Abstract
Let be a complex polynomial of fixed degree . In this paper we show that Cauchy’s method may fail to find all zeros of for any initial guess lying in the complex plane and we propose several ways to find all zeros of a given polynomial using scaled Cauchy’s methods.
1. Introduction
For a large class of computational problems a zero of a polynomial in the complex plane has to be found. Cauchy’s method [1] is an interesting candidate for the numerical solution. Let be a starting point for a zero of . We setIf is a simple root of and is selected in such way that the following criterion is satisfied,the Cauchy method has the following properties:(a)is an attractive fixed point of a map ().(b)For sufficiently close to , the sequence of iterates,will converge to the root and that is, the sequence (3) converges linearly.
However, someone may wonder about the basin of attractionof a root of and the relationship with the vector field and trajectories of the differential system:known as the continuous Cauchy’s method. It is assumed that discrete Cauchy’s method (1) may be interpreted as Euler step for the differential equation (6).
2. Examples and Graphics
Let us consider a famous example given by Cayley (1879, [2]). Letfor , , and , the three roots of , and color a point blue if converges to , color it green if converges to , and color it red if converges to Any remaining point gets coloured white (e.g., if ). Figure 1 shows basins of attraction for (7) with a value of , respectively, taken and for the region in Figure 1(a) and in Figure 1(b). Figure 2 indicates corresponding basins of attraction for the gradient method with and for the region in Figure 2(a) and in Figure 2(b). Note that the boundary of the basins of attraction for each root of exhibits a fractal structure (see Figures 1–5). These figures show also the role played by the parameter ; this parameter is used on one hand to enlarge the domains in the basins of attraction which contain the roots of the polynomial and on the other hand to ensure the convergence of the method. The question which arises naturally is of knowing what happens when in (1). The answer can be given by differential system (6). Denote , where To see this, let Rearranging yields Finally, letting we have
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We can see in Figure 3(a) that the fixed point is attractive while are repulsive; in Figure 3(b), the fixed point is repulsive while is attractive. It is clear that this qualitative behavior change is controlled by the sign of the parameter.
3. Cauchy’s Method as a Dynamical System
A critical point of a holomorphic map is usually a point, where the derivative vanishes. In particular, a critical points of are solutions of . Thus if is degree , then admits at the most different critical points.
Definition 1 (immediate basins). Let be an attracting fixed point of . The connected component of the basins of attraction that contains is called its immediate basins.
The following theorem is the main result in the study of basins.
Theorem 2 (Fatou, [3]). If is an attracting fixed point of , then the immediate basins contain at least one critical point.
We deduce directly from the last theorem that the number of attractive fixed points (attractive roots of ) of Cauchy’s method is at most equal to the degree of . Thus the Cauchy’s method cannot find all roots of the complex polynomial . In the subsequent two results we will explain this fact.
Theorem 3 (case of discrete Cauchy’s method). If is a root of with , then Cauchy’s method converges towards this root for some .
Proof. We have (where denotes the real part of complex number ) and, for , condition (2) is obtained.
Theorem 4 (case of continuous Cauchy’s method). If satisfies (6) for all with and is the root of when , then for some in .
Proof. Let the monic with degree with simple roots and we haveAssume that solves (6). Rearranging terms yieldsand integrating with respect to and in accordance with (14) Finally, exponentiation shows that
4. Computer Experiments with Scaled Cauchy’s Methods
To obtain the condition (2), we used the scaled complex of (1); that is,The continuous form is given bySubsequently, we have shown the basins of attraction and vector field according to three different choices of the function
Case 1 (). The function is not holomorphic and a similar analysis of (2) of Section 1 is not possible in ; then we consider the dynamics (18) with in . Considerwith The Jacobian of the transformation is expressed by with The eigenvalues of the matrix are and and are given by and in the singular points , , and , we have Thus the fixed point is attractive for , and then are attractive for . Figure 4(a) showed the basins of attraction of using (20) and , and Figure 4(b) represents the vectors field of (19).
Case 2 (). The dynamics (18) with can be formulated in asThe map has the following Jacobianwith In a similar way, the eigenvalues of the matrix are and at the singular points and thus we have
Case 3 (). In this caseThe Jacobian of the transformation is given bywith We notice that the transformation possesses the following fixed points:
At the points , the eigenvalues of , are equal to ; in this case the fixed points are attractive in the interval (, ) and is a indifferent fixed point (see Figure 6), since In the last case, the method,is the minimization process of the function given bycalled the discrete steepest descent method, since Note that is locally convex and the local convergence of (19) is established (see [4]).
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5. The Best Choice of -Parameter for Cauchy’s Method Leads to Newton’s Method
In order to have a condition more strict than (2), we seek holomorphic functions that yield the functional conditionwhen , and we can takeSubstituting (40) into (18), we get familiar Newton’s method; that is,Numerical investigations into the basins of attraction of (41) and their boundary for the cubic polynomial have been carried out, and pictures of these sets are well known [5–8] (see Figure 7).
6. Further Motivation
The method (1) is defined for any . However, we will only be concerned with a small real parameter . The study of the -parameter plane allows the identification of the singular points other than the fixed points, which are the periodic points.
Definition 5. Let satisfy and for . Then is a periodic point of period The set, is called a periodic orbit or a cycle (of period ).
Contrary to the fixed points of which are roots of the polynomial , the periodic points (and their orbit) are bad starting points for Cauchy’s method. In order to determine the existence of periodic points for (1), Theorem 2 indicates that it is necessary to follow the orbit of a critical point. The critical points of for when are and . In Figure 8(a) the global behaviour of the orbit of the critical point is pictured. The horizontal and vertical axes correspond to the real and imaginary part of the complex parameter in the region . The dark area in the picture is the subset of parameter values at which the orbit is bounded but does not converge to the fixed point of . Figure 8(b) is an enlargement of the region in the previous picture. In these figures, which were generated by examining the parameter values on grid, the self-similarity (fractal structures) of regions in parameter plan is obvious. In an analogous way to Mandelbrot sets (see [9]) and only for the example , the complex structure of the set exhibited in Figures 8-9 requires a more explicit study in forthcoming paper.
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Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author would like to thank Professor Ilhem Djellit for their help and valuable suggestions.