their eæcient computation V. DRAKOPOULOS University ofathens, Panepistimioupolis , Athens, GREECE S.

Transcription

1 Visualization on the Riemann sphere of Schríoder iteration functions and their eæcient computation V. DRAKOPOULOS Department of Informatics, Theoretical Informatics, University ofathens, Panepistimioupolis , Athens, GREECE S. GEORGIOU Department of Mathematics, University ofathens, Panepistimioupolis , Athens, GREECE Abstract: - The theory of Julia and Fatou holds for the complex plane as well as for the Riemann sphere, with minor modiæcations. The latter space seems a more natural approach to examine rational functions. Schríoder iteration functions, a generalization of Newton-Raphson method to determine roots of equations, are generally rational functions which, until now, are examined only in the complex plane. On the Riemann sphere we examine the Julia sets of the Schríoder functions constructed to converge to the nth roots of unity and the orbits of all free critical points of these functions as applied to a oneparameter family of cubic polynomials. Finally, we present a new algorithmic construction to maximize the computational eæciency of the Schríoder functions. Key-Words: - critical point, Fatou set, Julia set, Riemann sphere, Schríoder's method 1 Introduction The method of iteration is the prototype of all numerical methods for attacking the problem of solving ènonlinearè equations in one and several variables. The convergence of these methods, generally, depends upon the initial approximation to a root of our equation. In the special case of polynomial equations, good a priori knowledge of the desired root of the equation often is not available. Looking at the set of all the starting values from a geometrical point of view will deænitely help us to choose these initial approximations to a root and compare the convergence properties of various iterative methods. Ernst Schríoder ë 9 ë in 1870 described a method of ænding a rational iterating function of any desired ëorder of convergence" to determine roots of equations. For polynomial equations this involves the iteration of rational functions over the complex Riemann sphere which is described by the classical theory of Julia ë 8 ë and Fatou ë 6 ë and its subsequent developments, also of paramount importance in the context of Numerical Analysis. In what follows we abbreviate as f k the k-fold composition f æf æææææf, by region we mean a connected open set on the extended complex plane C = C ë f1g and if we have a rational function of the form Rèzè = P èzè=qèzè, where P èzè and Qèzè are complex polynomials with no common factors, the degree of R is deæned by degèrè = maxfdegèp è; degèqèg. It appears that convergence of the sequence of iterates z 0 ;Rèz 0 è;r 2 èz 0 è;::: is assured for every

2 choice of starting point z 0 on the extended complex plane, except when z 0 is a point of a certain nowhere dense perfect set, the Julia set JèRè. How ëclose" a starting point must be to the desired root depends on certain convergence conditions and how ëfast" does the method converge depends on the order of convergence of our iterative method. It may happen, however, that when choosing a starting point z 0 in a certain domain, convergence takes place not to a zero of our polynomial, but to a periodic orbit or cycle, that is a set of p 2 distinct points fa 1 ;:::;a p g such that Rèa 1 è=a 2 ;:::;Rèa p,1 è=a p ;Rèa p è=a 1 ; so that, in fact, for each k =1; 2;:::;p, z = a k is a solution of R p èzè =z. Hence, a point a is periodic if R p èaè =a for some pé0; it is repelling, indifferent or attracting depending on whether jèr p è 0 èaèj is greater than, equal to, or less than one. If jèr p è 0 èaèj =0,a is termed superattracting. Ifp =1, z is called a æxed point of R. The Julia set JèRè of a rational function R is the closure of all repelling periodic points of R and the Fatou set is its complement onc. Also, if a is an attracting æxed point of R then Aèaè=fz 2 C : lim k!1 R k èzè =ag is the basin of attraction of a. We also have the following important property: the boundary of Aèaè is JèRè. It follows that if R has several distinct attracting æxed points, then their basins of attraction share the same boundary, the Julia set of R. A portion of this paper was motivated by ë 5 ë and could be considered an extension of it to the Riemann sphere. We ærst outline the construction of the extended complex plane as a sphere. Then, we maximize the computational eæciency of the Schríoder functions by reconstructing them. Finally we examine on the Riemann sphere, with the aid of microcomputer-generated plots, the Julia sets of the Schríoder scheme as applied to the set of functions f n èzè =z n, 1 for n =2; 3;:::, their roots' basins of attraction and the dynamics of Schríoder maps applied to the one-parameter family of cubic polynomials presented in ë 2 ë. 2 Schríoder Iteration Functions revisited Schríoder iteration functions are a family of rational iteration functions which are designed to converge with order 2 to the zeros of a function f. We deæne the Schríoder iteration functions as èsee ë 7 ëè X,1 S èzè =z + c k èzèë,fèzèë k ; =2; 3;:::; è1è where and 1 f 0 èzè k=1 c k èzè= 1 1 d k,1 1 k! f 0 èzè dz f 0 èzè d k,1 1 d = dz f 0 èzè dz 1 d f 0 èzè dz 1 æææ d f 0 èzè dz íz í k,1 factors The coeæcients c k èzè are analytic functions in every subregion T of C for f 0 èzè 6= 0. For =2,we get the familiar NewtoníRaphson function S 2 èzè=nèzè =z, fèzè=f 0 èzè: We observe that Schríoder's iteration is Newtoní Raphson-like in form. The iteration sequence z n+1 = S èz n è; = 2; 3;:::; converges locally to the roots z æ i, i =1; 2;:::;degèfè, of fèzè = 0, as Oèjz n, z æ i j èèë 7 ë, Theorem 6.12c, p. 530è. The S are truncations of a general inænite series in f and it is easily seen that their construction requires the ærst, 1 derivatives of f. Thus, the above form of Eq. è1è is not so useful for generalized computations and programming of all the S terms. To eliminate this inconvenience we deem it necessary to introduce the following construction. Starting with h 1 èzè =1wehave and, in general, h1 èzè 0 1 f 0 èzè f 0 èzè h2 èzè ëf 0 èzèë f 0 èzè = = h 2èzè ëf 0 èzèë 3; h 3èzè ëf 0 èzèë 5 h k+1 èzè=h 0 kèzèf 0 èzè, è2k, 1èh k èzèf 00 èzè è2è :

3 for k =1; 2;:::;, 2: Thus, and Eq. è1è becomes,1 X S èzè =z + c k èzè = 1 h k èzè k! ëf 0 èzèë ; 2k,1 k=1 è,1è k k! h k èzè ëf 0 èzèë 2k,1 ëfèzèëk : è3è We observe that S uses, 1number of h k, k = 1; 2;:::;, 1, and 2: For = 2, the æxed point condition S èzè =z implies that fèzè =0: For é2, it implies that fèzè = 0, or,1 X k=1 è,1è k,1 k! h k èzè ëf 0 èzèë 2k,1ëfèzèëk,1 =0: è4è We shall refer to the zeros of Eq. è4è as extraneous æxed points. Their appearance may complicate the root-ænding procedure. For more informations about the behaviour of these points and speciæcally for the attracting ones which lead to pathological cycles, we refer to ë 3 ë. Suppose that f is a polynomial of degree d 2 and that its ærst derivatives exist. Then h k ;k = 1; 2;:::;, are polynomials and S are rational functions. The JuliaíFatou theory can describe the possible types of behavior of the iteration sequence èsè n 1 n=1. To analyze this behavior, the critical points of S will play a crucial role. Critical values of a function f are deæned as those values v 2 C for which fèzè =v hasamultiple root. The multiple root z = c is called the critical point of f. This is equivalent to the condition f 0 ècè = 0. The underlying reason for studying these special orbits rests in the following theorem of Fatou ë 6 ë: Theorem 1 If R is a rational function having an attracting cycle, then at least one critical point will converge to it. Among the critical points of S determined by the condition Sèzè 0 = 0 are the zeros z æ i which are also attracting æxed points of the S. These points are obviously not free to converge to any other attracting cycles. Other roots, which we shall call the free critical points, are available, however. Diæerentiating Eq. è3è with respect to z we have that the condition S 0 èzè = 0 implies that h èzèëfèzèë,1 =0: è5è Thus, the free critical points of S are the zeros of h. 3 Computational Techniques The computational eæciency of an iterative method is a measure of howmuch computation must be done to arrive atagiven accuracy in a root. Because the computational eæciency of Schríoder's method depends upon the number ofevaluations of f and its derivatives, we describe here how one can reduce these calculations and produce quickly all of the S terms. If we denote as æ k èzè = è,1èk h k èzè; k =1; 2;:::;, 1; k! Eq. è3è becomes S èzè=z + æ 1èzè f 0 èzè fèzè+ æ 2èzè ëf 0 èzèë 3ëfèzèë2 + æææ We deæne as + æ,1èzè ëf 0 èzèë 2,3 ëfèzèë,1 : F èzè =fèzè=f 0 èzè and Gèzè=fèzè=ëf 0 èzèë 2 ; therefore S èzè=z +èæ 1 èzè+èæ 2 èzè+æææ +æ,1 èzègèzèègèzèèf èzè: è6è è7è During the development of the images we compute the coeæcients of the polynomials f 0 ; æ k, k = 1; 2;:::;, 1, before the main pixel-scanning procedure begins. In the pixel-scanning procedure, for each iteration of a point z n,we compute, fèz n è and f 0 èz n è with Horner's scheme, F èz n è and Gèz n è from è6è and S èz n è from è7è. The case of the -parameter space of the cubic polynomials p èzè=z 3 +è, 1èz, =èz, 1èèz 2 + z + è; 2 C ; è8è

4 which will be described in Section 6, involves greater diæculty because the coeæcients of the polynomials f 0 and æ k depend on the parameter that changes for each pixel. So, f 0 and æ k must be computed from the beginning for each pixel in the scanning procedure. Something like that is timeconsuming: one æ k is evaluated recursively from the h k ;k =1; 2;:::;, 1, and in every iteration we apply Eq. è2è. In this case, the initial evaluation of æ k takes place in a symbolic way; that is, the information for the creation of each coeæcient of the polynomial æ k is stored according to the parameter. Fortunately, we have to multiply only a number èindependently of è by a speciæc power of, 1. That can be seen from the following. Let c i ;i=1; 2;:::, be complex numbers independent of or of a =, 1 and not excluding the case where some of them are equal. Note that h k are the polynomials presented in Section 2. We then have and we deæne p èzè =z 3 + az, p 0 èzè =c 1z 2 + a p 00 èzè =c 2z qèzè = z 3 + z, 1 q 0 èzè = c 1 z 2 +1 q 00 èzè=c 2 z: For the h k associated with q 0 and q 00 wehave that h 1 èzè =1 h 2 èzè =c 2 z h 3 èzè =c 3 z 2 + c 4 h 4 èzè =c 5 z 3 + c 6 z and so on. To compute the coeæcients of the polynomial h k èzè associated with p 0 and p00 we modify the above mentioned h k èzè for k =1; 2;:::;, 1so that h 1 èzè =1 h 2 èzè =c 2 z h 3 èzè =c 3 z 2 + c 4 a h 4 èzè =c 5 z 3 + c 6 az and so on. The above analysis of h k, k =1; 2;:::;, 1, generally gives that h k èzè = mx i=0 c i a i z k,1,2i ; where m =èk, 1è=2 ifk is odd and m =èk=2è, 1 if k is even. It is obvious from the above that one of the roots of h k èzè = 0, for even values of k, isz =0. Thus, if is even, the map S èp èzèè has z =0as a free critical point. 4 The World is Round To obtain a metric on C we identify C with the horizontal plane fèx 1 ;x 2 ;x 3 è 2 R 3 : x 3 = 0g in R 3 and proceed to construct the usual model for C as a sphere. Figure 1: Stereographic projection Let S be the sphere in R 3 with unit radius and centre at the origin so that C cuts S along the equator, that is S = fèx 1 ;x 2 ;x 3 è 2 R 3 : x x x 2 3 =1g and denote the point è0; 0; 1è èthe top point ofsè by B. This point is called the north pole. Another choice is to take the south pole at the origin. We now project each point z 2 C linearly towards èor away fromè B until it meets S at a point z æ distinct from B: this means that a line from B to a point z in the complex plane intersects the sphere at a point z æ. Ifjzj é 1 then z æ is in the northern hemisphere and if jzj é 1 then z æ is in the southern hemisphere; also for jzj =1,z æ = z. The map : z 7! z æ is called the stereographic projection of C into S. Wemay equivalently deæne : S nfbg!c with èx 1 ;x 2 ;x 3 è=èx 1 + ix 2 è=è1, x 3 è. Clearly, if jzj is `large', then z æ is `near' to B and with this in mind we deæne the projection è1è of1 to be B. This mapping is conformal èi.e., angle-preservingè, and so the corresponding conformal geometries are the same. With this deænition, the identiæcation of C with S allows us to use them interchangeably, because is a bijective map from C to S, and this

5 explains why C is also called the Riemann sphere: see Figure 1. We now use the bijection of C onto S to transfer the Euclidean metric èin R 3 è from S to a metric on C : this simply means that is deæned in the natural way by the formula èz; wè=jèzè, èwèj = jz æ, w æ j: A gentle exercise in Vector Geometry èwhich we omitè yields an explicit formula for, namely èz; wè= 2jz, wj p è1 + jzj 2 èè1 + jwj 2 è when z and w are in C, while for z in C, èz;1è = lim w!1 èz; wè= 2 p è1 + jzj 2 è : As èz; wè is the Euclidean length of the chord joining z æ to w æ, is called the chordal metric on C. 5 Julia Sets of Schríoder Functions applied to the Polynomials f n èzè =z n, 1 and their Zeros' Basins of Attraction In ë 5 ëwe examined in the complex plane the Julia sets of Schríoder functions S, for =2; 3;:::, constructed to converge to the nth roots of unity and these roots' basins of attraction. We have furthered this by examining the above mentioned sets on the Riemann sphere. With this projection the outcome is even more astonishing and useful than the usual examination in the complex plane. The projection of the image on the Riemann sphere can be accomplished after three fundamental steps: ærstly, we project our grid upon the Riemann sphere, secondly, we project the sphere on the plane using the stereographic projection described in Section 4 and, thirdly, we produce the basin of attraction or the Julia set. The ærst step can be accomplished by employing the normal projection according to ë 1 ë and observing the sphere from the south pole. The second step uses the fact that the interior of the south hemisphere is mapped to the interior of a large disk around the origin. The third step was computed by almost the same algorithm as described in Section 4 of ë 5 ë, except that we did not take a square as our starting value's grid, but a circle. We observe that, for a given polynomial f n, the complexity of the basin maps increases with the order of the S maps and with each increment a new set of n ëpetals" appears to be embedded in an inænitely self-similar fashion. To have an idea how this looks like on the sphere, we can rotate it using the width and length angles assigned to our program. First the y-rotation takes place and then the x-rotation. The white regions in some of the ægures, are not to be interpreted as part of a basin of attraction Aèz æ è. Since f 0 nè0è = 0, z = 0 is mapped to the point at inænity. Points near z =0may ærst be mapped many orders of magnitude away from it, whereupon a great number of iterations might be required to bring them back tothe z æ i, if at all. Therefore, these grid points would remain as part of the background which has been plotted as white. We can control these white regions with an overæow parameter added to the program for this special dynamic behaviour. Analysis of these plots that considers individually each iteration procedure S as applied to f n appears in ë 4 ë. For the Figures 2 to 4 the angles were, in the x-direction, O æ and, in the y-direction, 3O æ. The overæow parameter was 1E10 for n =2; 3 and 1E13 for n =4. Case 1. = 2: The basins of attraction for n = 2; 3 are presented in Figures 2èaè, 3èaè, respectively. The Julia set for n = 4 is presented in Figure 4èaè. Figure 2: Schríoder basins of attraction for the roots of f 2èzè = z 2, 1 on the Riemann sphere. Blue regions constitute Aè1è; yellow regions constitute Aè,1è; èaè S 2 method, èbè S 3 method, ècè S 4 method. Case 2. = 3: The basins of attraction for n = 2; 3 are presented in Figures 2èbè, 3èbè, respectively. The Julia set for n = 4 is presented in Figure 4èbè. Case 3. = 4: The basins of attraction for n =

6 Figure 3: Schríoder basins of attraction for the roots of f 3èzè =z 3, 1 on the Riemann sphere. Blue regions constitute Aè1è; yellow regions constitute Aè,0:5+i 0:5 p 3è; green regions constitute Aè,0:5, i 0:5 p 3è. èaè S 2 method, èbè S 3 method, ècè S 4 method. 2; 3 are presented in Figures 2ècè, 3ècè, respectively. The Julia set for n = 4 is presented in Figure 4ècè. Figure 4: Julia sets of the Schríoder functions applied to f 4èzè =z 4, 1 on the Riemann sphere. èaè S 2 method, èbè S 3 method, ècè S 4 method. 6 A Walk in Parameter Space We now focus attention on Schríoder iteration methods associated with the particular one parameter family of cubic polynomials è9è the zeros of which are z æ = 1, 1 zæ = è,1 +p 2 1, 4è=2 and z æ = 3 è,1, p 1, 4è=2. Note the -dependence of z æ and 2 zæ 3. The polynomials p are exactly the monic cubics whose roots sum to zero and which have 1 as a root. Since any cubic can be transformed into an p or into z 3 by an aæne change of variable and multiplication by a constant, analyzing Schríoder's method for a general cubic reduces essentially to analyzing it for the p 's. Each point =Reèè+iImèè represents a dynamical system with its own æxed points, possible attracting cycles and Julia sets. There are regions in the -parameter space where attracting periodic cycles exist in addition to the attracting æxed points associated with the zeros of p. Extra æxed points corresponding to the roots of Eq. è4è, shown to be repelling for = 1,may become attracting in regions of the -space. To detect the existence of attracting cycles which could interfere with the Schríoder search for the z æ i,we observe the orbits of the free critical points of the S functions. The free critical points c i ;i =1; 2;:::;, 1 for the ærst three S functions associated with p can be computed by Eq. è5è and are given below: =2,c 1 =0, =3,c 1;2 = æ p è, 1è=15, =4,c 1;2 = æ p è, 1è=6; c 3 =0: The free critical points for the ærst ten S functions associated with p are presented in ë 5 ë. The dynamics of each Schríoder map S on the extended complex parameter space was studied in much the same way as described in Section 5 of this paper and in Subsection 5.2 of ë 5 ë, except that the region of the complex -plane was represented by a circle grid. The grid point in-space was colored accordingly: blue for convergence to z æ 1, red for convergence to z æ 2 and green for convergence to z3. æ The resulting black areas represent regions in parameter space for which additional attracting cycles existed. The corresponding parameter space maps for the critical points c i+1 =,c i are obtained by a reæection about the real -axis. For the Figures 5, 7èaè and 7èbè the angles were, in the x-direction, O æ and, in the y-direction,,9o æ. The overæow parameter was 1E30. èaè èbè Figure 5: Iteration schemes for the one-parameter family of cubic polynomials p; èaè S 2 method, èbè S 3 method. Case 1. = 2: Figure 5èaè represents regions in parameter space for which the critical point c 1 is attracted to z æ i. Figure 6èaè is an enlargement ofa region in Figure 5èaè, but located on the dark side of this sphere. Case 2. = 3: Figure 5èbè represents regions in parameter space for which the critical point c 1 is

7 èaè èbè Figure 6: Magniæcations of Figures 6èaè and 6èbè revealing Mandelbrot sets for the S 2 and S 3 methods, respectively. Figure 7: èaè, èbè S 4 iteration scheme for the one-parameter family of cubic polynomials p. ècè A magniæcation of a region of Figure èbè revealing a Mandelbrot set for the S 4 method. attracted to z æ i. Figure 6èbè is an enlargement ofa region in Figure 5èbè. Case 3. = 4: Figures 7èaè and èbè represent regions in parameter space for which the critical points c 1 and c 3, respectively, are attracted to z æ i. Figure 7ècè is an enlargement of a region in Figure 7èbè showing a Mandelbrot-like set associated with c 3. Because the black shapes that we discovered in the -parameter space exhibit the morphology and the classical characteristics of Mandelbrot sets, it comes in a non-surprising fashion the following Deænition 1 The Mandelbrot set associated with the Schríoder function S for the cubic polynomials p is the set ë2ë J. H. Curry, L. Garnett and D. R. Sullivan, On the iteration of a rational function: Computer experiments with Newton's method, Communications in Mathematical Physics, Vol. 91, No. 2, 1983, pp. 267í277. ë3ë V. Drakopoulos, On the additional æxed points of Schríoder iteration functions associated with a one-parameter family of cubic polynomials, Computers & Graphics, Vol. 22, No. 5, 1998, pp. 629í634. ë4ë V. Drakopoulos and A. Bíohm, Basins of attraction and Julia sets of Schríoder iteration functions, in T. Bountis and Sp. Pnevmatikos èedsè, Order and chaos in nonlinear dynamic systems, Vol. IV, Pnevmatikos, ë5ë V. Drakopoulos, N. Argyropoulos and A. Bíohm, Generalized computation of Schríoder iteration functions to motivate families of Julia and Mandelbrot-like sets, SIAM Journal on Numerical Analysis, Vol. 36, No. 2, 1999, pp. 417í435. ë6ë M. P. Fatou, Sur les equations fonctionelles, Bulletin de la Societe Mathematique de France, Vol. 47, 1919, pp. 161í271; Vol. 48, 1920, pp. 33í94, 208í314. ë7ë P. Henrici, Applied and computational complex analysis, Vol. 1, Wiley, ë8ë G. Julia, Memoire sur l'iteration des fonctions rationelles, Journal de Mathematiques Pures et Appliquees, Vol. 8, 1918, pp. 47í245. ë9ë E. Schríoder, íuber unendlich viele Algorithmen zur Auæíosung der Gleichungen, Mathematische Annalen, Vol. 2, 1870, pp. 317í365. M èè =f 2 C : there exist attracting k-cycles, k =1; 2;:::; for S other than the roots z æ i of p èzè =0g: References ë1ë K.íH. Becker and M. Díoræer, Dynamische Systeme und Fraktale: Computergraæsche Experimente mit Pascal, 3. Auæage, Vieweg, 1989.