We consider an optimal fourth-order method for solving nonlinear
equations and construct polynomials such that the rational map arising from the method applied to these polynomials has an attracting periodic orbit of any prescribed period.
1. Introduction
Recently, an optimal fourth-order iterative method to find a simple root ρ, that is, f(ρ)=0 and f′(ρ)≠0, of a nonlinear equation f(x)=0, which is given byxn+1=xn-16f(xn)f′(xn)-5f′2(xn)+30f′(xn)f′(yn)-9f′2(yn),
where yn=xn-(2/3)(f(xn)/f′(xn)) was proposed in [1] and its dynamics behavior was investigated and analyzed in detail. By an optimal method, we mean a multipoint method without memory which requires n+1 functional evaluations per iteration, but achieves the order of convergence 2n [2].
The method (1.1) may be used to approximate both real and complex roots of the nonlinear equation with x0∈ℂ. If the initial guess x0 is chosen sufficiently near a zero of f, then an iterative method is expected to converge. This is, however, not true in general if there exists an attracting periodic orbit or cycle of period k≥2 (whose definition will be introduced below). If the initial guess happens to be chosen from the basin of attraction of an attractive periodic cycle, the sequence {xn}n=0∞ converges to the attractive cycle, not to a zero of f since any root of f(x)=0 may be considered as a period orbit of period 1. Thus the existence of attracting periodic cycles of period greater than or equal to 2 could interfere with an iterative method searching for a root of the nonlinear equation. As a result, it has been an important concern from practical aspect in iteration theory to construct specific polynomials for a given method such that the map arising from the iterative method applied to the polynomials has an attractive periodic orbit. In this direction there was some result in [3] where attractive periodic orbits of any prescribed period were constructed for some classical third-order methods. Motivated by this, in this paper we extend the construction of attractive periodic cycles of any prescribed period to higher-order iterative methods. To this end, we will recall some preliminaries, see for example Milnor [4], Amat et al. [5], and Chun et al. [1]. Let R:ℂ̂→ℂ̂ be a rational map on the Riemann sphere.
Definition 1.1.
For z∈ℂ̂ we define its orbit as the set
orb(z)={z,R(z),R2(z),…,Rn(z),…}.
Definition 1.2.
A point z0 is a fixed point of R if R(z0)=z0.
Definition 1.3.
A periodic point z0 of period m is such that Rm(z0)=z0 where m is the smallest such integer. The set of the m distinct points {z,R(z),R2(z),…,Rm-1(z)} is called a periodic cycle.
Remark 1.4.
If z0 is periodic of period m then it is a fixed point for Rm.
Definition 1.5.
If z0 is a periodic point of period m, then the derivative (Rm)′(z0) is called the eigenvalue of the periodic point z0.
Remark 1.6.
By the chain rule, if z0 is a periodic point of period m, then its eigenvalue is the product of the derivatives of R at each point on the orbit of z0, and we have
(Rm)′(z0)=(Rm)′(z1)=⋯=(Rm)′(zn-1),
that is, all the points of a cycle have the same eigenvalue.
We classify the fixed points of a map based on the magnitude of the derivative.
Definition 1.7.
A point z0 is called attracting if |R′(z0)|<1, repelling if |R′(z0)|>1, and neutral if |R′(z0)|=1. If the derivative is zero then the point is called superattracting.
Definition 1.8.
The Julia set of a nonlinear map R(z), denoted J(R), is the closure of the set of its repelling periodic points. The complement of J(R) is the Fatou set 𝔽(R).
By its definition, J(R) is a closed subset of ℂ̂. A point z0 belongs to the Julia set if and only if dynamics in a neighborhood of z0 displays sensitive dependence on the initial conditions, so that nearby initial conditions lead to wildly different behavior after a number of iterations. As a simple example, consider the map R(z)=z2 on ℂ̂. The entire open disk is contained in 𝔽(R), since successive iterates on any compact subset converge uniformly to zero. Similarly the exterior is contained in 𝔽(R). On the other hand if z0 is on the unit circle than in any neighborhood of z0 any limit of the iterates would necessarily have a jump discontinuity as we cross the unit circle. Therefore J(R) is the unit circle. Such smooth Julia sets are exceptional.
Lemma 1.9 (Invariance lemma (Milnor [4])).
The Julia set J(R) of a holomorphic map R:ℂ̂→ℂ̂ is fully invariant under R. That is, z belongs to J if and only if R(z) belongs to J.
Lemma 1.10 (Iteration Lemma).
For any k>0, the Julia set J(Rk) of the k-fold iterate coincides with J(R).
Definition 1.11.
If O is an attracting periodic orbit of period m, we define the basin of attraction to be the open set A∈ℂ̂ consisting of all points z∈ℂ̂ for which the successive iterates Rm(z),R2m(z),… converge towards some point of O.
The basin of attraction of a periodic orbit may have infinitely many components.
Definition 1.12.
The immediate basin of attraction of a periodic orbit is the connected component containing the periodic orbit.
Lemma 1.13.
Every attracting periodic orbit is contained in the Fatou set of R. In fact the entire basin of attraction A for an attracting periodic orbit is contained in the Fatou set. However, every repelling periodic orbit is contained in the Julia set.
2. Attractive Cycles Results
Let us consider the map Rf associated to fRf(z)=z+16f(z)f′(z)5f′2(z)-30f′(z)f′(y)+9f′2(y),
wherey=z-23f(z)f′(z),
for which the optimal method (1.1) is written asxn+1=Rf(xn),n≥0.
Toward the aim to construct periodic orbits of any prescribed period for the method (1.1), we have the following characterization.
Proposition 2.1.
Let Ω={x1,x2,…,xn} be a set of n distinct complex numbers, and let f be a complex analytic function. Then Ω is periodic orbit of period n of iteration function Rf if and only if
16f′(xi)f(xi)xi+1-xi=5f′2(xi)-30f′(xi)f′(yi)+9f′2(yi),i=1,2,…,n-1,16f′(xn)f(xn)x1-xn=5f′2(xn)-30f′(xn)f′(yn)+9f′2(yn),
where yi=xi-(2/3)(f(xi)/f′(xi)), i=1,2,…,n.
Proof.
Assume that Ω is a periodic orbit of Rf. Then Rf(xi)=xi+1 for i=1,2,…,n-1 and Rf(xn)=x1. We have
xi+1=xi+16f′(xi)f(xi)5f′2(xi)-30f′(xi)f′(yi)+9f′2(yi),i=1,2,…,n-1,x1=xn+16f′(xn)f(xn)5f′2(xn)-30f′(xn)f′(yn)+9f′2(yn).
Therefore, we obtain
xi+1-xi=16f′(xi)f(xi)5f′2(xi)-30f′(xi)f′(yi)+9f′2(yi),i=1,2,…,n-1,x1-xn=16f′(xn)f(xn)5f′2(xn)-30f′(xn)f′(yn)+9f′2(yn).
Conversely, suppose that f satisfies condition (2.4). Then we easily have
Rf(xi)=xi+16f′(xi)f(xi)(16f′(xi)f(xi))/(xi+1-xi)=xi+1,i=1,2,…,n-1,Rf(xn)=xn+16f′(xn)f(xn)(16f′(xn)f(xn))/(x1-xn)=x1.
Proposition 2.2.
For any positive integer n≥2, there exists a polynomial f(x) of degree less than or equal to 3n-1 for which Rf has a periodic orbit of period n.
Proof.
Let x1,x2,…,xn be a given a set of n distinct complex numbers, let z1,z2,…,zn be a set of n nonzero complex numbers, and let s1,s2,…,sn be another set of nonzero complex numbers such that
yi≠yj∀i≠j,yi≠xj∀i,j=1,2,…,n,152-9(5si2-16sizixi+1-xi)≥0,i=1,2,…,n-1,152-9(5sn2-16snznx1-xn)≥0,
where yi=xi-(2/3)(zisi).
Assume that there exists a polynomial f(x) such that
f(xi)=zi,i=1,2,…,n,f′(xi)=si,i=1,2,…,n,9f′2(yi)-30sif′(yi)+5si2=16sizixi+1-xi,i=1,2,…,n-1,9f′2(yn)-30snf′(yn)+5sn2=16snznx1-xn.
Then by Proposition 2.1, Ω={x1,x2,…,xn} is a periodic orbit of period n of Rf. Using (2.9) and condition (2.8) yields
f′(yi)=15±152-9(5si2-(16sizi/(xi+1-xi)))9i=1,2,…,n-1,f′(yn)=15±152-9(5sn2-(16snzn/(x1-xn)))9.
We now show that such a polynomial exists. For this, we use the Hermite interpolation procedure. We begin the construction by writing f(x) as
f(x)=∑i=13naifi(x),
where the functions fi(x) are polynomials defined as follows:
f1(x)=1,f2(x)=(x-x1),f3(x)=(x-x1)2,f4(x)=(x-x1)2(x-y1)2,f5(x)=(x-x1)2(x-y1)2(x-x2),f6(x)=(x-x1)2(x-y1)2(x-x2)2,f7(x)=(x-x1)2(x-y1)2(x-x2)2(x-y2)2,⋮f3n(x)=(x-x1)2(x-y1)2(x-x2)2(x-y2)2⋯(x-xn)2.
To determine the polynomial f(x) we must find suitable coefficients ai for i=1,2,…,3n. For this we must solve a linear system of 3n equations with 3n unknown of the form AX=B. The matrix A associated with the linear system of equations is given by
A=(f1(x1)f2(x1)f3(x1)⋯f3n(x1)f1′(x1)f2′(x1)f3′(x1)⋯f3n′(x1)f1′(y1)f2′(y1)f3′(y1)⋯f3n′(y1)f1(x2)f2(x2)f3(x2)⋯f3n(x2)f1′(x2)f2′(x2)f3′(x2)⋯f3n′(x2)f1′(y2)f2′(y2)f3′(y2)⋯f3n′(y2)⋯⋯⋯⋯⋯f1′(yn)f2′(yn)f3′(yn)⋯f3n′(yn))
and the column vectors X,B∈ℝn are given by X=(a1,a2,…,a3n)T and B=(f(x1),f′(x1),f′(y1),…,f(xn),f′(xn),f′(yn))T. The components of the upper triangle of the matrix A are zero, and the components of the diagonal of A are nonzero. Therefore, the linear system AX=B has a unique solution.
Example 2.3.
Let us construct a polynomial f(x) for which the iterative method Rf has a periodic orbit of period 2. For this, we let x1=0,x2=1,z1=f(x1)=2,z2=f(x2)=2,s1=f′(x1)=2 and s2=f′(x2)=2. Then we have y1=x1-(2/3)(z1/s1)=-2/3, y2=x2-(2/3)(z2/s2)=1/3, f′(y1)=(15±152-9(5s12-16s1z1/(x2-x1)))/9=22/3,-2/3, and f′(y2)=(15±152-9(5s22-16s2z2/(x1-x2)))/9=14/3,2. We construct a polynomial f(x)=∑i=16aifi(x), where f1(x)=1, f2(x)=x, f3(x)=x2, f4(x)=x2(x+(2/3))2, f5(x)=x2(x+(2/3))2(x-1), f6(x)=x2(x+(2/3))2(x-1)2. Then matrix A is given by
A=(f1(x1)00000f1′(x1)f2′(x1)0000f1′(y1)f2′(y1)f3′(y1)000f1(x2)f2(x2)f3(x2)f4(x2)00f1′(x2)f2′(x2)f3′(x2)f4′(x2)f5′(x2)0f1′(y2)f2′(y2)f3′(y2)f4′(y2)f5′(y2)f6′(y2)).
By evaluating A we obtain
A=(10000001000001-43000111259000128092590012389-13272081).
Now we consider the system of linear equations AX=B, where X=(a1,a2,a3,a4,a5,a6)T and B=(f(x1),f′(x1),f′(y1),f(x2),f′(x2),f′(y2))T. Here, we can take B=(2,2,22/3,2,2,14/3)T. Hence we have the following system:
(10000001000001-43000111259000128092590012389-13272081)(a1a2a3a4a5a6)=(2222322143)
whose solution is X=(2,2,-4,18/25,72/125,12582/625)T. Therefore, the polynomial f is given by f(x)=2+2x-4x2+(18/25)x2(x+(2/3))2+(72/125)x2(x+2/3)2(x-1)+(12582/625)x2(x+2/3)2(x-1)2 = 2+2x+(3132/625)x2-(14808/625)x4+(5872/625)x3-(8028/625)x5+(12582/625)x6. From (2.1), we obtain Rf(0)=1 and Rf(1)=0. So, Ω={0,1} is a periodic orbit of period 2 for the method Rf(x).
Proposition 2.4.
Let f(x) be a polynomial for which Rf has a periodic orbit of period n, say Ω={x1,x2,…,xn}. If f′′(xi)=0=f′′(yi) and f′(xi)=f′(yi) or f′′(xi)=0=f′′(yi) and 7f′(xi)=3f′(yi) for some i=1,2,…,n, then Ω is an superattracting periodic orbit of period n for Rf.
Proof..
Without loss of generality, we may assume that f′′(x1)=0=f′′(y1) and f′(x1)=f′(y1). By the chain rule, we have
(Rfn)′(x1)=Rf′(xn)Rf′(xn-1)Rf′(xn-2)⋯Rf′(x1).
Note that on differentiating (2.1) we have
Rf′(x)=[f′(x)(5f′2(x)-30f′(x)f′(y)+9f′2(y))2]-1×[f′(x)(5f′2(x)-30f′(x)f′(y)+9f′2(y))(21f′(x)2-30f′(x)f′(y))+9f′2(y)+16f(x)f′′(x)-16f(x)[(f′2(x)+2f(x)f′′(x))10f′3(x)f′′(x)-30f′2(x)f′′(x)f′(y)-2(f′2(x)+2f(x)f′′(x))(5f′(x)f′′(y)-3f′(y)f′′(y))]].
We therefore obtain Rf′(x1)=0. By (2.17) we conclude that (Rfn)′(x1)=0.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011–0025877).
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