On the Complex Dynamics of Continued and Discrete Cauchy’s Method

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.


Introduction
For a large class of computational problems a zero of a polynomial  in the complex plane C has to be found.Cauchy's method [1] is an interesting candidate for the numerical solution.Let  0 ∈ C be a starting point for a zero of .We set  +1 =   (  ) =   −  (  ) ,  ∈ R. ( If  is a simple root of () and  is selected in such way that the following criterion is satisfied,      1 −   ()      < 1, the Cauchy method has the following properties: (a)  is an attractive fixed point of a map   : C → C (C = C ∪ {∞}).

Examples and Graphics
Let us consider a famous example given by Cayley (1879, [2]).Let for  1 = 1,   .Figure 1 shows basins of attraction for (7) with a value of , respectively,   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  → 0 in ( 1).
The answer can be given by differential system (6).Denote Δ = , where  = sign().To see this, let Rearranging yields Finally, letting Δ → 0 (⇒  → 0) we have We can see in Figure 3(a) that the fixed point  1 is attractive while  2,3 are repulsive; in Figure 3(b), the fixed point  1 is repulsive while  2,3 is attractive.It is clear that this qualitative behavior change is controlled by the sign of the  parameter.

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    () = 1 −   () = 0. Thus if  is degree , then   admits at the most −1 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.
Proof.Let the monic with degree with  simple roots  1 ,  2 , . . .,   and we have Assume that () solves (6).Rearranging terms yields and integrating with respect to  and in accordance with (14 Finally, exponentiation shows that

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 by Subsequently, we have shown the basins of attraction and vector field according to three different choices of the function .
Note that  is locally convex and the local convergence of (19) is established (see [4]).

The Best Choice of 𝜆-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 condition      1 −  ()   ()      = 0, when () = 0, and we can take Substituting (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 () =  3 − 1 have been carried out, and pictures of these sets are well known [5][6][7][8] (see Figure 7).

Further Motivation
The method ( 1) is defined for any  ∈ C.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.
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  ̸ = 0 when () =  3 − 1 are  1 = − √ 3/3 √  and  2 = √ 3/3 √ .In Figure 8(a) the global behaviour of the orbit of the critical point  1 is pictured.The horizontal and vertical axes correspond to the real and imaginary part of the complex parameter  in the region [−1.2, 1.20] × [−1.20, 1.20].The dark area in the picture is the subset of 600 × 600 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 [−1.10,1.30] × [−1.20, 1.20] in the previous picture.In these figures, which were generated by examining the parameter values on 600×600 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 () =