Newton’s Method and the Newton Fractal Author: Anthony Chiu Supervisor: Dr. Graham Gough BSc. Computer Science and Mathematics School of Computer Science The University of Manchester Abstract Newton’s Method is a numerical method for root-finding, defined by f (xn ) xn+1 = xn − 0 f (xn ) When x0 is ‘sufficiently close’ to a root, Newton’s method usually converges quadratically. However, in some cases the choice of x0 will lead to Newton’s method converging to a root further away, or not converge to a root at all. When analysing this behaviour in the complex plane and colouring each point in the plane according to the root it converges to, it becomes apparent that the method exhibits fractal nature. These coloured objects are called “Newton fractals”. The system created for this project draws Newton fractals for many choices of f and has enabled some interesting investigations into Newton fractals. i Acknowledgements I would like to express my gratitude to my supervisor, Graham Gough, for his guidance throughout this project. I would also like to thank my good friend Simon Loach for proofreading this report and providing suggestions for improvements, James Montaldi for suggesting this project in the first place, and my family and friends for their constant support and encouragement. ii Contents 1 Introduction 1.1 Newton’s method . . . . . . . . . . . . . . . 1.1.1 Problems with Newton’s method . . 1.2 Newton fractals . . . . . . . . . . . . . . . . 1.2.1 Fractals . . . . . . . . . . . . . . . . 1.2.2 Basic construction of Newton fractals 1.3 The project . . . . . . . . . . . . . . . . . . 1.4 Report overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background 2.1 Overview . . . . . . . . . . . . . . . . . 2.2 Terminology . . . . . . . . . . . . . . . 2.2.4 Petals . . . . . . . . . . . . . . 2.2.5 Branches . . . . . . . . . . . . . 2.3 Existing systems . . . . . . . . . . . . 2.3.1 Features and capabilities . . . . 2.3.2 Comparison of existing systems 2.3.3 Required features . . . . . . . . 2.4 Proposal of features for a new system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . 6 . 6 . 6 . 7 . 8 . 8 . 10 . 10 . 10 3 Design 3.1 Overview . . . . . . . . . . . . . 3.2 Functional requirements . . . . 3.2.1 Component interactions 3.3 Non-functional requirements . . 3.4 Class diagram . . . . . . . . . . 3.5 GUI design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 12 12 13 13 14 14 . . . . . . . . 17 17 17 17 17 17 17 18 18 . . . . . . 4 Implementation 4.1 Overview . . . . . . . . . . . . . . 4.2 Technology choices . . . . . . . . 4.2.1 Programming language . . 4.2.2 Fractal drawing . . . . . . 4.2.3 String parsing . . . . . . . 4.2.4 Development environment 4.2.5 Dependencies . . . . . . . 4.2.6 Platform . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 2 2 2 3 5 Contents Contents 4.3 Drawing a fractal . . . . . . . . . . . . . . . 4.3.1 Time complexity . . . . . . . . . . . 4.4 Panning and zooming around the fractal . . 4.4.1 Panning . . . . . . . . . . . . . . . . 4.4.2 Zooming . . . . . . . . . . . . . . . . 4.5 Parsing strings as functions . . . . . . . . . 4.6 Evaluating functions . . . . . . . . . . . . . 4.7 Changing tolerance and maximum iterations 4.7.1 Increasing maximum iterations . . . 4.7.2 Decreasing maximum iterations . . . 4.7.3 Decreasing tolerance . . . . . . . . . 4.8 Orbits and periodic points . . . . . . . . . . 4.8.1 Drawing orbits . . . . . . . . . . . . 4.8.2 Periodic points . . . . . . . . . . . . 4.9 Exploring different branches . . . . . . . . . 4.9.1 Monomials with complex powers . . . 4.9.2 Selecting a branch in C++ . . . . . . 4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 18 19 20 20 21 22 23 23 24 24 24 24 24 25 25 26 27 5 Results 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Drawing basic fractals . . . . . . . . . . . . . . . . 5.2.1 Changing the function . . . . . . . . . . . . 5.2.2 Changing maximum iterations and tolerance 5.3 Panning and zooming . . . . . . . . . . . . . . . . . 5.4 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 More interesting functions . . . . . . . . . . . . . . 5.5.1 Trigonometric functions . . . . . . . . . . . 5.5.2 Polynomials with complex powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 28 28 29 30 30 35 35 35 37 6 Testing and Evaluation 6.1 Overview . . . . . . . . . . . . . . . . . . 6.2 Function parser . . . . . . . . . . . . . . 6.2.1 Creating mathematical functions 6.2.2 Polynomial derivative . . . . . . . 6.3 Fractals . . . . . . . . . . . . . . . . . . 6.3.1 Basic fractal drawing . . . . . . . 6.3.2 Panning, zooming etc. . . . . . . 6.3.3 Automatic root finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 38 38 38 38 38 38 40 41 . . . . . . 42 42 42 42 44 45 45 7 Conclusions 7.1 Overview . . . . . . . . 7.2 Achievements . . . . . 7.2.1 Targets met . . 7.2.4 Missing features 7.3 Further work . . . . . 7.3.1 Function parser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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Contents 7.3.2 Contents More multivalued functions . . . . . . . . . . . . . . . . . . . 45 8 Appendix A: Project Plan 47 Bibliography 52 v Chapter 1 Introduction 1.1 Newton’s method Newton’s Method (or the Newton-Raphson method) is a numerical method for finding the roots of a function f , defined by xn+1 = xn − f (xn ) . f 0 (xn ) If f is a function of the real numbers Newton’s method can be illustrated graphically by drawing the tangent line to f at (xn , f (xn )). xn+1 takes the value where the tangent line intersects the x-axis. An example is illustrated in Figure 1.1. Figure 1.1: Graphical representation of Newton’s method. 1.1.1 Problems with Newton’s method With a ‘well chosen’ initial guess of x0 , Newton’s method converges quadratically. If x0 is not ‘well chosen’, Newton’s method may not converge to the nearest root. In some cases, Newton’s method will not converge at all. This usually occurs when x0 is in the vicinity of a local extremum [14]. The first graph in Figure 1.2 shows that when xn is near a local extremum it may not converge to the root to the nearest root. 1 Chapter 1. Introduction 1.2. Newton fractals The second graph shows the worst case scenario for Newton’s method, where (xm , f (xm )) is a local extremum. If xm is not a root, it is clear that the tangent line will never meet the x-axis, so Newton’s method will not converge at all1 . Figure 1.2: Points near critical points are not ‘well-behaved’. 1.2 Newton fractals The behaviour of Newton’s method can be captured in a coloured diagram. The colour shows which root a point converges to, and the shading shows how quickly it converges. We will see in Example 1.2.3 that these coloured diagrams are fractals. 1.2.1 Fractals Although there is no universally accepted definition of a fractal [1], it is possible to identify one by some properties. Two of these properties [10] are: • A fractal is self-similar - subsets “look like” rescaled versions of larger parts of the fractal. • A fractal has an arbitrarily fine structure - more detail is revealed when magnified. 1.2.2 Basic construction of Newton fractals Given a domain D and a function f : D → D, assign each root ri of f with a colour ci . Choose a colour cnot to represent non-convergence. For each2 x ∈ D we perform Newton’s method on f . If x converges to a root ri in s steps, assign x with the colour ci , adjusting its brightness of the colouring depending on s. If x does not converge to a root, assign x with the colour cnot . This method creates an object which demonstrates the dynamical behaviour of Newton’s method. When analysed in a little detail it is easy to see that this object is actually a fractal. m) For these points we know that f 0 (xm ) = 0, so it impossible to compute ff0(x (xm ) . 2 Of course it is usually impossible to go through all elements in D. It is much easier to consider the points as pixels on a computer screen. 1 2 Chapter 1. Introduction 1.3. The project Example 1.2.3. Define f : R → R by f (x) = −2x3 + 9x2 − 11x + 3. This has the graph shown in Figure 1.3, which shows√f has three real roots at z0 = 1.5, √ 1 z1 = 2 (3 − 5) ≈ 0.381966 and z2 = 12 (3 + 5) ≈ 2.61803. Assign the following colours to these roots: Root z0 z1 z2 Colour red blue green 10 5 -1 1 2 Figure 1.3: Graph for f (x) = −2x3 + 9x2 − 11x + 3. If we construct a Newton fractal using the method described above, we obtain the one-dimensional object shown in Figure 1.4. When magnifying the object at x = 1, another green ‘strip’ appears followed by another blue ‘strip’, and so on. This shows the self-similar nature of the object and so this object is indeed a fractal. Example 1.2.4. Define f : C → C by f (z) = −2z 3 + 9z 2 − 11z + 3. This is the same function with the same roots as the previous example but we are now working in the complex plane. Constructing a Newton fractal in the same method, we obtain the object shown in Figure 1.5. The set of real numbers is a subset of the complex numbers, so the pattern of colours in Figure 1.4 reappears on the real line in Figure 1.5. As with the fractal in Example 1.2.3, magnifying the object at z = 1 reveals more detail. 1.3 The project Different choices for the function f creates different Newton fractals. The differences can be quite startling, but there are not many examples of Newton fractals in the public domain. Existing systems that draw Newton fractals are introduced in Section 2.3. These systems have restrictions on the number and types of functions it can draw Newton fractals for. This project focuses on a system which was created to draw Newton fractals for a wide range of functions, and to analyse the results in more detail. 3 Chapter 1. Introduction 1.3. The project ↓ ↓ ↓ Figure 1.4: Newton fractal for f (x) = −2x3 + 9x2 − 11x + 3 on R (top). Successive images show magnifications on the fractal at the point x = 1. The fractals have been stretched to make it easier to see the details. Figure 1.5: Newton fractal for f (z) = −2z 3 + 9z 2 − 11z + 3 on C. 4 Chapter 1. Introduction → 1.4. Report overview → → Figure 1.6: Magnifying the fractal centred at z = 1. 1.4 Report overview Chapter 2 introduces the important mathematical concepts behind the project. This chapter also introduces existing fractal systems which can also draw Newton fractals. Chapter 3 outlines the requirements of the system and the desired underlying structure of the system. Chapter 4 discusses the main technique used to draw Newton fractals and how to make various features efficient. Chapter 5 illustrates the functioning system and describes how it is intended to be used in practice. Chapter 6 outlines how the various components of the system were tested. Chapter 7 summarises the achievements of the project and how it could be improved. 5 Chapter 2 Background 2.1 Overview This chapter outlines the possible features of the system. Before this is possible it is necessary to introduce the mathematical background and a couple of existing fractal systems. 2.2 Terminology Definition 2.2.1. A (discrete time) dynamical system is a map φ : X → X on a state space X. We will denote this by {X; φ}. Suppose Newton’s method is used to find the roots of a function f : C → C. }. Then the dynamical system we will consider is {C; z − ff0(z) (z) Notation 2.2.2. Let {X; φ} be a dynamical system. Denote the n-th iterate of φ by φn (z0 ) = φ ◦ φ ◦ . . . ◦ φ(z0 ) = zn {z } | n times Definition 2.2.3. Let {X; φ} be a dynamical system. The orbit of a point x ∈ X is {φn (x)}∞ n=0 , the sequence of points obtained from continually applying φ. For Newton’s method, the orbit of the initial guess z0 is the sequence of successive approximations to a root. 2.2.4 Petals Let φ(z) = z − ff0(z) describe the Newton iteration for a function f : C → C. Given (z) the Newton fractal for f (z), a path-connected subset P ⊂ C is (informally) called a ‘petal’ if for all p ∈ P , φn (p) → r as n → ∞, where f (r) = 0. Informally, a petal is a subset of a Newton fractal whose points all converge to the same root. The word ‘petal’ derives from the petal-like shape of these subsets of the Newton fractal for f (z) = z 3 − 1 (see Figure 2.1). 6 Chapter 2. Background 2.2. Terminology Figure 2.1: ‘Petals’ of the Newton fractal for f (z) = z 3 − 1. 2.2.5 Branches Suppose f : C → C is a complex multivalued function. For the purposes of this project, a branch point of f is the point at infinity or a singularity of f . A curve joining two branch points is known as a branch cut of f . It is important to note that the values of f (z) are discontinuous either side of a branch cut. A branch of f is a restriction on the range of f , over which it is a single-valued function. A branch is characterised by a choice of branch cut. For this project, a branch will be defined by a straight line joining a singularity of f with the point at infinity. Example 2.2.6. Let f (z) = ln z = ln |z| + i(arg z + 2mπ), m ∈ Z. This is a multivalued function with singularity at z = 0. By choosing the range in which i(arg z+2mπ) must lie, the values of f (z) become unique for all z. There are infinitely many possible branch cuts, two of which are shown in Figure 2.2. Branch cut Branch cut 0 0 Figure 2.2: Possible branch cuts of ln z. Values of ln z across the branch cuts are discontinuous. 7 Chapter 2. Background 2.3 2.3. Existing systems Existing systems There are several systems capable of rendering Newton fractals and other dynamical systems. Fractal Science Kit (FSK) by Hilbert and Ultra Fractal (UF) by Phreakware both are feature rich and are more flexible systems than the others available. 2.3.1 Features and capabilities Fractal Science Kit Figure 2.3: Fractal Science Kit Fractal Science Kit [7] allows the user to create many different types of fractals. It has such a large number of customisation options available that it is clearly targeted at experts who are familiar with the subject. Its features include: • Show the orbit of points. The user clicks on a point on the plane and the system shows the point’s orbit. • Many types of fractals are available such as Mandelbrot, Newton and Sierpinski. • Two colouring methods - ‘standard’ or orbit trap - both of which can be edited. Standard The colour of a point is determined by the number of iterations of the dynamical system before it meets the stopping criterion. It is important to note that, for FSK, the colours do not indicate which root the point converged to. 8 Chapter 2. Background 2.3. Existing systems Orbit trap Orbit traps can create interesting images, but for the purposes of this project we will not be discussing or exploring them. • Save and load settings, which are stored in XML files. This is a particularly useful feature because of the vast number of options available. Ultra Fractal Figure 2.4: Ultra Fractal Ultra Fractal [12] has fewer options than FSK and is a lot easier to use as well. It seems that its main purpose is for creating fractal art. Its features include: • Option to auto-adjust the maximum number of iterations, which changes as the user zooms in or out. This means that, unlike a lot of other systems, UF can achieve a very deep level of zoom without losing detail. • Fractal images are interlaced as they are being calculated. This is useful when many iterations are being performed. • The user can specify the exponent, p ∈ C, and a root, c ∈ C, for the Newton fractal, where the function f is f (z) = z p − c. • Animation creation. The user can move around the fractal or change some variables, which can then be animated. • Save and load settings using proprietary ufr files. 9 Chapter 2. Background 2.3.2 2.4. Proposal of features for a new system Comparison of existing systems Both FSK and UF have basic functions such as exporting the fractal details to XML and ufr files respectively. Both systems can also change the variables which control the tolerance or maximum number of iterations. They both can also change the colouring gradient of the fractal, although they both only colour according to the number of iterations and not the root the Newton fractal converges to. FSK has a lot more variation in the fractal types than UF. Each fractal type can be tweaked further and transformations applied as well. Its interface for changing the properties is complicated and can be overwhelming for a new user. Conversely, UF’s interface is a lot simpler, but also gives the user fewer options. However, UF does allow the user to manually edit the fractal template files, which enables the user to change any parameter used to generate the fractal. The zoom depth of FSK is inferior, and appears to lose accuracy for deep levels of zoom regardless of the number of iterations set. 2.3.3 Required features To investigate the behaviour of Newton fractals, some features found in the existing systems will be needed, along with features not found in either system. These are: Given a function, draw the corresponding Newton fractal. The existing systems are able to draw different Newton fractals, but they are limited to preset functions with some adjustable parameters. UF allows the user to freely edit the Newton iteration, but this requires the user to change the template code and might not be suitable for non-expert computer users. Changeable variables at runtime. Both systems are able to change variables such as the maximum number of iterations and tolerance. Ability to view different branches of multivalued functions. UF appears to be the only system that handles multivalued functions. However, it only shows the principal branch of the function. Orbits of points/subsets of the fractal. FSK is the only system that shows the orbits of points. However, because its colouring schemes do not indicate which root points converge to, this feature is not very useful or interesting. Animation. Ultra Fractal is capable of animating various changes made to the fractal. The animations it produces are not very sharp because each frame is calculated while the animation is running. 2.4 Proposal of features for a new system The proposed system should include the following features: 10 Chapter 2. Background 2.4. Proposal of features for a new system • Given a user-specified function f and its derivative f 0 , draw its Newton fractal. It would be desirable for the user to only supply f , and then for the system to calculate f 0 and the roots. • Changeable variables at runtime. This will include the tolerance value and maximum number of iterations, which will be auto-adjustable. • View different branches of multivalued functions. The user should be able to choose the values to which the arguments of the branch points will be restricted. • Orbits of points/subsets of the fractal. This feature will be similar to Fractal Science Kit’s orbit feature, and will be extended to investigate if points in ‘petals’ share similar orbits. • Animation of gradual changes to certain parameters of the function. • A simple colouring system that can be customised to show the root a point converges to or the number of iterations. This may be extended to include other kinds of colouring schemes based on other criteria. 11 Chapter 3 Design 3.1 Overview This chapter outlines the requirements of the system and describes the intended structure of the finished system. 3.2 Functional requirements After researching the existing fractal systems in the previous chapter, the following functional requirements were identified for a new system: Parse string as a mathematical function The key feature missing from the other fractal systems is the ability to draw Newton fractals for a user-specified function. A string parser was designed to handle real polynomials initially, followed by complex polynomials and then additions such as trigonometric and exponential functions. Draw fractal Edit fractal parameters This includes the tolerance and maximum number of iterations values. Pan/zoom around fractal Basic functions for exploring the fractal. View orbits of points Edit branch of the function Save/load fractal settings This feature was not implemented due to lack of time (see Section 7.2.4). It was desirable for the system to be able to save the settings (the function, tolerance, maximum number of iterations, the complex plane coordinates, all pixel data) to file. It could then have enabled the user to save interesting views of a fractal to file and then load it with minimal time spent recalculating the fractal. 12 Chapter 3. Design 3.3. Non-functional requirements Create fractal animation This feature was not implemented due to lack of time (see Section 7.2.4), and was possibly too ambitious to implement because of other subject commitments. Changes to parameters of functions such as coefficients or powers can show dramatically different fractals. If the gradual changes to the parameters is animated, it could provide interesting animations. 3.2.1 Component interactions Figure 3.1 shows how the different functional requirements interact with each other. ts rea d wri s/ tes se s set es lu va Fractal plotter Fractal settings file nts pri ines def ts n poi on cti Colour New met ton's hod alu es un sf uses sv set Orbit plotter Edit parameters s et String parser set fram s es Pan/Zoom Animator Image file Figure 3.1: How components interact with each other 3.3 Non-functional requirements The following non-functional requirements were identified to aid the design of the system: Screen resolution Over time, many computer users adopt higher display resolutions. This trend can be verified by the display statistics for visitors to W3Schools, see Table 3.1. Although the fractal system is not a web application, these statistics give a good insight into a potential users’ display resolution. Date Jan 2009 Jan 2008 Higher 1024 × 768 800 × 600 640 × 480 Unknown 57% 36% 4% 0% 3% 38% 48% 8% 0% 6% Table 3.1: Browser display statistics for W3Schools [11] From this it is clear that the system should be designed for resolutions of 1024 × 768 and higher. 13 Chapter 3. Design 3.4. Class diagram Performance Newton’s method is performed for each point on the screen. This means that drawing a fractal involves a lot of calculations, particularly for non-convergent points. The system needs to be as efficient as possible and to reuse any information from previous calculations. Usability The user should be able to make quick changes to the function and other parameters. The interface should be kept simple. 3.4 Class diagram The important classes for the system were identified and an outline class diagram of the system was detailed as in Figure 3.2. This was largely followed except the system only allowed one instance of a Fractal. This was for simplicity purposes because of the lack of experience using Qt to show tabs. Given more time, this could be implemented. An addition to this design involved storing more data in each Pixel. For more information, see Chapter 4. 3.5 GUI design A GUI was needed for the benefit of the user, typically a mathematician. It was decided that all the features of the system should be available on the same screen, reducing the number of menus. A GUI design (see Figure 3.3) helped with ensuring that all the important features were accounted for. As noted above, multiple instances of Fractals was not implemented. The toolbar of buttons across the top was omitted because not all features were implemented. Instead, all features were included in the panel on the right. The progress bar on the bottom was replaced by the fractal being rendered gradually. 14 Chapter 3. Design 3.5. GUI design Fractal -f: mathfunction -derivative: mathfunction -roots: list<Root> -bottomLeft: complex -topRight: complex -maxIterations: unsigned int -tolerance: double -screenSize: pair<int, int> -autoAdjust: bool -displayBuffer: Pixel[][] -branchLowerBound: double +setFunction(newF:string): bool +setDerivative(newDeriv:string): bool +addRoot(root:complex): bool +removeRoot(root:complex): bool +clearRoots(): void +setBottomLeft(bl:complex): void +setTopRight(tr:complex): void +save(filename:string): bool +load(filename:string): bool +setScreenSize(width:int, height:int): bool +setAutoAdjust(state:bool): void +draw(): void +drawOrbit(c:complex): void +setBranchLowerBound(b:double): void +getBranchLowerBound(): double 0..* displays creates 1 FractalPlotter 1 -fractals: list<Fractal> +addFractal(): bool +addFractal(filename:string): bool + removeFractal(f:Fractal): void Animation 0..* -frames: Fractal[] -speed: double +addFrame(f:Fractal): void +clearFrames(): void Pixel -root: Root -numIterations: int 1..* -colour: Colour +getRoot(): Root +getNumIterations(): int +getColour(): Colour +setRoot(r:Root): void +setNumIterations(n:int): void +setColour(c:Colour): void 1 1 0..* colours characterised by 1 1..* Colour Root -colour: Colour -value: complex +getColour(): Colour +getValue(): complex +setColour(col:Colour): void +setValue(value:complex): void colours 0..* 1 Figure 3.2: Class diagram 15 -red: double -green: double -blue: double +getRed(): double +getGreen(): double +getBlue(): double +setRed(r:double): void +setGreen(r:double): void +setBlue(r:double): void Chapter 3. Design 3.5. GUI design Newton Plot File Help New fractal Open Save Fractal1 Pointer Move Zoom in Zoom out View orbit Anim. Create frame anim. Step size Fractal2 Function a t c I l a r F e g a m f(z)= z^3 - 1 f'(z)= 3z^2 Roots -0.5+0.866i, 1, -0.5-0.866i Options Tolerance 0.01 Max iterations 20 Auto adjust Location Bottom left -2 - 2i Top right 2 + 2i Fractal Width 512 Height 768 Branch -3.14159 <= arg(z) < 3.14159 Rendering... Figure 3.3: GUI design 16 Chapter 4 Implementation 4.1 Overview This stage of the project involved the implementation of the ideas planned in the design stage. This chapter focuses on the main ideas behind the calculating of Newton fractals and the techniques used to create an efficient system. 4.2 4.2.1 Technology choices Programming language The fractal system was written in C++. Although I had a more exposure to Java, I decided to use C++ because I wanted more experience with it. The system also needed to be as efficient as possible because of the number of calculations involved in Newton’s method. This made C++ more ideal than Java. 4.2.2 Fractal drawing I had no previous experience creating a graphical user interface (GUI) for a C++ application, so the Qt framework was chosen because of its relative ease of use. OpenGL was chosen for the actual drawing of the fractals because of past experience. This ultimately became the QtOpenGL module when the fractal drawing was integrated with Qt. 4.2.3 String parsing I had no prior experience of string parsing, so I was recommended to use Flex for the lexical analysis of a string and GNU Bison for the actual parsing. 4.2.4 Development environment The integrated development environment (IDE) of choice was Visual Studio. A Qt add-in for Visual Studio also made it easier to understand the basics of creating Qt applications. 17 Chapter 4. Implementation 4.3. Drawing a fractal Eclipse was later chosen as the IDE to use in Linux (see Section 4.2.6). 4.2.5 Dependencies Aside from the obvious dependency on the Qt and QtOpenGL libraries, the only external libraries used were the Boost C++ libraries. These were primarily used as part of the string parser. 4.2.6 Platform It was always desirable, but not essential, for the system to be cross-platform. This is reflected in the choices of C++, OpenGL and Qt, which are all available on multiple platforms. The system was initially developed in Windows but problems with the slow speed of the system meant that development switched to Linux where there were no such problems. Qt’s qmake, which generates makefiles for Qt applications, made the transition to Linux relatively straightforward. The reason for the speed problem in Windows was never identified, but these problems were later resolved by resetting the project settings in Visual Studio. Consequently, the system runs in both Windows and Linux with little speed difference. 4.3 Drawing a fractal A Newton fractal is drawn using the calculate fractal function shown in Listing 4.1. In short, the function performs Newton’s method for each pixel on the screen using the find root function. This returns the root the point converged to and the number of iterations it took to converge. This information is used to set the colours of a screen buffer. The colour of a point p is determined by: • p.root - the root it converges to. • p.numIts - the number of iterations taken to converge. Each root is characterised by a Colour, an RGB triplet. Suppose we have p.root.colour = (r, g, b). Denote a brightness factor bri = 1 - (p.numIts / maxIterations ). Then p.colour = (bri * r, bri * g, bri * b). This leads to a fractal where brighter areas converge quickly and darker areas converge slowly. By default, non-convergent points are coloured black. 4.3.1 Time complexity Suppose we have a screen w pixels wide by h pixels high. Set m as the maximum number of Newton iterations. Then the best case time complexity is O(wh) and the worst case time complexity is O(whm). It is clear that if we have a large screen and/or m is large, then this method will be extremely slow. However, it is important to note that it is the most accurate way of drawing a Newton fractal. 18 Chapter 4. Implementation 4.4. Panning and zooming around the fractal void F r a c t a l : : c a l c u l a t e f r a c t a l ( ) { point info ptinfo ; // Blank the screen . b l a n k f l i s t ( ∗ c u r r e n t L i s t , Colour : :NONROOT) ; // Go through each pixel on screen . for ( int j = s c r e e n H e i g h t − 1 ; j >= 0 ; −−j ) { f o r ( int i = 0 ; i < s c r e e n W i d t h ; ++i ) { // Perform Newton ’s method . Sets colour info , etc. // to_complex converts a screen point to its complex // representation . ptinfo = find root ( t o c o m p l e x ( i , j , p l a n e , screenWidth , s c r e e n H e i g h t ) , maxIterations ) ; // Set the pixel information to the fractal buffer . s e t f l i s t (∗ c u r r e n t L i s t , i , j , p t i n f o ) ; } // Draw gradually . i f ( j % 256 == 0 ) updateGL ( ) ; } updateGL ( ) ; } Listing 4.1: Function for drawing a fractal To overcome this problem, the system reuses information when navigating around the fractal (Section 4.4), evaluates polynomials using Horner’s rule (Section 4.6), reuses information when certain variables are changed (Section 4.7) and detects periodic points to avoid unnecessary iterations (Section 4.8). 4.4 Panning and zooming around the fractal The screen represents a subset V of the complex plane which is represented by the class variable plane in the Fractal class. plane consists of a pair of complex numbers representing the bottom left and top right coordinates of V . The values of plane change when panning or zooming around the fractal. We have seen that calculating a fractal can be slow, so the system was designed to avoid recalculating known information where possible. 19 Chapter 4. Implementation 4.4.1 4.4. Panning and zooming around the fractal Panning Let A ⊂ C denote the original view of the complex plane and let γ : C → C be a translation defined by γ(z) = z + d, d ∈ C. Then the panned view A0 = γ(A). If, as in Figure 4.1, A ∩ A0 6= ∅ then the colour information in this overlapping region can be reused when drawing A0 . Figure 4.1: Panning around the plane. The pixel information in the regions where A and A0 overlap can be reused. This observation is used in the panning function. If the point was represented in the original view, the pixel information is copied from the old screen buffer to its new position on the next screen buffer. The memcpy C++ function is used for directly copying the pixel information, making this process more efficient. The colours of the other points are calculated in the usual way using Newton’s method. The use of a second screen buffer means that more memory is used. Unfortunately, this is necessary because the system would otherwise be extremely slow. 4.4.2 Zooming A similar observation can be made for zooming out. In Figure 4.2, all of the information in the original view A is contained in the zoomed-out view A0 . In a similar implementation to the panning function above, all pixel information in A can be copied to its new position in A0 . Figure 4.2: Zooming out from A shows a wider view of the fractal, A0 . In this case A0 contains all of the points in A, so we can reuse all of this information. However, there is a noticeable difference between the panning and zooming implementations. For panning, all the information in the overlapping region is reused 20 Chapter 4. Implementation 4.5. Parsing strings as functions whereas after zooming out, the original view is displayed using a smaller part of the screen. So although information is reused, some of it is discarded. This causes a slightly less accurate representation of the fractal to be drawn, which can be seen in Figure 4.3. Therefore, this method of zooming out is an optional feature. Figure 4.3: Difference between reusing information (left) and recalculating everything (right). Notice the different bands of the green region at the bottom are less ‘smooth’ when reusing information. For zooming in to the fractal, there is no immediately obvious technique for improving the speed, so the system performs Newton’s method on every point. 4.5 Parsing strings as functions The string to mathematical function parser was implemented using Flex, a lexical analyser, and GNU Bison, a parser generator. I had no experience with string parsing and time restrictions made it difficult to learn Flex and Bison. Incorporating Bison in C++ code was particularly difficult. Therefore, a Flex Bison C++ Template [6] was used to write the parser. The lexical analyser (scanner) reads in the input string. It has a set of rules for recognising numbers and mathematical terms. These rules determine which token to return to the parser. The parser has a set of grammar rules, each comprising of a pattern of tokens. If one of the grammar rules matches, the parser builds a mathematical function accordingly. The scanner and parser are brought together by the Driver class. The Driver acts as an intermediate class between the Fractal class and the parser. The parser was developed incrementally, at each stage adding support for: 1. Real polynomials. 2. Complex polynomials (including complex powers). 3. Trigonometric, exponential and logarithmic terms. 21 Chapter 4. Implementation 4.6. Evaluating functions At the end of the implementation stage, the parser handles a linear combination of monomial, sin(z), cos(z), tan(z), exp(z) and log(z) terms. There is scope to improve this - see Section 7.3.1 for more information. 4.6 Evaluating functions The function f (z) supplied by the user consists of a linear combination of monomial, trigonometric, exponential and logarithmic terms. Newton’s method is called multiple times per pixel and the value of f (zn ) and f 0 (zn ) is required for the (n + 1)th iteration. This requires the evaluation of a function to be as efficient as possible. The system has a Polynomial class which is made up of Monomials. To evaluate a Polynomial at the point z0 , one can either: 1. evaluate all the Monomials at z0 , and sum the results. 2. use Horner’s rule. Definition 4.6.1. (Horner’s rule) [13] Let n X p(z) = ai z i = a0 + a1 z + a2 z 2 + . . . + an z n i=0 be a polynomial, where ai ∈ C (0 ≤ i ≤ n). By factoring out powers of z, p(z) can be rewritten p(z) = a0 + z(a1 + z(a2 + . . . + z(an−1 + an z) . . .)) In general, method (1) involves more multiplications than Horner’s rule. In the fractal system, Horner’s rule has been implemented for evaluating polynomials with real powers. All other terms are evaluated individually. The implementation of Horner’s rule uses an array, called terms, of complex numbers to store the monomial terms. Each index represents the power of the monomial and each element represents the coefficient. For example, the polynomial p(x) = 3 − 2z + 4z 3 will be stored in terms as follows: index 0 1 2 3 ... coefficient 3 -2 0 4 . . . max index 0 Note that terms has a fixed maximum index, so polynomials of a higher order have to be evaluated using the other method. This array allows direct access to the coefficients. This is quicker than sorting the Monomials in order of power and then calling the accessor function to retrieve the coefficient. The highest power (or order) of the polynomial is also stored so that only the necessary multiplications are performed. The implementation of Horner’s rule is shown in Listing 4.2. 22 Chapter 4. Implementation 4.7. Changing tolerance and maximum iterations s t d : : complex<f l o a t > v a l u e = 0 . 0 f ; // Start from highest power . for ( int i = o r d e r ; i >= 1 ; −−i ) { // Add the next coefficient and multiply by z. v a l u e = ( v a l u e + terms [ i ] ) ∗ z ; } // Add the constant term. v a l u e += terms [ 0 ] ; Listing 4.2: Horner’s rule 4.7 Changing tolerance and maximum iterations The maximum iterations and tolerance may be adjusted at runtime. These are denoted by the class variables maxIterations and tolerance respectively. The screen buffer is a two-dimensional array of Pixels. A Pixel stores • colour - the Colour of the pixel. • root - the Root to which it converges. • numIterations - the number of iterations performed. • lastPt - the last point in the orbit. • isPrePeriodic - whether it is a pre-periodic point (see Section 4.8). There is a lot of information which can be reused to minimise the number of repeated calculations. This is demonstrated below with regards to changing the tolerance and maximum iterations, and in Section 4.8 for some non-convergent points. Note that if the maximum iterations is changed, this changes the value of the brightness factor mentioned in Section 4.3. This means that, even if no additional calculations are necessary, the colour of all points will still have to be calculated again. 4.7.1 Increasing maximum iterations Suppose maxIterations is increased from m to n. Then all Pixels p with p.numIterations <= m have already converged. For these points, it is sufficient to copy the old information and only calculate the new colour. The remaining Pixels q have q.numIterations > m, i.e. q did not converge. Note that the system has stored the last point in the orbit of q, so the first m iterations do not need to be performed again. Therefore, only a maximum of (n m) iterations needs to be performed for each q. 23 Chapter 4. Implementation 4.7.2 4.8. Orbits and periodic points Decreasing maximum iterations Suppose maxIterations is decreased from m to n. Then all Pixels p with p.numIterations <= n still converge under the new maxIterations . Again, it is sufficient to copy the old information and calculate the new colour. The remaining Pixels q have q.numIterations > n and therefore do not converge under the new maxIterations . q.colour can be set to black instantly. However, the first n iterations will need to be performed again in order to set a new q.lastPt . I decided it was important to store q.lastPt because of the tremendous speed benefit it brings when increasing the maximum iterations. Of course this adds extra computation time when decreasing the maximum iterations, which is unfortunate. 4.7.3 Decreasing tolerance If the tolerance is decreased, more iterations will need to be performed for most Pixels p. As with increasing the maximum iterations, p.lastPt is used to avoid performing the first p.numIterations iterations. When the tolerance is increased, there is no obvious method to speeding up the calculation. Therefore the whole fractal is recalculated. 4.8 4.8.1 Orbits and periodic points Drawing orbits The orbit (see Definition 2.2.3) of a point p is determined by performing Newton’s method for p. The orbit is stored as a vector of complex numbers, denoted here as orb. The system simply draws the orbit as a set of connected lines from orb[i] to orb[i+1] for each i < orb.size()-1. This is illustrated in Figure 4.4. 4.8.2 Periodic points In the Newton fractal for f (z) = z 3 − 2z + 2 there are many ‘large’ black areas - subsets which do not converge. As seen in Figure 4.5 the orbits of all points in these black areas eventually oscillate between one black region and another. It turns out that, with enough iterations, this settles into a period 2 orbit where the orbit oscillates between the points 0 and 1. This behaviour is discussed in more detail in Section 7.2.1. The points that oscillate between each other are called periodic points. The points which eventually settle into a periodic orbit are called preperiodic points. Note that periodic points may have period greater than 2. It is clear that periodic and preperiodic points will never converge to a root. If not handled appropriately, each of these points will take maxIterations iterations before Newton’s method terminates. To avoid this, when performing Newton’s method the system stores the orbit in a temporary vector. After each iteration, the system checks whether the current 24 Chapter 4. Implementation 4.9. Exploring different branches Figure 4.4: The orbit of a point in f (z) = z 3 − 1. point is a point in the last m points of the orbit (m a fixed natural number). If it is, a periodic point has been detected and Newton’s method terminates. 4.9 Exploring different branches Due to a lack of time, the only support for multivalued functions is for polynomials with complex powers. C++ has no native support for multivalued functions, so the C++ function pow(x, y) only returns the value of xy for the principal branch. Similarly, abs(z) returns the argument of z in the principal branch. 4.9.1 Monomials with complex powers α Consider a monic monomial f (z) = z α , α ∈ C. Using z α = elog z it can be shown that f (z) is a multivalued function: f (z) = z α = rα eiα(θ+2πn) (4.1) where r = |z|, n ∈ Z. A different choice of n selects a different branch with the cut along the negative real axis. 25 Chapter 4. Implementation 4.9. Exploring different branches Figure 4.5: The orbit of a point in f (z) = z 3 − 2z + 2. 4.9.2 Selecting a branch in C++ The new pow function in Listing 4.3 was written to implement Equation 4.1. complex<f l o a t > pow ( complex<f l o a t > z , complex<f l o a t > p , int n ) { return pow ( abs ( z ) , p ) ∗ exp ( p ∗ complex<f l o a t >(0.0 f , a r g ( z ) + ( 2 . 0 f ∗ PI ∗ ( f l o a t ) n ) ) ) ; } Listing 4.3: Selecting branches To allow a branch cut which is not along the negative real axis, the system has to use two choices of n. Suppose we set lb to be the lower bound of the branch. We can define lowerN, the smaller possible n, by lowerN = floor((lb + PI)/(2*PI)). It is then possible to determine what value of n to use by observing Figure 4.6. All points which lie between lb and the negative real axis uses n = lowerN. Otherwise, n = lowerN + 1. The code for this is in Listing 4.4. int F r a c t a l : : g e t p o i n t n ( complex<f l o a t > z ) { // The actual argument of z. f l o a t argZ = a r g ( z ) + ( 2 . 0 f ∗ PI ∗ branchN ) ; 26 Chapter 4. Implementation 4.10. Summary Up p Lo er b o we r b und ou nd Figure 4.6: The choice of n depends where the point is in C. // Decide on choice of n depending on where z lies. i f ( argZ >= l b ) return lowerN ; else return lowerN + 1 ; } Listing 4.4: Choosing the correct n. Like trigonometric functions, computing complex powers is very time consuming. When the user changes branches, as long as lb is not changed by more than 2π, it is possible to reuse information like when panning and zooming. The only recalculations required are the points with argument between the new and old values of lb. 4.10 Summary The basic ideas behind drawing a Newton fractal are very simple. This chapter has focused on the techniques used to make the features of the system as efficient as possible. 27 Chapter 5 Results 5.1 Overview This chapter describes how the system is intended to work in practice. 5.2 Drawing basic fractals As mentioned in Section 3.5, the GUI was designed to include all features on the same screen. Figure 5.1 shows the system after it starts, displaying the Newton fractal for f (z) = z 3 − 1 by default. Figure 5.1: The Newton Plot GUI. 28 Chapter 5. Results 5.2.1 5.2. Drawing basic fractals Changing the function The function used to draw the Newton fractal can be changed. As expected, this is done by changing the f (z) field. As shown in Figure 5.2, the user can also choose to specify the derivative in the f 0 (z) field and the roots in its respective field. Using the values of f (z), f 0 (z) and roots in 5.2, the user can click the Set button (or press Enter) to draw the Newton fractal in Figure 5.3. Figure 5.2: Fields related to the function. Figure 5.3: The Newton fractal for f (z) = z 3 − 2z + 2. For polynomials it is also possible to leave the f 0 (z) field blank because it can be calculated automatically. Likewise, it is possible to leave the roots field blank for any function. In this case, Newton’s method is used to approximate the roots instead. 29 Chapter 5. Results 5.3. Panning and zooming If either field is left blank, these values are printed in blue to inform the user the values were calculated by the system. This is shown in Figure 5.4. Figure 5.4: f 0 (z) and the roots have been calculated by the system. For some context, the “Show roots” checkbox may be used to display the roots of f (z). The “Show axes” checkbox shows the real and imaginary axes. Figure 5.5 shows the axes and the roots as white crosses. 5.2.2 Changing maximum iterations and tolerance The user is free to change the maximum iterations at runtime using the field in Figure 5.6. Increasing this the maximum iterations to 50 creates a brighter fractal (Figure 5.7) whereas decreasing it to 10 iterations creates many dark ‘non-convergent’ areas (Figure 5.8). Similarly, changing the tolerance is also possible at runtime. An extreme example is shown in Figure 5.9, where the tolerance is set to 1.2. 5.3 Panning and zooming There are a few different ways to explore a fractal: Panning • Arrow keys pan in their respective directions. • Dragging with the right mouse button pans the camera in that direction. 30 Chapter 5. Results 5.3. Panning and zooming Figure 5.5: Showing the roots and axes. Figure 5.6: Fields to tweak the maximum iterations and tolerance. 31 Chapter 5. Results 5.3. Panning and zooming Figure 5.7: Increasing to a maximum of 50 iterations. Figure 5.8: Decreasing to a maximum of 10 iterations. 32 Chapter 5. Results 5.3. Panning and zooming Figure 5.9: Increasing tolerance to 1.2. Zooming • The + and - keys zoom in and out respectively. • The mouse wheel scrolled up and down zooms in and out respectively. • Dragging the left mouse button draws a box. The system zooms to the region in the box. It is also possible to state the bottom left and top right coordinates of the fractal to show, as in Figure 5.10. Figure 5.10: Fields to adjust the coordinates of the fractal. Examples of panning, zooming in and zooming out are show in Figures 5.11, 5.12 and 5.13 respectively. 33 Chapter 5. Results 5.3. Panning and zooming Figure 5.11: Panning across the fractal. Figure 5.12: Zooming into an area. 34 Chapter 5. Results 5.4. Orbits Figure 5.13: Zooming out. 5.4 Orbits There are two methods for viewing the orbits of points on the fractal. Middle click The user middle clicks on a point on the fractal. The orbit for this point is drawn. Mouse move The orbit is drawn for the point under the mouse pointer. The orbit changes as the mouse pointer is moved. An example of an orbit is shown in Figure 5.14. The orbit starts in a blue region near the top right and converges to the root in the blue region at the bottom. 5.5 5.5.1 More interesting functions Trigonometric functions By specifying f (z) = sin(z) we get a very interesting fractal in 5.15. sin(z) has infinitely many roots and so it is helpful that the user does not need to specify any roots at all. The system also uses a fixed palette of nine colours for up to nine roots. When a fractal has more roots, the palette ‘wraps around’ and reuses colours. There is no feature that allows the user to select their own colours, so this ensures every root is assigned a colour. Of course, if colours are reused, there will be petals of the same colour which converge to different roots. The user can overcome this problem by using the orbits feature to determine which root these regions converge to. 35 Chapter 5. Results 5.5. More interesting functions Figure 5.14: The orbit of a point on the fractal. Figure 5.15: The default (left), far away (centre) and close (right) views of f (z) = sin(z). 36 Chapter 5. Results 5.5.2 5.5. More interesting functions Polynomials with complex powers Suppose the user specifies a polynomial with complex powers. The lower bound of the branch can be set in the “Branch” field, as shown in Figure 5.16. The resulting Newton fractal will have a discontinuity across the branch cut. Figure 5.16: The lower bound of the branch defines the whole branch. by The user can click on the “-” or “+” buttons to decrease or increase the branch respectively. π 6 Figure 5.17: Branches of f (z) = z 3−i − 1: 0 ≤ arg(z) < 2π (left), (centre), π3 ≤ arg(z) < 7π (right). 3 37 π 6 ≤ arg(z) < 13π 6 Chapter 6 Testing and Evaluation 6.1 Overview This chapter outlines the methods used to test various components of the system. 6.2 Function parser The function parser is an important part of the system. A test program was written to test the parser individually before it was integrated into the fractal system. The test program was given a string as input. If successfully parsed, a corresponding instance of a mathfunction was created. 6.2.1 Creating mathematical functions The parser was improved incrementally, adding support for more types of functions each time. At each stage a set of test data was used to test the parser. The test program was expected to output a string representation of the parsed function and its value at arbitrarily chosen real and complex number. The expected output for each stage is listed in Table 6.1 which verifies that the function parser works as expected. 6.2.2 Polynomial derivative The test program also calculated the derivatives of polynomials with the expected output in Table 6.2. The actual output of the test program matches these expected values. 6.3 6.3.1 Fractals Basic fractal drawing The system is capable of drawing the Newton fractal for a wide range of functions. It is sufficient to test functions whose Newton fractals are in the public domain. 38 Chapter 6. Testing and Evaluation f(z) 1 z z^2 z^3 2z -z -3z 3z^3 -z^4 + 4z + 1z^2 + 1 z + z z^i z^(1+i) z^1+i (1+i)z 1+iz iz^(2-i) sin(z) cosz -tan z ln(z) 2log(z) e^z iexp^(z) 6.3. Fractals f(2) f(1-i) 1 1 2 1−i 4 −2i 8 −2 − 2i 4 2 − 2i −2 −1 + i −6 −3 + 3i 24 −6 − 2i −3 9 − 6i 4 2 − 2i 0.769 + 0.639i 2.063 + 0.745i 1.538 + 1.278i 2.808 − 1.318i 2+i 1 2 + 2i 2 1 + 2i 2+i −14.802 + 12.295i 0.858 − 0.310i 0.909 1.298 − 0.635i −0.416 0.834 + 0.989i 2.185 −0.272 + 1.084i 0.693 0.347 − 0.785i 1.386 0.693 − 1.571i 7.389 1.469 − 2.287i 7.389i 2.287 + 1.469i Table 6.1: Test data for the math function parser. f(z) 1 z z^2 z^3 2z -z -3z 3z^3 -z^4 + 4z + 1z^2 + 1 z + z z^i z^(1+i) z^1+i (1+i)z 1+iz iz^(2-i) z^(1+i) - z^2 f’(z) 0 1 2z 3z^2 2 -1 -3 9z^2 -4z^3 + 4 + 2z 2 iz^(-1+i) (1 + i)z^i 1 1 + i 1 + i (1+2i)z^(1-i) (1+i)z^i - 2z Table 6.2: Test data for polynomial derivatives. 39 Chapter 6. Testing and Evaluation 6.3. Fractals The examples in Figures 6.1 and 6.2 show that the fractals drawn by the system are very similar to the existing images. It is therefore possible to verify the method for drawing fractals is ‘correct’. Figure 6.1: Comparing Newton fractals for z 3 − 2z + 2 from Wikipedia [9] (left) and my system (right). Figure 6.2: Comparing Newton fractals for z 8 + 15z 4 − 16 from Wikipedia [5] (left) and my system (right). 6.3.2 Panning, zooming etc. Let A denote a fractal after it has been panned. Let B denote the fractal in the same position with all the points recalculated. Then the correctness of panning can be tested simply by comparing A with B. The same is true for zooming and changing values such as tolerance and maximum iterations. 40 Chapter 6. Testing and Evaluation 6.3.3 6.3. Fractals Automatic root finding The Newton fractals with the roots entered manually or found automatically can be compared to check the latter is correct. Of course, there will be slight differences between the two fractals because the roots are approximated. Figure 6.3 shows this slight discrepancy, but the fractals are mostly the same. Figure 6.3: Comparing Newton fractals for z 3 − 1 with roots entered manually (left) and found using Newton’s method (right). 41 Chapter 7 Conclusions 7.1 Overview The project has been largely successful, but not all targets were met. This chapter summarises the achievements and discusses how the system could be improved. 7.2 Achievements The system has a number of features which make it a useful tool for exploring Newton fractals. However, not all features have been implemented and the system has scope for improvement and further developments. The first few months of the project were spent trying to find interesting research ideas from mathematicians in the School of Mathematics. Some members of staff were away for a period of time, so no progress was made at this time and the project fell behind. In the end, the ideas suggested required a lot of background reading in an area I was unfamiliar with, so these ideas were not used. If the aims of the project had been identified earlier, more time could have been spent on the implementation. I underestimated the time required to understand the string parsing and, given the lack of time, was probably a little ambitious with some features in the project proposal. 7.2.1 Targets met The system meets the requirements in Section 3.2 except for those indicated. In particular, the string parser is able to parse a linear combination of (real and complex) monomials, sin z, cos z, tan z, ln z, ez and |z|. This allows users to investigate the Newton fractals for a wide range of functions. Many of the features such as panning and changing the maximum iterations have been optimised. This satisfies the performance (non-functional) requirement (Section 3.3). Points on the fractal are coloured according to the root it converges to. The system is capable of showing the orbit of any point on the fractal. These two features make it easy to verify both features are working correctly. The orbit feature also 42 Chapter 7. Conclusions 7.2. Achievements demonstrates an interesting theorem on dynamical systems which has been applied to the Newton fractal in Theorem 7.2.2. Theorem 7.2.2. (Non-Wandering Domain Theorem) (Sullivan) [4] Let F be the Newton fractal for f : C → C, where the Newton iteration is φ(z) = z − ff0(z) . Suppose P ⊂ F is a ‘petal’. Then P is eventually periodic under (z) iterations of φ. That is, there exists some S ∈ N such that φS (P ) is a periodic point of φ. If all points p ∈ P converge to a root, the orbit of P ends with the petal containing the root, which has period 1. Figure 7.1 shows the orbits of three distinct points in the same petal. Although these points do not require the same number of iterations to converge, it is clear that each point visits the same petals. One can say that the set of points in the petal behave similarly. Figure 7.2 shows points in a non-convergent petal also share similar orbits. The difference is that these non-convergent petals eventually settle into a period 2 orbit. Figure 7.1: Points in the same petal visit the same petals. Figure 7.2: Points in non-convergent petals eventually settle into a period 2 orbit. Orbits with period greater than 2 Figure 7.2 shows a Newton fractal where the non-convergent petals eventually settle into a period 2 orbit. While testing the system I have not encountered any Newton fractals with period 3 (or greater) petals. 43 Chapter 7. Conclusions 7.2. Achievements There is an interesting result by Li and Yorke which states that, in a dynamical system {X; φ}, a period 3 orbit implies “sensitive dependence on initial conditions” (SDIC) [2]. This means that under iterations of φ any two points x, x0 ∈ X, no matter how close they are, will (eventually) behave differently from each other. This is also a property of a chaotic dynamical system. From our observations from Theorem 7.2.2 we know that all points in the same petals behave similarly. So if we have a Newton fractal which has a period 3 orbit we surely cannot have SDIC at the same time. The following result provides a little more insight into petals of orbit greater than 3. Theorem 7.2.3. (Sharkovskii’s theorem) [3] Order the natural numbers as follows: 3 ≺ 5 ≺ 7 ≺ 9 ≺ . . . ≺ 2 · 3 ≺ 2 · 5 ≺ . . . ≺ 22 · 3 ≺ 22 · 5 ≺ . . . . . . ≺ 23 · 3 ≺ 23 · 5 ≺ . . . ≺ 24 · 3 ≺ 24 · 5 ≺ . . . ≺ 23 ≺ 22 ≺ 2 ≺ 1 Suppose φ has a period p orbit. Then p ≺ q implies the existence of a period q orbit. Sharkovskii’s theorem shows that a period 3 orbit implies orbits of all periods and thus some kind of chaotic nature. We can see that petals with an orbit of period 22 = 4 may be possible in a Newton fractal, although I have yet to find any such fractals. Branches of functions The ability to change the branch of a multivalued function is a feature not seen in any other fractal system. This is probably due to restrictions on the family of functions allowed, such as f (z) = z p − 1, p ∈ N. Ultra Fractal allows p to take imaginary values, but does not give the option to select a different branch. This is probably because its main purpose is for creating fractal art, which means viewing different branches is a low priority feature. If the bounds of the branch are changed gradually, the system reveals more of a Newton fractal beyond a branch cut. It is quite a nice way to demonstrate why a branch cut is needed and how the choice of branch affects the fractal. Gradually adjusting the roots sometimes introduces a new discontinuity which is not the branch cut (as in Figure 7.3). This appears to be because multivalued functions can have multiple roots, so changing the branch may ‘cover’ up a root. Points will no longer be able to converge to this root. This discontinuity shows that the same point can behave differently under different branches. Handling the branches elegantly appears to be a complicated problem. With a lot more time available, there are some possibilities for improving this feature. More details are described in Section 7.3.2. 7.2.4 Missing features Although the system allows the user to conduct some research into Newton fractals, it does not meet all the requirements set in the Design stage. 44 Chapter 7. Conclusions 7.3. Further work Figure 7.3: Gradually changing the branch (left to right) may create new discontinuities. Both Fractal Science Kit and Ultra Fractal could save the settings (the function, tolerance, maximum iterations and complex plane coordinates) of a fractal to file. A similar feature was due to be implemented in the last few weeks of the implementation stage, but was deferred because of a lack of time. The fractal animation feature was also not implemented for similar reasons. Near the end of the implementation, the branches feature took priority because it was deemed more interesting. This was a good decision because the two omitted features already exist in other fractals systems whereas the branches feature does not. 7.3 Further work Aside from the missing features, there are some other possible areas where the system could be improved further. 7.3.1 Function parser The number of fractals the system can draw is limited by the types of mathematical functions the system can parse. Although it can parse a linear combination of trigonometric functions, this is still quite limiting. For example, if we set f (z) = tan z the system would require the user to set f 0 (z) = sec2 z, which is not a linear combination of trigonometric terms. The next natural step for the system would be to improve the function parser to recognise many more kinds of functions. 7.3.2 More multivalued functions The only multivalued functions the system can handle only handle are polynomials with complex powers. The complex analogue of functions such as log(z) could be 45 Chapter 7. Conclusions 7.3. Further work implemented next. The anomalies observed in Section 7.2.1 regarding the fractals of multivalued functions also raises a question: If a point converges to different roots under different branches, how does it behave when all branches are allowed? For example, instead of drawing the complex plane with branch cuts, could a Newton fractal be drawn on a Riemann surface such as in Figure 7.4? Figure 7.4: The Riemann surface for the multivalued function ln z [8] 46 Chapter 8 Appendix A: Project Plan The original project plans and revisions follow. The first revision was made after the direction of the project became more clear. The second revision was made after the Christmas / exam break. This revision includes a lot more detail and was helpful with checking my progress. 47 Project Plan Project Plan Milestone / Deadline Search for related work Read related work Finalise direction of project Poster Design Reading Week Seminar slides Seminar period Implementation (basic) Testing (basic) Christmas revision month Exams Implementation (flexible) Testing (flexible) Testing (final) Investigate fractals I Investigation report I Investigate fractals II Investigation report II Presentation of results Report sample Easter revision month Project report Week Beginning (2009) Week Beginning (2010) 27/09 04/10 11/10 18/10 25/10 01/11 08/11 15/11 22/11 29/11 06/12 13/12 20/12 27/12 03/01 10/01 17/01 24/01 31/01 07/02 14/02 21/02 28/02 07/03 14/03 21/03 28/03 04/04 11/04 18/04 25/04 02/05 Project Plan - Revision 1 Project Plan Milestone / Deadline Search for related work Read related work Finalise direction of project Poster Design Reading Week Seminar slides Seminar Implementation (basic) Testing (basic) Christmas revision month Exams Implementation (flexible) Testing (flexible) Testing (final) Investigate fractals I Investigation report I Investigate fractals II Investigation report II Presentation of results Report sample Easter revision month Project report Week Beginning (2009) Week Beginning (2010) 27/09 04/10 11/10 18/10 25/10 01/11 08/11 15/11 22/11 29/11 06/12 13/12 20/12 27/12 03/01 10/01 17/01 24/01 31/01 07/02 14/02 21/02 28/02 07/03 14/03 21/03 28/03 04/04 11/04 18/04 25/04 02/05 Project Plan - Revision 2 Project Plan Milestone / Deadline Search for related work Read related work Finalise direction of project Poster Design Reading Week Seminar slides Seminar Implementation (basics) Imp'tion (poly parser) Testing (basic) Christmas revision month Exams Imp'tion (incorporate Qt) Imp'tion (incorporate parser) Imp'tion (basic tweaking) Imp'tion (pan, zoom, colours) Imp'tion (orbits) Imp'tion (advanced parsing) Imp'tion (period points) Imp'tion (auto root finding) Imp'tion (auto iterations) Imp'tion (load /save to file) Testing (final) Presentation of results Report sample Easter revision month Project report Week Beginning (2009) Week Beginning (2010) 27/09 04/10 11/10 18/10 25/10 01/11 08/11 15/11 22/11 29/11 06/12 13/12 20/12 27/12 03/01 10/01 17/01 24/01 31/01 07/02 14/02 21/02 28/02 07/03 14/03 21/03 28/03 04/04 11/04 18/04 25/04 02/05 Bibliography [1] Kathleen T. Alligood, Tim Sauer, and James A. Yorke. Chaos- An Introduction to Dynamical Systems, pages 149–150. Springer Science + Business Media, 233 Spring Street, New York City, New York, 10013, United States of America, 1996. [2] Kathleen T. Alligood, Tim Sauer, and James A. Yorke. Chaos- An Introduction to Dynamical Systems, page 32. Springer Science + Business Media, 233 Spring Street, New York City, New York, 10013, United States of America, 1996. [3] Kathleen T. Alligood, Tim Sauer, and James A. Yorke. Chaos- An Introduction to Dynamical Systems, page 135. Springer Science + Business Media, 233 Spring Street, New York City, New York, 10013, United States of America, 1996. [4] Michael Barnsley. Fractals Everywhere, page 278. Academic Press Limited, 24-28 Oval Road, London, NW1 7DX, United Kingdom, 1993. [5] ‘Bhappynick’. File:Timelapse34.jpg. http://en.wikipedia.org/wiki/File: Timelapse34.jpg, May 2006. [6] Timo Bingmann. Flex Bison C++ Template/Example (released under the Do What The Fuck You Want To Public License (WTFPL)). http://idlebox. net/2007/flex-bison-cpp-example/, April 2010. [7] Ross Hilbert. Fractal Science Kit - Fractal Generator. fractalsciencekit.com/, March 2010. http://www. [8] Jan Homann. File:Riemann surface log.jpg. http://en.wikipedia.org/wiki/ File:Riemann_surface_log.jpg, August 2007. [9] Henning Makholm. File:newton z3-2z+2.png. http://en.wikipedia.org/ wiki/File:Newton_z3-2z%2B2.png, August 2007. [10] James Montaldi and David Broomhead. Math34042: Discrete time dynamical systems lecture notes. http://www.maths.manchester.ac.uk/ pag/dynsyst/3fractals.pdf, 2009. [11] Refsnes Data. W3Schools Browser Display Statistics. http://w3schools.com/ browsers/browsers_display.asp, November 2009. [12] Frederik Slijkerman. Ultra Fractal: Advanced Fractal Animation Software. http://www.ultrafractal.com/, March 2010. 51 Bibliography Bibliography [13] Eric W. Weisstein. “Horner’s Rule.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HornersRule.html, April 2010. [14] Eric W. Weisstein. “Newton’s Method.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NewtonsMethod.html, April 2010. 52

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