CONJUGACIES CLASSES OF SOME NUMERICAL METHODS ∗

. This fact makes the Newtonmethod one of the most used for approximating the roots of a poly-nomial. In [19] Haeseler and Peitgen discuss the basin of attraction ofthe roots and give an overview of complex dynamics of rational mapson the Riemann sphere. For an extensive and comprehensive review ofthe theory of iteration of rational maps, see that given by Blanchard in[7], and that by Blanchard and Chiu in [8], and for a treatment of thetheory see Milnor [18], and Emerenko and Lyubich [16]. Concerning ahistory of complex dynamics, the reader can consult, e.g., Alexander[1] and the reference therein.Forthestudyofotheriterationfunctionsusedforﬁndingarootofapolynomial, see[21]and[22]inwhichthedynamicsoftheSchr¨oderandK¨onig iteration functions is studied. The parameter space associatedto Schr¨oder and K¨onig iteration functions of the one–parameter familyof cubic polynomials


Introduction
Let p(z) be a complex polynomial.The Newton iteration function associated to p(z) is The function N p defines a rational map on the Riemann sphere C = C ∪ {∞} and, thus, defines a discrete dynamical system z n+1 = N p (z n ) .If the seed z 0 is chosen "near" a root α of p(z) , then the sequence (z n ) n∈N converges quadratically to the root α , since α is a superattracting fixed point of N p .This fact makes the Newton method one of the most used for approximating the roots of a polynomial.In [19] Haeseler and Peitgen discuss the basin of attraction of the roots and give an overview of complex dynamics of rational maps on the Riemann sphere.For an extensive and comprehensive review of the theory of iteration of rational maps, see that given by Blanchard in [7], and that by Blanchard and Chiu in [8], and for a treatment of the theory see Milnor [18], and Emerenko and Lyubich [16].Concerning a history of complex dynamics, the reader can consult, e.g., Alexander [1] and the reference therein.
For the study of other iteration functions used for finding a root of a polynomial, see [21] and [22] in which the dynamics of the Schröder and König iteration functions is studied.The parameter space associated to Schröder and König iteration functions of the one-parameter family of cubic polynomials p A (z) = z 3 + (A − 1)z − A, is studied in [22].In [11] This kind of study was initiated by Curry, Garnett and Sullivan for the Newton iteration function associated to the one-parameter family p A (z) of above.Here parameter regions where extraneous attractive periodic cycles exist are described.This feature is also observed for the Schröder iteration function associated to p A (z) (see [21]) as well as for the König iteration function associated to p A (z) (see [22]).A more recent study on this subject is given in [12].Another well known iteration function for finding roots of a polynomial is the Halley iteration function; for a study of this iteration function see [6] and [5] and the references therein.
In Section 3 the study of the dynamics of Newton iteration function associated to a degree three generic polynomial is reduced (by conjugacy) to the study of the dynamics of the Newton iteration function of the corresponding parameter value.The same is done for the Halley iteration function as well as for Schröder and König iteration functions in sections 4, 5, and 6, respectively,.

Preliminaries
We recall the following definition.
Definition 1.Let f and g be two maps from the Riemann sphere into itself.An analytic conjugacy between f and g is an analytical diffeomorphism h from the Riemann sphere into itself such that h • f = g • h (conjugacy equation) .

Remark.
A conjugacy h between two maps f and g preserves the basic feature of the dynamics; for example, h sends periodic points of f into periodic points of g and so on.Now we recall the concept of order of convergence of an analytical function in a neighborhood of an attracting fixed point.Let F : C → C be an analytic function, and let z 0 ∈ C .We define the sequence of iterates of z 0 by F as (z n ) n∈N where z n+1 = F (z n ) , n = 0, 1, 2, . . . .Assume that lim n→∞ z n = z .The error associated to the n -th iterate is given by e n = z n − z .If m is the first positive integer such that F (m) (z) = 0 we have In this case we say that F is an order m iteration function.For example, generically, the Newton iteration function has order 2 .

The Newton Iteration Function
We have the following useful and well known result ( [13]).
We restrict our attention to degree three and four generic polynomials.For non generic polynomials (that is, polynomials with multiple roots) the Newton iteration function associated to such polynomials does not have good properties.For example, in a neighborhood of a multiple root, the convergence of iterations to it is only linear.For a study of the Newton method for multiple roots see [14] and [15].
For the sake of completeness, we now give a result for the Newton iteration function for generic quadratic polynomials.Proposition 1.Let p(z) = az 2 + bz + c , with a = 0 , be a generic quadratic polynomial, and let q(z) = z 2 − A where A = b 2 − 4ac .Then N p and N q are conjugated.

Remark
The study of the dynamics of the Newton iteration function associated to a generic quadratic polynomial was done by A. Cayley in [9] and [10].
Let us consider generic cubic polynomials; that is, polynomials with their three roots distinct.
Theorem 2. Let p(z) be a generic cubic polynomial, and let a , b and c be its roots.Assume that they are ordered as follows: 0 and T is a conjugacy between N p and N q (q(z) = q λ,ρ (z)); that is, Proof.The proof follows from Theorem 1 of above and a straightforward computation.Note that, for three degree generic polynomials, the parameter λ of above is not essential in the study of the dynamics of the Newton map for three degree generic polynomials.

Now we show how our one-parameter family qρ (z) = z(z−ρ)(z−1) may be reduced to the well known family p
Hence setting µ = ( 2−ρ 3 ) 3 and A = 1−ρ−2ρ 2 (ρ−2) 2 , the family qρ (z) may be written as follows: by the Scaling Theorem.On the other hand, considering the family p A (z µ .Therefore, we have reduced the study of the dynamics of N qρ to the study of the dynamics of N p A , which is simple.A study of the A-parameter space was done in [11].
Let us consider generic four degree polynomials; that is, polynomials with their four roots distinct.
that is, the Newton iteration functions of q λ,ρ,µ (z) and of p(z) are conjugated), where q λ,ρ,µ (z Proof.The proof follows from Theorem 1 of above and a straightforward computation.Hence to study the dynamics of the Newton iteration function of a four degree generic polynomial, it suffices to study the dynamics of the Newton iteration function of the corresponding polynomial q ρ,µ (z) .A study of the (ρ, µ)-parameter space may be done.

The Halley Iteration Function
The Halley iteration function of a map is next in the order of simplicity of the root finding methods after the Newton iteration function.
Let p(z) be a complex polynomial.The Halley iteration function associated to p(z) is Let α be a simple root of p(z) .Then we have: where For two degree generic quadratic polynomials we have the following result.
Proposition 2. Let p(z) = az 2 + bz + c , with a = 0 , be a generic quadratic polynomial, and let q(z) = z 2 − A where A = b 2 − 4ac .Then H p and H q are conjugated.
Proof.We have On the other hand, we have p which completes the proof.
Proposition 3. (Scaling theorem for the Halley iteration function) Let p(z) be a complex polynomial, and let that is, H p and H q are conjugated by T .
Proof.We have .
Next let q(z) = b p(z) where b is a constant.An easy computation then yields H q = H p ; that is, the identity map is a conjugacy between the maps H q and H q , thus their dynamics are equivalent.
Using the above results we may now prove the following result.

König Iteration Functions
Now we study the König iteration functions.Let p(z) be a generic complex polynomial (that is, all of its roots distinct), and let m be a positive integer greater or equal than two, the König iteration function K m (p) is an iteration function of order m defined as follows For example, for m = 3 and m = 4 , we have It is clear from equation ( 3) that the construction of K m,p requires the computation of the first m − 1 derivative of p(z) .From [2], letting h 1 (z) = 1 and h k+1 (z) = h k (z)p(z) − kh k (z)p (z) , for k = 1, 2, . . ., m − 1 , we have

Thus equation (3) becomes
, where a = 0 is a constant, an easy computation shows that K m,q = K m,p .
Proof.We give the proof by induction, for m = 2 we have K 2,p = N p .Now On the other hand, we have In fact, by induction, the assertion is true for k 5) we obtain and the proof is complete.

Schröder Iteration Functions
We now study the Schröder iteration function.
Let p(z) be a generic complex polynomial (that is, all of its roots distinct), and let m be a positive integer greater or equal than two.
The Schröder iteration function S m (p) is an iteration function of order m .
We now show how to obtain the Schröder iteration function (for more details see [22]).If |h| is small enough, then , n = 1, 2, . . . .The function B(z) may be considered as a formal power series in the variable h whose coefficients depend on z .If b 1 (z) = F (z) = 0 , then B(z) may be inverted and . .(Lagrange-Bürmann formula see [17]).Now using the above approximation of an analytic function, we may define the Schröder iteration function of order m = 2, 3, . . .for a polynomial p(z) as follows: It is easy see that , and that the coefficients c n (z) are analytic functions in every region R ⊂ C where p (z) = 0 .
For example, the order 3 and order 4 Schröder iteration functions of p(z) are given by and Note that S 2,p = N p ; that is, the Schröder iteration function of order 2 corresponds to the Newton iteration function.
Theorem 1. (Scaling theorem for Schröder iteration functions) Let p(z) be a polynomial and let T (z) = αz + β de an affine map, and let q(z) = p • T (z) , then for m = 3, 4 , • • •, we have T is a conjugacy between S m,q and S m,p .
Since, q • T −1 (z) = p(z) we have q (T −1 (z)) = αp (z) and the above expression becomes We prove (9) by induction.It is clear for k = 1 , and for k = 2 and k = 3 the verification is a straightforward computation.Suppose that and the proof is complete.
We now have the next two results.Their proofs follow from Theorems 5 and 6, a straightforward computation.
we have that T is a conjugacy between S m,q λ,ρ and S m,p as well as between K m,q λ,ρ and K m,p .Furthermore, if q(z) = z(z − 1)(z − ρ) , for m = 2, 3, 4, • • • , we have that S m,q ρ and S m,q λ,ρ are conjugated, and so are K m,q ρ and K m,q λ,ρ .Therefore T is a conjugacy between S m,q and S m,p as well as between K m,q and K m,p , for m = 2, 3, 4, • • • . .For m = 3, 4 , we have that T is a conjugacy between S m,q and S m,p as well as between K m,q and K m,p .Furthermore if q(z) = z(z − 1)(z − ρ)(z − µ) , for m = 3, 4 , we have that S m,q and S m,q are conjugated, and so are K m,q and K m,q .
be a four degree generic polynomial.Assume that the roots of p(z) are distinct and ordered as follows: 0 ≤ |a| ≤ |b| ≤ |c| ≤ |d| .Let T (z) = (d − a)z + a , and let q