Attracting Periodic Cycles for an Optimal Fourth-Order Nonlinear Solver

Abstract

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 by𝑥_𝑛+1=𝑥_𝑛−16𝑓𝑥_𝑛𝑓^′𝑥_𝑛−5𝑓^′2𝑥_𝑛+30𝑓^′𝑥_𝑛𝑓^′𝑦_𝑛−9𝑓^′2𝑦_𝑛, (1.1) where 𝑦_𝑛=𝑥_𝑛−(2/3)(𝑓(𝑥_𝑛)/𝑓^′(𝑥_𝑛)) 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 2^𝑛 [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 {𝑥_𝑛}^∞_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∶ℂ→ℂ be a rational map on the Riemann sphere.

Definition 1.1. For 𝑧∈ℂ we define its orbit as the set

orb(𝑧)=𝑧,𝑅(𝑧),𝑅²(𝑧),…,𝑅^𝑛(𝑧),…. (1.2)

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 𝑅^𝑚(𝑧₀)=𝑧₀ where 𝑚 is the smallest such integer. The set of the 𝑚 distinct points {𝑧,𝑅(𝑧),𝑅²(𝑧),…,𝑅^𝑚−1(𝑧)} is called a periodic cycle.

Remark 1.4. If 𝑧₀ is periodic of period 𝑚 then it is a fixed point for 𝑅^𝑚.

Definition 1.5. If 𝑧₀ is a periodic point of period 𝑚, then the derivative (𝑅^𝑚)^′(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 𝑅^𝑚′𝑧₀=𝑅^𝑚′𝑧₁=⋯=𝑅^𝑚′𝑧_𝑛−1, (1.3) 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 ℂ. 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 ℂ. 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∶ℂ→ℂ 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(𝑅^𝑘) 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∈ℂ consisting of all points 𝑧∈ℂ for which the successive iterates 𝑅^𝑚(𝑧),𝑅^2𝑚(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 𝑅_𝑓 associated to 𝑓𝑅_𝑓(𝑧)(𝑧)𝑓=𝑧+16𝑓^′(𝑧)5𝑓^′2(𝑧)−30𝑓^′(𝑧)𝑓^′𝑦+9𝑓^′2𝑦, (2.1) where2𝑦=𝑧−3𝑓(𝑧)𝑓^′(𝑧), (2.2) for which the optimal method (1.1) is written as𝑥_𝑛+1=𝑅_𝑓𝑥_𝑛,𝑛≥0. (2.3)

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,…,𝑥_𝑛} 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 𝑅_𝑓 if and only if 16𝑓^′𝑥_𝑖𝑓𝑥_𝑖𝑥_𝑖+1−𝑥_𝑖=5𝑓^′2𝑥_𝑖−30𝑓^′𝑥_𝑖𝑓^′𝑦_𝑖+9𝑓^′2𝑦_𝑖,𝑖=1,2,…,𝑛−1,16𝑓^′𝑥_𝑛𝑓𝑥_𝑛𝑥₁−𝑥_𝑛=5𝑓^′2𝑥_𝑛−30𝑓^′𝑥_𝑛𝑓^′𝑦_𝑛+9𝑓^′2𝑦_𝑛, (2.4) where 𝑦_𝑖=𝑥_𝑖−(2/3)(𝑓(𝑥_𝑖)/𝑓^′(𝑥_𝑖)), 𝑖=1,2,…,𝑛.

Proof. Assume that Ω is a periodic orbit of 𝑅_𝑓. Then 𝑅_𝑓(𝑥_𝑖)=𝑥_𝑖+1 for 𝑖=1,2,…,𝑛−1 and 𝑅_𝑓(𝑥_𝑛)=𝑥₁. We have 𝑥_𝑖+1=𝑥_𝑖+16𝑓^′𝑥_𝑖𝑓𝑥_𝑖5𝑓^′2𝑥_𝑖−30𝑓^′𝑥_𝑖𝑓^′𝑦_𝑖+9𝑓^′2𝑦_𝑖𝑥,𝑖=1,2,…,𝑛−1,₁=𝑥_𝑛+16𝑓′𝑥_𝑛𝑓𝑥_𝑛5𝑓^′2𝑥_𝑛−30𝑓′𝑥_𝑛𝑓′𝑦_𝑛+9𝑓^′2𝑦_𝑛. (2.5) Therefore, we obtain 𝑥_𝑖+1−𝑥_𝑖=16𝑓^′𝑥_𝑖𝑓𝑥_𝑖5𝑓^′2𝑥_𝑖−30𝑓^′𝑥_𝑖𝑓^′𝑦_𝑖+9𝑓^′2𝑦_𝑖𝑥,𝑖=1,2,…,𝑛−1,₁−𝑥_𝑛=16𝑓^′𝑥_𝑛𝑓𝑥_𝑛5𝑓^′2𝑥_𝑛−30𝑓^′𝑥_𝑛𝑓^′𝑦_𝑛+9𝑓^′2𝑦_𝑛. (2.6) Conversely, suppose that 𝑓 satisfies condition (2.4). Then we easily have 𝑅_𝑓𝑥_𝑖=𝑥_𝑖+16𝑓^′𝑥_𝑖𝑓𝑥_𝑖16𝑓^′𝑥_𝑖𝑓𝑥_𝑖/𝑥_𝑖+1−𝑥_𝑖=𝑥_𝑖+1𝑅,𝑖=1,2,…,𝑛−1,_𝑓𝑥_𝑛=𝑥_𝑛+16𝑓^′𝑥_𝑛𝑓𝑥_𝑛16𝑓^′𝑥_𝑛𝑓𝑥_𝑛/𝑥₁−𝑥_𝑛=𝑥₁. (2.7)

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 𝑅_𝑓 has a periodic orbit of period 𝑛.

Proof. Let 𝑥₁,𝑥₂,…,𝑥_𝑛 be a given a set of 𝑛 distinct complex numbers, let 𝑧₁,𝑧₂,…,𝑧_𝑛 be a set of 𝑛 nonzero complex numbers, and let 𝑠₁,𝑠₂,…,𝑠_𝑛 be another set of nonzero complex numbers such that 𝑦_𝑖≠𝑦_𝑗𝑦∀𝑖≠𝑗,_𝑖≠𝑥_𝑗∀𝑖,𝑗=1,2,…,𝑛,15²−95𝑠²_𝑖−16𝑠_𝑖𝑧_𝑖𝑥_𝑖+1−𝑥_𝑖≥0,𝑖=1,2,…,𝑛−1,15²−95𝑠²_𝑛−16𝑠_𝑛𝑧_𝑛𝑥₁−𝑥_𝑛≥0, (2.8) where 𝑦_𝑖=𝑥_𝑖−(2/3)(𝑧_𝑖𝑠_𝑖).
Assume that there exists a polynomial 𝑓(𝑥) such that 𝑓𝑥_𝑖=𝑧_𝑖𝑓,𝑖=1,2,…,𝑛,_′𝑥_𝑖=𝑠_𝑖,𝑖=1,2,…,𝑛,9𝑓^′2𝑦_𝑖−30𝑠_𝑖𝑓^′𝑦_𝑖+5𝑠²_𝑖=16𝑠_𝑖𝑧_𝑖𝑥_𝑖+1−𝑥_𝑖,𝑖=1,2,…,𝑛−1,9𝑓^′2𝑦_𝑛−30𝑠_𝑛𝑓^′𝑦_𝑛+5𝑠²_𝑛=16𝑠_𝑛𝑧_𝑛𝑥₁−𝑥_𝑛. (2.9) Then by Proposition 2.1, Ω={𝑥₁,𝑥₂,…,𝑥_𝑛} is a periodic orbit of period 𝑛 of 𝑅_𝑓. Using (2.9) and condition (2.8) yields 𝑓^′𝑦_𝑖=15±15²−95𝑠²_𝑖−16𝑠_𝑖𝑧_𝑖/𝑥_𝑖+1−𝑥_𝑖9𝑓𝑖=1,2,…,𝑛−1,_′𝑦_𝑛=15±15²−95𝑠²_𝑛−16𝑠_𝑛𝑧_𝑛/𝑥₁−𝑥_𝑛9. (2.10)
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)=^3𝑛_𝑖=1𝑎_𝑖𝑓_𝑖(𝑥), (2.11) where the functions 𝑓_𝑖(𝑥) are polynomials defined as follows: 𝑓₁(𝑥)𝑓=1,₂(𝑥)=𝑥−𝑥₁,𝑓₃(𝑥)=𝑥−𝑥₁²,𝑓₄(𝑥)=𝑥−𝑥₁²𝑥−𝑦₁²,𝑓₅(𝑥)=𝑥−𝑥₁²𝑥−𝑦₁²𝑥−𝑥₂,𝑓₆(𝑥)=𝑥−𝑥₁²𝑥−𝑦₁²𝑥−𝑥₂²,𝑓₇(𝑥)=𝑥−𝑥₁²𝑥−𝑦₁²𝑥−𝑥₂²𝑥−𝑦₂²,⋮𝑓_3𝑛(𝑥)=𝑥−𝑥₁²𝑥−𝑦₁²𝑥−𝑥₂²𝑥−𝑦₂²⋯𝑥−𝑥_𝑛². (2.12)
To determine the polynomial 𝑓(𝑥) we must find suitable coefficients 𝑎_𝑖 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 𝑓𝐴=¹𝑥₁𝑓₂𝑥₁𝑓₃𝑥₁⋯𝑓_3𝑛𝑥₁𝑓^′₁𝑥₁𝑓^′₂𝑥₁𝑓^′₃𝑥₁⋯𝑓^′_3𝑛𝑥₁𝑓^′₁𝑦₁𝑓^′₂𝑦₁𝑓^′₃𝑦₁⋯𝑓^′_3𝑛𝑦₁𝑓₁𝑥₂𝑓₂𝑥₂𝑓₃𝑥₂⋯𝑓_3𝑛𝑥₂𝑓^′₁𝑥₂𝑓^′₂𝑥₂𝑓^′₃𝑥₂⋯𝑓^′_3𝑛𝑥₂𝑓^′₁𝑦₂𝑓^′₂𝑦₂𝑓^′₃𝑦₂⋯𝑓^′_3𝑛𝑦₂𝑓⋯⋯⋯⋯⋯_′1𝑦_𝑛𝑓^′₂𝑦_𝑛𝑓^′₃𝑦_𝑛⋯𝑓^′_3𝑛𝑦_𝑛 (2.13) and the column vectors 𝑋,𝐵∈ℝ^𝑛 are given by 𝑋=(𝑎₁,𝑎₂,…,𝑎_3𝑛)^𝑇 and 𝐵=(𝑓(𝑥₁),𝑓′(𝑥₁),𝑓′(𝑦₁),…,𝑓(𝑥_𝑛),𝑓′(𝑥_𝑛),𝑓′(𝑦_𝑛))^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 𝑅_𝑓 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(5𝑠²₁−16𝑠₁𝑧₁/(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)=∑⁶_𝑖=1𝑎_𝑖𝑓_𝑖(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 𝑓𝐴=¹𝑥₁𝑓00000_′1𝑥₁𝑓^′₂𝑥₁𝑓0000_′1𝑦₁𝑓^′₂𝑦₁𝑓^′₃𝑦₁𝑓000₁𝑥₂𝑓₂𝑥₂𝑓₃𝑥₂𝑓₄𝑥₂𝑓00_′1𝑥₂𝑓^′₂𝑥₂𝑓^′₃𝑥₂𝑓^′₄𝑥₂𝑓^′₅𝑥₂0𝑓^′₁𝑦₂𝑓^′₂𝑦₂𝑓^′₃𝑦₂𝑓^′₄𝑦₂𝑓^′₅𝑦₂𝑓^′₆𝑦₂. (2.14) By evaluating 𝐴 we obtain 4𝐴=10000001000001−3000111259000128092590201389−13272081. (2.15)
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: 10000001000001−43000111259000128092590201389−132720𝑎81¹𝑎₂𝑎₃𝑎₄𝑎₅𝑎₆=2222322143 (2.16) 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 𝑅_𝑓(0)=1 and 𝑅_𝑓(1)=0. So, Ω={0,1} is a periodic orbit of period 2 for the method 𝑅_𝑓(𝑥).

Proposition 2.4. Let 𝑓(𝑥) be a polynomial for which 𝑅_𝑓 has a periodic orbit of period 𝑛, say Ω={𝑥₁,𝑥₂,…,𝑥_𝑛}. If 𝑓^′′(𝑥_𝑖)=0=𝑓^′′(𝑦_𝑖) and 𝑓^′(𝑥_𝑖)=𝑓^′(𝑦_𝑖) or 𝑓^′′(𝑥_𝑖)=0=𝑓^′′(𝑦_𝑖) and 7𝑓^′(𝑥_𝑖)=3𝑓^′(𝑦_𝑖) for some 𝑖=1,2,…,𝑛, then Ω is an superattracting periodic orbit of period 𝑛 for 𝑅_𝑓.

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 𝑅^𝑛_𝑓^′𝑥₁=𝑅^′_𝑓𝑥_𝑛𝑅^′_𝑓𝑥_𝑛−1𝑅^′_𝑓𝑥_𝑛−2⋯𝑅_𝑓′𝑥₁. (2.17) Note that on differentiating (2.1) we have 𝑅_𝑓′(𝑥)=𝑓^′(𝑥)5𝑓^′2(𝑥)−30𝑓^′(𝑥)𝑓^′𝑦+9𝑓^′2𝑦²⁻¹×𝑓^′(𝑥)5𝑓^′2(𝑥)−30𝑓^′(𝑥)𝑓^′𝑦+9𝑓^′2𝑦21𝑓^′(𝑥)²−30𝑓^′(𝑥)𝑓^′𝑦+9𝑓^′2𝑦+16𝑓(𝑥)𝑓^′′(𝑥)−16𝑓(𝑥)10𝑓^′3(𝑥)𝑓^′′(𝑥)−30𝑓^′2(𝑥)𝑓^′′(𝑥)𝑓^′𝑦−2𝑓^′2(𝑥)+2𝑓(𝑥)𝑓^′′(𝑥)5𝑓^′(𝑥)𝑓^′′𝑦−3𝑓^′𝑦𝑓^′′𝑦. (2.18) We therefore obtain 𝑅^′_𝑓(𝑥₁)=0. By (2.17) we conclude that (𝑅^𝑛_𝑓)′(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).