
1;k with the origin 0 via a sequence of innitely many Julia points, depicted
by the thick dashed lines. The two images 2 and 3 again have images obeying
theorem 2.24, each connecting a Julia point of order 2 with one of order 1 via
innity. The innite chain of Julia points is depicted by a thin dashed line, and
the straight line denotes the connection between start and end point of the chain.
It is important to note that the dependencies between Julia points in Fig. 4 do
not correlate with the dependencies dictated by the inverse Newton mapping, i.e.
the three second-order Julia points in the tree are not direct images of the one
directly above. The images of the axes could not be depicted straightforwardly
on a tree that displays the mapping dependencies. However, the nodes of such a
tree and the tree depicted in Fig. 4 are identical. All the remaining branches of
the tree are reached by images of the remaining two axes.
2.4 Generation via a Rotational Basis
In [6], a generation principle was stated for the third-order Newton fractal. Its
main property was that the solutions to (2.3) for = 3 could be expressed via a
basis of three vectors, involving only rotations and additions of the vectors. In
this section, we will generalise this principle for > 3, thereby explaining the
general structure of the th -order Newton fractal. Although this explanation in
principle yields the same results that can be obtained via a classical root analysis,
it provides a more elegant theory for the generation of the fractal. We will start
with proving the existence of a solution for the inverse Newton mapping that can
be expressed via a rotational basis.
We state
Theorem 2.26 The set of solutions to the inverse Newton mapping of order can be expressed via vectors rm 2 R2 in the following fashion.
24
One solution 1 is obtained by
1 =
X
m
rm:
(2.54)
The remaining ( 1) solutions k are obtained by
X
k = r0 + Rk;m rm
(2.55)
m1
with Rk;m being the rotation matrix about an angle of
2mk .
According to the theorem, the solution for = 4 would be
1 = r0 + r1 + r2 + r3
2 = r0 + R90r1 + R180r2 + R270r3
(2.56)
3 = r0 + R180r1 + R0r2 + R180r3
4 = r0 + R270r1 + R180r2 + R90r3
with R' denoting a rotation by ' degrees. The solution for = 3 as stated in
[6] follows immediately from the theorem as well.
Proof Noting that the vectors in theorem 2.26 can be expressed as complex
numbers
rk = [rk]x + i [rk]y ;
k = [k ]x + i [k ]y ;
rk 2 C;
k 2 C;
(2.57)
the rotation matrices can be written as
Rk;m = e mki :
(2.58)
2
We abbreviate the principal rotation by
! = e i :
(2.59)
2
Using this notation, the solution to (2.3) can be written as
X
1 = rm
m X
k = r0 + ! km rm
m1
or in matrix form Vr = 2
66 11
66 1
66
66 1
64 ...
1
1
..
.
1 ! (
1
!2
!4
!6
!
!2
!3
1)
..
.
! 2(
!3
!6
!9
1)
(2.60)
..
.
! 3(
1)
...
...
...
...
...
...
1
! ( 1)
! 2( 1)
! 3( 1)
! (
..
.
1)2
32
77 6
77 66
77 66
77 66
75 64
2 3
3
77
66 12 77
77
6 7
77 = 666 34 777 : (2.61)
66 . 77
.. 775
4 .. 5
.
r0
r1
r2
r3
r(
1)
25
The existence of the k is guaranteed by the fundamental fact that any polynomial
can be written in factorised form. From (2.61), we can see that there exists a unique
set of rk , if the matrix V is invertible. As it is of Vandermonde type, its determinant
d can be written in terms of ordered dierences of the second-row entries [9]
d=
Y1
i=0;i>j
! i ! j 6= 0:
(2.62)
The determinant cannot be zero as i 6= j and therefore dierences of the dierent nth
roots of unity are considered. None of these dierences is zero, therefore their product
cannot be zero. Hence, the matrix is invertible and a unique set of rk exists. 2
We are also able to state three further observations.
Lemma 2.27 The matrix V is its own inverse with each element inverted, scaled
by a factor of 1 .
Proof We consider the following expression for a; b 2 N, noting that ! = 1
X1
k=0
!ak !
bk
=
X1
k=0
! (a
b)k
(a b)
= ! (a b) 1 = ab :
!
1
(2.63)
For the part a = b, we made use of the l'Hospital rule to obtain
! x 1 = lim ! ( 1)x = :
lim
(2.64)
x!0 ! x 1 x!0
Comparing (2.63) with the matrix entries in (2.61), we see that the sum is equivalent
to the product of row a and elementwise inverted column b of V and therefore, by the
denition of the matrix inverse, V 1 = 1 [vij 1]. 2
Lemma 2.28 Applying the rotations in V to a set of equal vectors rm = r, a
closed polygon is obtained.
Proof To prove the lemma in the above form, it has to be shown that the sum
of the elements in each row containing rotations (i.e. rows 1; 2; . . . ; 1) is zero.
Summing the elements of the bth row (a special case of the previous proof), we obtain
X1
b
! bk = !! b 11 = 0b :
k=0
(2.65)
This yields the desired result for b 1. 2
Lemma 2.29 For prime, the rst principal minor of V only consists of permutations of its rst row. Therefore, the ( 1) rotational solutions of (2.3) are
obtained by permuting the rotation matrices Rk;m and applying them to the rm .
26
For = 5, we get for the solution
1 = r0 + r1 + r2
2 = r0 + R72r1 + R144r2
3 = r0 + R144r1 + R288r2
4 = r0 + R216r1 + R72r2
5 = r0 + R288r1 + R216r2
with R' denoting a rotation by ' degrees.
+
+
+
+
+
r3
R216r3
R72r3
R288r3
R144r3
+
+
+
+
+
r4
R288r4
R216r4
R144r4
R72r4
(2.66)
Proof For notational simplicity, we will consider a principal minor M of V which
is composed of the exponents in V. As we are concerned about rotations, and ! = 1,
we are interested in each element of M modulo . We note that M is a ( 1) ( 1)
matrix of the form
2 1 mod 2 mod 3 mod 66 2 mod 4 mod 6 mod 66 3 mod 6 mod 9 mod M=6
..
..
..
64
.
.
.
( 1) mod 2( 1) mod 3( 1) mod ...
...
...
...
...
3
( 1) mod 2( 1) mod 77
3( 1) mod 777
..
75
.
( 1)2 mod We prove the lemma from the following three assertions, each of which will be discussed
and proved in detail.
No two rows of M are equal.
No row of M contains a zero, therefore all ( 1) elements of a row are contained
in f1; 2; . . . ; 1g.
No two elements in a row of M are identical, therefore all elements in
f1; 2; . . . ; 1g appear in a row.
These three assertions state the lemma in their combination. We will now give the
proof for each assertion separately.
The rst assertion is trivial, as every row starts with a dierent element.
By denition, the rst row does not contain any zero, and therefore satises the
second assertion. By the elementary properties of the modulo operator, we can write
a mod = k , a = + k; k 2 f1; 2; . . . ; 1g; 2 N0
(2.67)
and from the denition of M for the nth element in the bth row
mnb = nb;
n; b 2 f1; 2; . . . ; 1g:
(2.68)
We now assume that there exists an element in M that is zero. Setting k = 0 and
equating the above, we obtain
= nb , n = b :
(2.69)
27
By comparing (2.67) and (2.68), we see that for an element of M, the range of is
bound from above by
b 1:
(2.70)
As is prime by assumption, b does not factor it and the denominator in (2.69) does
not decrease by cancelling a common factor. As, from the above, is less than b, it
cannot oset the denominator and the righthand side term in (2.69) will always be a
true fraction. However, by denition, the lefthand side is an integer and therefore the
assumption k = 0 leads to a contradiction. This proves the second assertion.
As for the third assertion, it is again obvious from the denition of M that its
rst row, being an increasing series with maximum value 1, has no two common
elements. The basic condition for any element mnb being k is from (2.67) and (2.68)
2 N0 :
(2.71)
We now assume that there is dierent element mhb in the same row equalling k, there + k = nb;
fore satisfying
+ k = hb;
2 N0 :
(2.72)
Combining these two equations, we obtain
( ) = (n h)b:
(2.73)
We immediately obtain n = h if = , therefore violating the assumption that the
two elements are at dierent positions. For the rest of the discussion, it can therefore
be assumed that 6= . It is also obvious from the positivity of and b that the
signs of the bracketed expressions in (2.73) are matching. We now divide (2.73) by b
to obtain
(2.74)
n h = ( ) b :
By being prime, it cannot be factored by b and therefore the denominator of the
righthand side is not decreased. Applying the argument of the second assertion, both
and are bounded from above by (b 1), and so is their dierence (in modulus).
Therefore the expression in brackets cannot oset b and the righthand side expression
stays a true fraction. As the dierence of two integers is an integer, the lefthand side
is integer and therefore (2.73) cannot be satised, thus proving the non-existence of
two identical elements in a row by contradiction. 2
After having proved the existence and general properties of the rotational
basis, we now investigate its appearance for the case of symmetric polynomials.
This notes that all the results so far have been for general polynomials of degree
and therefore can also explain the properties of the fractals associated with
them via the use of a rotational basis.
To obtain results for the symmetric polynomials, we rst approximate the
corresponding Newton polynomial to determine the solution and then, by using
the previous results on the rotational basis, derive its properties.
28
Rearranging (2.3), we obtain
1
1:
1
z
z
z
0 =
From this equation, we consider three dierent cases.
z0 large, z large
In this case, the above equation can be written as
1
z = 1 z0 z 1 1 z0:
This is the one solution determining the global scale factor.
z0 large, z small
Rewriting (2.75) in a suitable fashion, we obtain
1 :
z 1 = 11
z z0 z0
(2.75)
(2.76)
(2.77)
This yields the remaining ( 1) solutions for large z0, all having a distance
from the origin of approximately
(2.78)
r p1z :
0
As
1
p p 1 > 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, dening
= e i ;
2
it follows that
(2.81)
1
2 #
1
= 1 z0; pz ; pz ; . . . ; pz
(2.82)
0
0
0
is obtained. Solving (2.61) for r, using lemma 2.27 and 2.28, this yields
3
2
2 z 3
z
1
77
66
66 z 1 77
z
!
1
1
6
+
p 7
(2.83)
r = 666 1 ! .. 1 z 777 jz j1 6666 . 1 7777 :
.
75
64
.
4 . 5
1
1p
z +! z
1 ! 1 z
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:
X1
rb = 1 z0 + p1 z ! bk k
0 k=1
(2.84)
b
1
!
1
= 1 z0 + pz ! b 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,
2bi + 2i = 2mi , b + = m:
(2.85)
1
1
This cannot be true for b; m 2 N0 and therefore the denominator cannot
be zero.
z0 small
Dening a principal rotation
= e i ;
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
X1
! 1
1
rb = p
! kb 2k+1 = p
1 k=0
1 (! b
2) 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
b + 1 = m; m 2 N ;
(2.91)
0
has to hold. As 0 b < , this can only be satised 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
!
X1 X1
X
bk
k =
! rb = r0:
(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
rm = p 1m 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 -
250
230
210
204 203 -
205
204
202 200 190 160 -
203
202
200
190
155 154 153 152 -
160
155
154
153
150 130 110 105 -
152
150
130
110
104 103 102 100 90 60 55 -
105
104
103
102
100
90
60
54 53 52 50 -
55
54
53
52
30 10 54-
50
30
10
5
32BELOW
4
3
2
1.0
0.5
0.0
-0.5
-1.0
-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 identied
as transition states of the general cubic fractal.
3.1 Denitions 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 denitions 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 :
d = aa1 3aa22 ; c = a32aa21 27
(3.5)
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 = d3 ; 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
d
c
c
d +c :
t
t
s1 = 2 + 27 + 4 s2 = 2
(3.8c)
27 4
It is easy to see from this representation that there is a critical value
s2
(3.9)
drs = 3 c4
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
0;k = d3 ; k = 1; 2
(3.10)
3
3
3
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
p
1
(
z
0)
z1 = 2 ' + p' + z0 ;
(3.14a)
p q
z2 = 4p1 ' [z0 p']2 i 3 '2 (z0)2 ;
(3.14b)
p q
z3 = 4p1 ' [z0 p']2 + i 3 '2 (z0)2 ;
(3.14c)
with the nonlinearity
r
h
i
3
' = (z0) 2 (c dz0) + 2 (c dz0) c dz0 (z0)3 :
(3.15)
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
1
(
z
p
0)
b1 = 2 '; b23 = 2p' ; t = 12 z0;
(3.17)
it can be immediately veried 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 inuence 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 = 32 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 quantied 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 reective 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;1 = 23
(3.23)
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
(3.24)
3z2z0 dz0 0 , z2 3d :
We note that this constitutes the equation denining the primary Julia points
0;k. To determine the scale factor, we perturb (3.24)
(3.25)
z2 = d3 + ;
yielding in a rst-order approximation, assuming jj 3d :
s
s d
d
3
(3.26)
z = i 3 i 3 1 2d + . . . :
Dropping higher-order terms again, we expand
! d
3
2z i 3 2 d9 :
(3.27)
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
1
1
3
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 = 23 :
(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 conrms 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 innity. 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 denition of the fractal, it is obvious that it must contain the
primary Julia points. Also, it must 'contain' innity - meaning that its branches
must extend towards innity. This is conrmed by the far-eld approximation
(3.19) stating furthermore that each branch approaches innity 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 denition 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 innity. 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 innity 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 innity with a near-zero angle on the Riemann sphere. For
this purpose, we recall that the primary Julia points are mapped into innity by
the Newton mapping (2.2) and therefore are locally conformal images of innity
41
under the inverse Newton mapping. In fact, by the same argument, every point
of the fractal is a locally conformal image of innity. Therefore, if the fractal
branches joined at innity 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 innity - therefore a
non-vanishing angle has to exist at innity 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 classied 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 innity. Each branch is located in a half-plane and
is symmetric with respect to the real axis. Each branch approaches innity 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 innity jointly in the negative half-plane. They split in the vicinity of the positive
primary Julia point, from where they independently and symmetrically approach
innity 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 innity 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 innity 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 specic 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
r
p
z = p1z 3c :
(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
(3.40)
3;s = 23 :
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)
r
1 = 2 2c
(3.41a)
r
2;3 = 2c :
(3.41b)
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
a2z2 + a1z + a0 = 0
(3.46)
can be normalised into
z2 + ac = 0
0
with
(3.47)
2
(3.48)
z 7! z a21 ; c = a0 a41 :
The associated Newton polynomial is, according to (2.3)
z2 2z0z + ac = 0
(3.49)
0
with solutions for the inverse Newton mapping
s
z1;2 = z0 (z0)2 ac :
(3.50)
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
1
2
2
z1;2 = 4 3z0 22 9(z0) + 122 z0 122 :
(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, innity 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 conned
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;1 = 23
(3.53)
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 = 23 :
(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 innity. 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 32 z0
z1z2 = 222 32 z0 , z1 f z1 :
(3.55)
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
z + z0 = 25 2:
(3.57)
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
(3.58)
1;2 = 4 5 i 5 15 :
It is of course possible to conrm 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 = dz
(3.59)
0
To obtain the total dierential, we dierentiate (3.45) implicitly
dz (3z 2 ) dz 3z 3 = 0:
4z dz
(3.60)
0
2
2
dz0
0
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 inuence 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 = p2 :
(3.62)
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 conrm 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 inuence
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] dened in the following fashion.
Denition 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))
(4.1)
D = alim
!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 (unlled 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
dened 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.
a
2
3
4
5
6
7
8
9
10
11
12
13
[ 0:8; 0] [
box counts Na(A)
8
16
34
87
249
638
1653
4627
12 601
34 203
93 266
253 276
0:3; 0]
[ 5; 5] [ 5; 5]
total boxes Na(A) total boxes
8
224
1600
21
600
6400
65
1636
25 600
260
4469
102 400
1040
12 174
409 600
4017
32 932 1 638 400
15 785
89 586 6 553 600
63 140
243 660 26 214 400
252 560 665 179 104 857 600
1 007 985
4 027 433
16 109 732
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 New-
ton 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
a
2
3
4
5
6
7
8
9
10
11
12
13
[0:75; 1:5] [0; 1:8]
[0; 2] [ 2; 2]
box counts Na(A) total boxes Na(A) total boxes
10
24
28
128
23
90
56
512
44
348
108
2048
87
1392
218
8192
188
5568
478
32 768
390
22 176
984
131 072
768
88 512
1946
524 288
1572
354 048
3968 2 097 152
3201
1 416 192 8076 8 388 608
6451
5 663 232 16 262 33 554 432
13 069
22 64 856 33 022 134 217 728
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
Equations
k
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 conned 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 innite 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
> 1 = 6b 2b2 + 4b2 12b1b3 :
(4.8)
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 conrm 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 dened 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 Scientic
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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61
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