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TECHNICALNOTE

A NOTE ON CHAOS AND HALLEYSMETHOD

CLIFFORD A. PICKOVER

This note illustrates very simple graphics techniquesfor visualizing a large class of graphically interestingmanifestations of chaotic behavior arising from com-plex analytic dynamics . Another goal of this note is togive a flavor of the subject of recurrence relations andchaos. As background, research over the past decadehas made it clear that many systems of physical, biolog-ical, electrical, and chemical interest exhibit highly un-stable or chaotic behavior [2-51, [14]. Chaos theoryoften involves the study of how complicated behaviorcan arise in systems which are based on simple rules,and how minute changes in the input of a system canlead to large differences in the output. Chaotic behaviorin such systems is generally irregular and disorderly-examples include weather patterns, some neurologicaland cardiac activity, and certain electrical networks o fcomputers [2], [5], [14].Numerical methods use numbers to simulate m athe-matical p rocesses, which in turn usually simulate real-world situations [a]. The choice of a particular algo-rithm influences not only the process of computing butalso how we are to understand the results when theyare obtained. In this technical note, I address the pro-cess of solving equations of the form f(x) = 0. Theproblem of finding the zeros of a continuous functionby iterative methods occurs frequently in science andengineering [l], 171, [8], [13]. These approximationtechniques start with a guess and successively improveupon it with a repetition of similar steps. The graphs in0 1988 ACM 0001.0782/88/1100-1326 $1.50

this note give an indication of how well one of theseiterative methods, Halleys method, works in order togain insight as to when Halleys method can be reliedupon and when it behaves strangely. Halleys method isof interest theoretically because it converges rapidlyrelative to many other methods. Interesting past workincludes a study of the iterates of a related method,Newtons method [I], for cubic polynomials. Otherwork suggests that computer graphics can play a role inhelping mathematicians form the intuitions needed toprove theorems about convergence of points in thecomplex plane [9], [ll], [15], [16].Let F(x) be a complex-valued function of the complexvariable x. The Halley map is the functionH(x): x,+1 = xn - [F(x.)/F(x,, 1 - (F$,)(F($l)] . (1)

This iteration is used to find the zeros of F and isclearly derived in [i]. I consider rational functions, F,that are analytic in the complex plane C . x,I) is a zeroof F and a fixed point of H: H(x,) = x,. The basin ofattraction of xZ s the set of all po ints whose forwardorbits by H converge to an x2.To simplify the discussion, I consider, as an example,the one-parameter polynomial

x7 - 1 = 0. (21Polynomials are useful for theoretical study since apolynomial of degree M has exactly M zeros, and wetherefore know when we have found all the zeros. Thispolynomial has seven roots at x = e2*im/7. n this note,

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Technical Note

the graphical behavior of Halleys method results fromforward iteration. An initial point on the complex planeis selected and iterated N times. Traditionally a point isconsidered to have converged ifI x,+1 - Xn12 c e, (3)

where E s a small value. Often, to verify that this crite-rion for ending the iteration has actually allowed thesystem to converge to a root, ) F(X) ) < t is used inconjunction with (3). However, in this note the follow-ing nonstandard test was used:

I I (&+I) I 2 - I Xn * I c 6 (4)The value oft was 0.0001 . The iteration in (1) wasperformed on 4 million initial param eter values in aZOOO-by-2000 oint square grid. The vertical axis is thereal axis; the horizontal axis is the imaginary axis.Note that the values being graphed are not the valuesof the function H. What is graphed are the values of nfor which x,, satisfies (4). The plots are created by map-ping the value of n to color using a color lookup table.For some figures, the lookup table is created by hand tocreate contour lines and to visually emphasize differentregions of behavior of the function. For other figures,the lookup table was automatically generated fromthree sine waves (one each for red, green, and blue),each with random phase and frequency. This producessmooth correlated changes in the color. A good way todisplay these images on a black and white device is tosimply alternate even and odd pixel-intensities withblack and white colors. For similar approaches, see [I],PII PI.Figure 1 shows a graph of Halleys Map for x7 - 1 =0. The basins of attraction for the roots of the equationare displayed for various initial values of (x0) in thecomplex plane (between -2.5 and 2.5 in the real andimaginary directions). The colors make visually obviousthe relative speed of convergence of different startingpoints. The seven central light pink colored regionscontain the roots and correspond to starting pointswhere convergence is achieved rapidly (within threeiterations). Initial guesses in the tear-shaped coloredbasins fanning out from the roots are safe; that is, anystarting points selected from these regions come closeto a root within a small number of iterations. Light grayregions converge much more slowly, and behaviornear the white radial boundaries is considerably morecomplicated than in the tear-shaped basins. Note thevarious nodules along these high-iteration radialbranches. These borders consist of elaborate swirls thatcan pull Halleys method into any one of the sevenroots. In this vicinity, a tiny shift in starting point canlead to widely divergent results.Using (4) produces the whisker-like projectionsaround each contour, and these whiskers generallypoint to the root (or to regions of fast convergence).Therefore, directionality can be easily understood byobserving the contour plots. Figure 2 is a magnificationof one of the nodules in Figure 1, and it gives a visualindication of the complexity of the behavior of Halleys

FIGURE1. Halleys Map for x - 1 = 0. The Basins of Attractionfor the Roots of the Equation are Displayed Graphically for VariousInitial Values of x0 in the Complex Plane. Colored (Nongray)Regions Correspond to Starting Points where Convergence isRapidly Achieved. Note the Various Tiny Nodules along the WhiteHigh-iteration Radial Branches.map when applied to a simple function. The five cen-tral dark bulls-eye regions converge rapidly to a solu-tion, and by testing the value of x,, after N iterations,one can determine to which root these areas converge.For example, the small central basin converges to theroot at (1, 0). This magnification also reveals additionalminiature copies of the large nodule. I have found that

FIGURE2. Magnification of one of the nodules in Figure 1. Here,Yellow and Green Regions are Slow to Converge to a Solution.

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Technical Note

FIGURE3. Example Nodule from a Slightly More ComplicatedFunction, x7 + x2 - 1 = 0. Yellow and Green Regions are Slow toConverge to a Solution.this self-similarity at all size scales is characteristic ofHalleys plot for polynomial equations. The concept ofself-similarity in both nature and mathematics hasbeen discussed previously in [l], [9], [ll]. Here yellowand green starting points are slow to converge. Figure 3is an example nodule from a slightly more complicatedfunction, x7 + x2 - 1 = O. The complexity and richnessof resultant forms contrasts with the simplicity of theformula being solved. Like Figure 2, yellow and green

FIGURE4. Same as Figure 3, Except Using a Different Color Tableto Emphasize High-iteration Regions.

regions are slow to converge, and blue and black areasconverge most rapidly. By altering the coloration, onecan make the high iteration regions more obvious (seenas yellow in Figure 4).Among the methods available for the characteriza-tion of complicated mathematical and physical phe-nomena, computers with graphics are emerging as animportant tool.* In this technical note, Halley maps ofthe equation 2 - 1 = 0 are presented, and their behav-ior ranges from stable points to chaotic fluctuations.The system becomes irregular in well-defined regions.The chaotic portion of the maps, while exhibiting com-plicated behavior, is composed of various underlyingself-similar structures. The beauty and complexity ofthese drawings correspond to behavior which no onecould fully have appreciated or suspected before theage of the computer. This complexity makes it difficultto objectively characterize structures such as these,and, therefore, it is useful to develop graphics systemsthat allow the maps to be followed in a qualitative andquantitative way. Provocative avenues of future re-search include extension to nonpolynomial equationsand to related root-finding numerical methods such asMullers method, Aitkens method, and the secantmethod. A report such as this can only be viewed asintroductory; however, it is hoped that the techniquesdescribed will provide a useful tool and stimulate fu-ture studies in the characterization of complicated be-havior o f other numerical methods which are beingused in many branches of modern science with increas-ing frequency. For papers in the authors fifteen-part Mathematics and Beauty series, see,for example: Pickover, C. (1988) Overrelaxation and Chaos. Physics Letters A.July 30(3): 125-128. Pickover. C. (1986) Mathematics and beauty : time-discretephase planes associated with the cyclic system, {l(t) = -f(y(t)), J(t) = f(x(tJ)l,Computers and Graphics, 11(2), 217-226; Pickover, C. (1988) A note on rend-ering chaotic Re peller Distance-Towers. Compule rs in Physics, May/June2(3): 75-76; Pickover, C. (1988) Pattern formation and chaos in networks,Communicat ions of the ACM, Febru ary 31(Z), 136-151; Pickover, C. (1988) Fromnoise comes beauty: textures reminiscent of rug weavings and wood grainsspring from simple formulas (Mathematics and Beauty X): Conzputet GraphicsWorld. March, ll(3): 115-116: Pickover. C. (1988) A Note on Renderi ng 3-DStrange-Attractors. Computers and Graphics, 12(Z): 263-267; Pickover. C.(1988) Symmetry, Beauty an d Chaos in Chebyshevs Paradise. The VisualComputer: An International Journal of Computer Graphics. 191, in press. For other works of the author on the use of graphic representations formaking complicated data easier to understand, see, for example: Pickover. C.(1989) Computers, Pattern, Chaos. and Beauty. Springer-Verlag; Pickover, C.(1984) Spectrographic representations of globular protein breathing motions.Science. 223: 181: Pickover, C. (1985) On the use of computer generatedsymmetrized dot-pat terns for the visual characterization of speech waveformsand other sampled data. J. Acoust. Sot. Am.. 80(3): 955-960; Pickover, C.(1987) DNA Vectorgrams: representa tion of cancer gene sequence s as move-ments along a 2-D cellular lattice, IBM J, Res. DEW..31: Ill-11% Pickover, C.(1985) On the educational uses of computer-generated cartoon faces. Journalof Educational Technology Systems, 13: 185-198.REFERENCES1. Benzinger, H.. Burns. S., and Palmore, J. Chaotic complex dynamicsand Newtons Method. Physics Letters A. 119, 9 (19871, 441-445.2. Campbell, D., Crutchfield, J., Farmer, D., and Jen, E. Experimentalmathematics: The role of computation in nonlinear science. Com-mm. ACM 28. [1985], 374-389.3. Crutchfield. J., Farmer, J., and Packard, N. Chaos. Scien. Amer. 255,(1986). 46-57.4. Devaney. R. Chaotic bursts in nonlinear dynamic systems. Science235. (1986). 342-345.5. Fisher, A. Chaos: The ultimate asymmetry. Mosaic 26, 1 (Jan.-Feb.1965). 24-30.

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6. Gleick, J. Chaos: The Making of a New Science.Viking: New York, NY,1987.7. Grove, W. Brief Numerical Methods. Prentice-Hall: Englewood Cliffs,NJ, 1966.8. Hamming, R. Numerical Methods for Scientists and Engineers. DoverPublications: New York, NY, 1973.9. Mandelbrot, B. The Fractal Geometry of Nature. Freeman: San Fran-cisco, CA, 1984.10. May, R. Simple mathematical models with very complicated dynam-ics. Nature 261. (1976). 459-467.11. Peitgen, H.. and Richter, P. The Beauty of Fractals. Springer: Heidel-berg. 1986.12. Peterson. I. Portraits of equations. ScienceNews 132. 12 (Dec. 1987),184-186 (and cover picture).13. Peterson. 1. Zeroing in on chaos. ScienceNews 131, 9 (1987), 137-139.14. Peterson. 1.Tovine with a touch of chaos. Science News 129. (Mav1986), 277-278: - . _15. Pickover, C. Biomorphs: Computer displays of biological forms gen-erated from mathematical feedback loops. ComputerGraphics Forum5.4 (Dec. 1986), 313-316.16. Pickover, C.. and Khorasani, E. Computer graphics generated fromthe iteration of algebraic transformations in the complex plane.Computers and Graphics 9, 2 (1985), 147-151.

CR Categories and Subject Descriptors: G.1 [Numerical Analysis]:G.1.2 Approximation, G.1.5 Roots of Nonlinear Equations: F.2 [Analysisof Algorithms and Problem Complexity]: F.2.1. Numerical Algorithms

and Problems; 1.3 Computer Graphics]: Picture/Image generation: I.4[Image Processing]: EnhancementGeneral Terms: Algorithms. Design, Experimentation, TheoryAdditional Key Words and Phrases: Chaos, computer art, computergraphics, experimental mathematics . fractals. iteration, Newtonsmethod, numerical analysis, symmetryABOUT THE AUTHOR:CLIFFORD A. PICKOVER is a research staff member at theIBM T.J. Watson Research Lab at Yorktown Heights, New York.He received his Ph.D. from Yale Universitys Department ofMolecular Biophysics and Biochemistry. He is author of thebook Computers, Pattern, Chaos and Beauty to be published bySpringer-Verlag. Authors present address: Clifford A. Pick-over, IBM Thomas J. Watson Research Center, YorktownHeights, NY 10598.Permission to copy without fee all or part of this material is grantedprovided that the copies are not made 01distributed for direct commer-cial advantage, the ACM copyright notice and the title of the publicationand its date appear, and notice is given that copying is by permission ofthe Association for Computing Machinery. To copy otherwise, or torepublish, requires a fee and/or specific permission.

Perlis l Wilkes- Hamming- MinskyWilkinson l McCarthy* DijkstraBachman-Knuth- Newell-SimonRabin l Scott l Backus l Floydlverson l Hoare*Codd-CookThompson l Ritchie- WirthKarPWhat do these prominent computer scientistshave in common?Theyre all recipients of the ACMsrevered Turing Award, and theirunique insights have been compiledfor you in ACM Turing AwardLectures: The First Twenty Years:1966-1985. Its the first book in theAnthology Series from the newlyformed ACM Press (a unique collab-oration between ACM and Addison-Wesley Publishing Company).After introductions from RobertAshenhurst (Anthology Series editor)and Susan Graham, ACM TuringAward Lectures presents the pro-vocative lectures delivered by the 23

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