We investigate the complex dynamics of a triparametric family of optimal fourth-order multiple-root solvers by analyzing their basins of attraction along with extensive study of Möbius conjugacy maps and extraneous fixed points applied to a prototype quadratic polynomial raised to the power of the known integer multiplicity m. A 600×600 uniform grid centered at the origin covering 6×6 square region is chosen to display the initial points on each basin of attraction according to a coloring scheme based on their orbit behavior. With illustrative basins of attractions applied to various test polynomials and the corresponding statistical data for convergence as well as a number of comparisons made among the listed methods, we confirm our investigation and analysis developed in this paper.
1. Introduction
Many researchers [1–7] have shown their interest in the dynamics of iterative methods locating the multiple roots [8, 9] of a nonlinear equation. To ensure the convergence of an iterative method in a root-finding problem [10], it is very important to take a good initial value [11–17] close to the desired zero of the given nonlinear equation under consideration. In connection with such a choice of a good initial value, we pay a special attention to the complex dynamics for a number of optimal fourth-order multiple-root finders by investigating their basins of attraction.
Definition 1.
Let xnn=0∞ be a sequence converging to α and let en=xn-α be the nth iterate error. If there exist real numbers p and a nonzero constant b such that the following error equation holds(1)en+1=benp+Oenp+1,then b or b is called the asymptotic error constant and p is called the order of convergence [18].
Definition 2.
Let d be the number of distinct functional or derivative evaluations per iteration. The efficiency index [19] is defined by EI=p1/d, where p is the order of convergence.
In our study, all the listed methods have the same EI=41/3≈1.587 agreeing with Kung-Traub optimality [19] conjecture. We investigate the basins of attraction of a number of iterative methods with various test polynomials. Typical fourth-order multiple-root finders developed by Kanwar et al. [20], Soleymani and Babajee [21], and Shengguo et al. [6] are conveniently denoted by Kan, Sol, and Li for later use. Besides, by extending the work of Geum and Kim [22, 23], we propose a triparametric family of optimal fourth-order methods Yk’s whose developments will be described in Section 2. They are listed below in their respective order.
Yk:(5)yn=xn-2mm+2fxnf′xn,xn+1=yn-af′xn+bf′ynfxn-cf′xn+df′ynFyn,whereFyn=fxn+yn-xnλf′xnf′ynf′xn+ρf′yn,where a, b, c, d, λ, and ρ are parameters to be chosen for fourth order of optimal convergence [19, 24]. Typical cases of methods Yk’s are presented in Table 1 for 1≤k≤6 with selected parameters λ, ρ, and d.
Typical methods with selected parameters (λ,ρ) and constants (a,b,c,d).
We describe the main theorem regarding the convergence behavior of proposed family of methods (5) and select parameters a, b, and c for the quartic convergence with the aid of Taylor expansion and symbolic computation of Mathematica [25].
Theorem 3.
Let f:C→C have a zero α with integer multiplicity m≥1 and be analytic in a small neighborhood of α. Let x0 be an initial guess chosen in a sufficiently small neighborhood of α. Let κ=m/(m+2)m,γ=2m/(m+2), and λ,ρ,d∈R be free constant parameters. Let a=(1/8m+2γκλ)τ-mm+24κ3γλ-ρρ2+κ[8dm+2γλγλ-ρ+m3(-2m2+5m+8γλ + 3m+22ρ)], b=mm+23κ(1+κρ)/8-d, c=-(m4+κ3m3m+2-8dγλρ + 3m2m+22κ2ρ2+mm+23κ3ρ3)/8γκλ, and τ=m4m+2-3m2m+23κ2γλ-ρρ. Then iterative method (5) is of order four and defines a triparametric family of iterative methods with the following error equation: (6)en+1=ψ4en4+Oen5,n=0,1,2,…,where ψ4=((m8+2m+6m2+4m3+m4+-8+12m+14m2+14m3+6m4+m5κρ)/3m4(m+1)3(m + 2)m+m+2κρ)θ13-(1/m(m+1)2(m+2))θ1θ2+(m/(m+2)3(m+1)(m+3))θ3, θj=f(m+j)(α)/f(m)(α) for j∈N, and en=xn-α.
Proof.
Using Taylor’s series about α, we have (7)fxn=emβ1+A1e+A2e2+A3e3+A4e4+Oen5,(8)f′xn=em-1βm1+B1e+B2e2+B3e3+B4e4+Oen5,where β=f(m)(α)/m!, Ak=(m!/m+k!)θk, Bk=(m-1!/m+k-1!)θk, and θk=f(m+k)(α)/f(m)(α) for k∈N.
Dividing (7) by (8) gives us(9)fxnf′xn=1men-K1en2-K2en3+K3en4+Oen5,where K1=-A1+B1, K2=-A2+A1B1-B12+B2, and K3=-A3+A2B1-A1B12+B13 + A1B2 − 2B1B2+B3.
Letting t=1-γ/m with the above relation (9), we have(10)yn=xn-γfxnf′xn=α+te+K11-ten2+K21-ten3+K31-ten4+Oen5,f′yn=mtm-1+mtmB1+K1m-1+t-mtt2e+12mtm-3K12m-2m-1t-12-2B1K1mt-1t2+2tB2t3+K2m-1+t-mten2+16mtm-4-K13m-3m-2m-1t-13+3B1K12m-1mt-12t2-6K1t-1t-tK2m-2m-1t-1+B2m+1t3+6t2K3m-1+t-mt+t-B1K2mt-1+B3t3en3+124mtm-5K14m-4m-3m-2m-1t-14-4B1K13m-2m-1mt-13t2+12K12t-12t-K2m-3m-2m-1t-1+B2mm+1t3-24K1t-1t2-K3m-2m-1t-1+tB3m+2t3+B1K2mm-1+t-mt+12t2K22m-2m-1t-12-2B2K2m+1t-1t3+2tK4m-1t-1+t-B1K3mt-1+B4t4e4+Oen5.Substituting (7)–(10) into (5), we get the error equation:(11)en+1=ψ1en+ψ2en2+ψ3en3+ψ4en4+Oen5,where ψ1=t-at+tmρ+t-mbtt+tmρ+dt+ctmt+tm-γλ+ρ/mt+tmρ and coefficients ψi(i=2,3,4) depend on the parameters t, a, b, c, d, λ, and ρ and the function f(x).
Solving ψ1=0 and ψ2=0 for a and b, we have(12)a=tm-b+dt-m+dtγλ-ct+tm-γλ+ρt+tmρ,(13)b=-mtmt-11+mt-1+t-dt2+2dt1+mρ+t2mcγλ+dρ-γλ+ρt+tmρ2.After substituting a and b into ψ3=ψ31θ12+ψ32θ2, we solve ψ3=0. Due to the fact that ψ3 is independent of θ1 and θ2, solving ψ31=ψ32=0 for c and t, we have(14)t=mm+2,c=-12t-mt+tmρ3Ω12mt2+t+Ω2t+m21t-3+4t2-2dtmρΩ1t+tmρ3,where Ω1=(t-1)2(1+m(t-1)+t)3γλ and Ω2=m3t-11+2t-1t.
Applying t=m/(m+2) into (12)–(14) with γ=2m/(m+2) and κ=(m/(m+2))m, we have the following relations:(15)a=18m+2γκλm4m+2-3m2m+23κ2γλ-ρρ-mm+24κ3γλ-ρρ2+κ8dm+2γλγλ-ρ+m3-28+m5+mγλ+3m+22ρ,b=18-8d+mm+23κ1+κρ,c=-m4+κ3m3m+2-8dγλρ+3m2m+22κ2ρ2+mm+23κ3ρ38γκλ.Thanks to symbolic computation of Mathematica [25], we reach the error equation below:(16)en+1=ψ4en4+Oen5,where ψ4=ψ41θ13+ψ42θ1θ2+ψ43θ3 with(17)ψ41=m8+2m+6m2+4m3+m4+-8+12m+14m2+14m3+6m4+m5κρ3m4m+13m+2m+m+2κρ,ψ42=-1mm+12m+2,ψ43=mm+23m+1m+3,completing the proof.
Remark 4.
If λ=-1/κγ, ρ=-1/κ, and d=0 are selected, then we find relations:(18)a=-m32m+2,b=0,c=-m. In this case, proposed method (5) reduces to method Li given by (4).
3. Conjugacy Maps and Dynamics
Multipoint iterative methods [19] solving a nonlinear equation of the form f(x)=0 can be generally written as a discrete dynamical system(19)xn+1=Rfxn,where Rf is the iteration function. We begin by writing (5) in the form of a complex discrete dynamical system:(20)zn+1=Rfzn=zn-fznf′znHfzn,where Hf(zn)=γ+(a+cσ)+(b+dσ)(v/s), γ=2m/(m+2), σ=1-γλs/(v+ρs), s=f(zn)/f′(zn), and v=f(zn)/f′(yn); a, b, and c are given, respectively, by (12), (13), and (14); λ,ρ,d∈C are free parameters.
Definition 5.
Let F:X→X and G:Y→Y be two functions (dynamical systems). One says that F and G are conjugate if there is a function h:X→Y such that h∘F=G∘h. Then the map h is called a conjugacy [26].
Remark 6.
Note that a conjugacy indeed preserves the dynamical behavior between the two dynamical systems; for example, if F is conjugate to G via h and ξ is a fixed point of F, then h(ξ) is a fixed point of G.
Furthermore, if h is a homeomorphism, that is, if F is topologically conjugate to G via h, and ζ is a fixed point of G, then h-1(ζ) is a fixed point of F. Also, we find G=h∘F∘h-1 and Gn=(h∘F∘h-1)∘(h∘F∘h-1)∘⋯∘(h∘F∘h-1)=h∘Fn∘h-1. If F and G are invertible, then the topological conjugacy h maps an orbit (21)…,F-2x,F-1x,x,Fx,F2x,…,of F, onto an orbit (22)…,G-2y,G-1y,y,Gy,G2y,…,of G, where y=h(x) and the order of points is preserved. Hence, the orbits of the two maps behave similarly under homeomorphism h or h-1.
Via Möbius conjugacy map M(z)=(z-A)/(z-B), z,A≠B, A,B∈C∪{∞}, considered by Blanchard [27], Rf in (20) is conjugated to J satisfying(23)Jz;A,B,λ,ρ,d=Hz;A,B,λ,ρ,dDz;A,B,λ,ρ,d,when applied to a quadratic polynomial f(z)=(z-A)(z-B)m raised to the power of m, where H and D are polynomials with no common factors whose coefficients are generally dependent upon parameters A, B, λ, ρ, and d. The following theorem favorably indicates that J is dependent only on ρ but independent of parameters A, B, λ, and d.
Theorem 7.
Let f(z)=(z-A)(z-B)m with m∈N and M(z)=(z-A)/(z-B), A≠B, A,B∈C∪{∞}. Then Rf(z;λ,ρ,d) is conjugate to J(z;ρ) satisfying (24)Jz;ρ=z·r1μ+1+κρτ-r2ωr3μ+z1+κρτ-r4ω,where r1=m3+(m3-4m-8z)κρ, r2=4(z-κρ)+m[2+m(m+3)(1+κρ)], r3=-8κρ+z[m3+(m − 2)m(m+2)κρ], r4=4+(m+2)z[m(m+1)+(m-1)(m+2)κρ], μ=β12m-2β22m-2β2, τ=m4m-2(m + 2)3z+14mκ2, ω=2m2m-1(z+1)2mβ1m-1β2m-1βκ, β1=mz+m-γ, β2=m+z(m-γ), β=mz+12-2zγ, κ=m/(m+2)m, and γ=2m/(m+2).
Proof.
Since the inverse of M(z) is easily found to be M-1(z)=(Bz-A)/(z-1), we find after a lengthy computation with the aid of Mathematica [25] symbolic capability: (25)Jz;ρ=M∘Rf∘M-1z=z·Hz;ρDz;ρ,where H(z;ρ)=r1μ+(1+κρ)τ-r2ω and D(z;ρ)=r3μ+z(1+κρ)τ-r4ω are polynomials of degree at most 4m+1 in z with a single free parameter ρ∈C. This gives the desired result, completing the proof.
The result of Theorem 7 enables us to discover that z=0 (corresponding to fixed point A of Rf or root A of f(z)=(z-A)(z-B)m) and z=∞ (corresponding to fixed point B of Rf or root B of f(z)) are clearly two of their fixed points of the conjugate map J(z;ρ), regardless of ρ-values. Besides, by direct computation, we find that z=1 is a strange fixed point [28–30] of J (that is not a root of f(z)=(z-A)(z-B)m) due to the fact that J(1;ρ)=1, regardless of ρ-values.
We now seek further strange fixed points including z=1 (corresponding to the original convergence to infinity in view of the fact that M-1(1)=∞ or M(∞)=1). To do so, we first investigate some properties of J(z;ρ)=z·(Hz;ρ/Dz;ρ) stated in the following theorem.
Theorem 8.
Let H(z;ρ) and D(z;ρ) be given by (25). Then the following hold:
The leading highest-order term of H(z;ρ) is given by -(8m7m-4/m+23m-2)[(m+2)/mm-1+ρ]z4m+1, provided that ρ≠-(m+2)/mm-1.
H(z;ρ) has a factor z3, provided that ρ≠-(3m+4)(5m2+4)γ/2(15m3+20m2+12m-8)κ.
H(1;ρ)=D(1;ρ), and H(1;ρ)/D(1;ρ)=1, provided that ρ≠-(m32m-γ4m-2 + τ1-m3+3m2+2m+4τ2)/κm3-4m-82m-γ4m-2+τ1-m-1m+22τ2, with τ1 = 42m-1m4m-2(m+2)3κ2 and τ2=4mm2m-1(2m-γ)2m-1κ.
J(z;ρ) approaches ∞ as z tends to ∞, provided that ρ≠-(m+2)/mm-1.
Proof.
After a lengthy computation and careful algebraic treatments with the aid of Mathematica, (a) and (c) follow without difficulty. For the proof of (b), we directly compute the values of H(0;ρ)=H′(0;ρ)=H′′(0;ρ)=0 and H′′′(0;ρ)≠0. The proof of (d) follows from the fact that J(∞;ρ)=∞, by using (a) along with a highest-order term of D(z;ρ) having degree at most 4m+1.
We now will begin with locating the fixed points of the iteration function J(z;ρ). Let ϕ(z;ρ)=z-J(z;ρ), whose zeros are the desired fixed points of J. The result of Theorem 8 shows that z=0 and z=1 are the roots of ϕ. Hence the expression of ϕ(z;ρ) will take the following form: (26)ϕz;ρ=zz-1·Ψz;ρTz;ρ,where Ψ(z;ρ)=m7qz+14+m2r1β12mβ22mβ2-2m6qz+14(γ-3)+8qz2γ4 − 2m2m+1(z + 1)2mβ1m+1β2m+1βκν0-4mqzγ3ν1+2m2qγ2ν2+m5qz+12ν3+2m4qz+12ν4+m3qγν5 and T(z;ρ) = m7qzz+14-2m6qzz+14(γ-3)+8qz3γ4-2m2m+1r4z+12mβ1m+1β2m+1βκ − 4mqz2γ3ν1 + m2(r3β12mβ22mβ2+2qzγ2ν2)+m5qzz+12ν3+2m4qzz+12ν4+m3qzγν5 are polynomials in z with q=z+14mm4mκ2(1+κρ), ν0=-4(1+κρ)+m(2+m(m+3)(1+κρ)), ν1=4z+12-3zγ, ν2=4(z + 1)2(1+4z+z2)-12zz+12γ+3z2γ2, ν3=12z+12-12(z+1)2γ+(1+4z+z2)γ2, ν4=4z+12 − 12z+12γ+3(1+4z+z2)γ2-zγ3, and ν5=-16z+14+12z+12(1+4z+z2)γ − 12zz+12γ2 + z2γ3 and with β1, β2, β, r1, r3, and r4 given in Theorem 7.
As a result, z=0, z=1, and z=∞ are clearly the fixed points of J. Among these fixed points, z=1 is a strange fixed point that is not the root A or B. Further strange fixed points are possible from the roots of Ψ(z;ρ). The following theorem describes some properties of ϕ(z;ρ).
Theorem 9.
Let ϕ(z;ρ) be given by (26). Then the following hold:
Ψ(1/z;ρ)=(1/z4m+1)Ψ(z;ρ) for z≠0, regardless of ρ-values.
Ψ(z;ρ) has double roots at z=-(m+2)/m and z=-m/(m+2); that is, it has a factor z+(m+2)/m2z+m/(m+2)2, provided that ρ≠-1/κ.
T(1/z;ρ)=(1/z4m+5)T~(z;ρ) for z≠0, regardless of ρ-values, where T~(z;ρ)=(T(z;ρ)-δ0)/z, δ0=8m2(z2-1)β1m+1β2m+1βκ(m2m-1(z+1)2m+β1m-1β2m-1βρ).
T(z;ρ) has also double roots at z=-(m+2)/m and z=-m/(m+2); that is, it has a common factor z+(m+2)/m2z+m/(m+2)2 as shown in Ψ(z;ρ), provided that ρ=-1/κ.
ϕ(1/z;ρ)=-((z-1)/z), and (Ψz;ρ/T~z;ρ), for z≠0, provided that ρ=-1/κ.
Proof.
Via careful algebraic treatments and symbolic computation with the aid of Mathematica, (a), (c), and (e) follow without difficulty. For the proof of (b), we directly compute the values of Ψ(-(m+2)/m;ρ)=Ψ′(-(m+2)/m;ρ)=0 and Ψ′′(-(m+2)/m;ρ)=24(m+1)m3γκ2m+12(1+κρ)≠0, for ρ≠-1/κ. In view of the relations, Ψ(z)=z4m+4Ψ(1/z), Ψ′(z)=z4m+2[4(m+1)zΨ(1/z)-Ψ′(1/z)], and Ψ′′(z)=z4m[4(4m2+7m+3)z2Ψ(1/z)-2(4m+3)zΨ′(1/z)+Ψ′′(1/z)], we also find Ψ(-m/(m+2);ρ)=Ψ′(-m/(m+2);ρ)=0 and Ψ′′(-m/(m+2);ρ)=24(m+1)m3γκ6(m+1)2(1+κρ)≠0. The proof of (d) follows from the fact that T(-(m+2)/m;ρ)=T′(-(m+2)/m;ρ)=0 and T′′(-(m+2)/m;ρ)=∞, for any ρ. We also find T(-m/(m+2);ρ)=T′(-m/(m+2);ρ)=0 and T′′-m/m+2;ρ=-24m+5m3m+12κ2(1+κρ)≠0, for ρ≠-1/κ. We also find T-m/(m+2);ρ=T′-m/(m+2);ρ=0 and T′′-m/(m+2);ρ=-24m+3m3m+12γ2κ6(1+κρ)≠0, for ρ≠-1/κ.
With the use of properties of ϕ(z;ρ), we now consider some strange fixed points along with their stability for selected values of m=1 and m=2.
To continue our investigation of dynamics behind iterative map (20) applied to a quadratic polynomial raised to the power of m, f(z)=p(z)=(z-A)(z-B)m, we will describe the fixed points of J in (25) and their stability. In view of the fact that M(z) is a fixed point of J for a fixed point z of Rp with M-1(z)=(zB-A)/(z-1), we are interested in the explicit form of ϕ(z;ρ)=z-J(z;ρ) for m∈{1,2} below:(27)ϕz;ρ=-zz-1Ψ1zq1z,ifm=1,-zz-1Ψ2zq2z,ifm=2,where we conveniently denote(28)Ψ1z=91+ρ+9z41+ρ+3z11+7ρ+3z311+7ρ+z254+34ρ,q1z=91+ρ+12z2+ρ+z221+13ρ,Ψ2z=42+ρ+4z82+ρ+4z17+8ρ+4z717+8ρ+3z4236+97ρ+z2256+113ρ+z6256+113ρ+z3548+230ρ+z5548+230ρ,q2z=42+ρ+10z512+5ρ+4z15+7ρ+z620+9ρ+6z352+21ρ+z2188+81ρ+z4280+111ρ.
This enables us to discover that z=0 (corresponding to fixed point a of Rp or root A of p(z)) and z=∞ (corresponding to fixed point B of Rp or root B of p(z)) are clearly two of their fixed points regardless of m. To find further strange fixed points, we solve equations ϕ(z;ρ)=0 in (27) for z with typical values of m∈{1,2}.
We now investigate further strange fixed points including z=1 (corresponding to the original convergence to infinity in view of the fact that M-1(1)=∞ or M(∞)=1). By direct computation, we will describe the roots of ϕ(z;ρ)=0 for m∈{1,2}. To this end, we first check the existence of ρ-values for common factors (divisors) of Ψi(z) and qi(z). Besides, qi(z) will be checked if it has a divisor (z-1) or z. The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.
Theorem 10.
Let m=1 in (27). Then the following hold:
If ρ=-3, then ϕ(z;ρ)=-(z-1)z(z2+z+1) and the strange fixed points z are given by z=1 and z=(-1±3)/2.
If ρ=-1, then ϕ(z;ρ)=-(z-1)z(3z2+5z+3)/(2z+3) and the strange fixed points are given by z=1 and z=(-5±i11)/6.
If ρ=-3/5, then ϕ(z;ρ)=-(z-1)z(z+1)(3z2+11z+3)/(11z+3) and the strange fixed points are given by z=±1, z=(-11±85)/6.
If ρ=-27/17, then ϕ(z;ρ)=-z(15z4+z3+z+15)/(z+15) and the strange fixed points are given by z=(1/60)-1-1801±i1798-21801 and z=(1/60)-1+1801±i1798+21801.
Let ρ∉{-3,-1,-3/5,-27/17}. Then Ψ1(1/z)=z-4Ψ1(z) holds for z≠0. Hence, if z≠0 is a root of Ψ1(z;ρ), then so is 1/z.
Proof.
(a)–(c) Suppose that Ψ1(z)=0 and q1(z)=0 for some values of z. Observe that parameter ρ exists in a linear fashion in all coefficients of both polynomials. By eliminating ρ from the two polynomials, we obtain the relation G(z)=z(1+z)(3+2z+3z2)=0. Hence, any root of G is a candidate for a common divisor of Ψ1(z) and q1(z). Substituting all the roots of G into Ψ1(z)=0 and q1(z)=0, we find required relations for ρ and solving them for ρ, we find ρ=-3,-1. The remaining part of the proof is straightforward. (d) If z is a divisor of q1(z), then q1(0)=9(1+ρ)=0 yielding ρ=-1, which is already handled in (b). If (z-1) is a divisor of q1(z), then q1(1)=2(27+17ρ)=0, yielding ρ=-27/17. Then remaining proof is trivial. (e) By direct substitution, we find Ψ1(1/z)=z-4Ψ1(z) without difficulty. Hence Ψ1(1/z)=0 if and only if Ψ1(z)=0 for z≠0. This completes the proof.
Theorem 11.
Let m=2 in (27). Then the following hold:
If ρ=-2, then ϕ(z;ρ)=-(z-1)z(2z4+9z3+15z2+9z+2)/(z3+7z2+7z+2) and the strange fixed points are given by z=1, z=-1.80198±0.880308i, and z=-0.448023±0.21887i.
If ρ=-4, then ϕ(z;ρ)=-(z-1)z1+z3/(2z+1) and the strange fixed points are given by z=1 and z=-1 (triple).
If ρ=-988/409, then ϕ(z;ρ)=-z(170z8+951z7+1735z6+777z5-516z4+777z3+1735z2 + 951z+170)/(178z5+258z4-955z3-1735z2-951z-170) and the strange fixed points are given by z=-2.35462,-0.424696,-1.68698±0.790067i,-0.486146±0.227678i,0.765729±0.643163i.
Let ρ∉{-2,-4,-988/409}. Then Ψ2(1/z)=z-8Ψ2(z) holds for z≠0. Hence, if z≠0 is a root of Ψ2(z;ρ), then so is 1/z.
Proof.
The proofs immediately follow from the same argument as used in the proofs of Theorem 10.
As a result of Theorem 9 (a), we find the fixed points of J(z;ρ), that is, the roots of ϕ(z;ρ) explicitly as stated in the following corollary.
Corollary 12.
Let z∉{0,1} be a root of ϕ(z;ρ), that is, a root of Ψi(z;ρ) for 1≤i≤2 in (27). Suppose that Ψ(z;ρ) and qi(z) have no common factors for some suitable ρ-values. Then the roots of ϕ(z;ρ) for 1≤i≤2 are explicitly given by the following:
The four roots of Ψ1(z;ρ) are explicitly found to be (29)z11=-11+t1+7ρ+i2-11t1+t2-7t1ρ121+ρ,z21=1z11,z31=-11+t1-7ρ-i211t1+t2+7t1ρ121+ρ,z41=1z31,
where t1=-23-3ρ(18+5ρ) and t2=23+ρ(94+55ρ).
The eight roots of Ψ2(z;ρ) are explicitly found to be (30)z12=12-s1--4+s12,z22=1z12,z32=12-s2--4+s22,z42=1z32,z52=12-u1--4+u12,z62=1z52,z72=12-u2--4+u22,z82=1z72,
where s1=(1/2)(c1-8+c12-4c2), s2=(1/2)(c1+8+c12-4c2), u1=(1/2)(d1-8+d12-4d2), u2=(1/2)(d1+8+d12-4d2), c1=(17+8ρ--7-6ρ-C2)/2(2+ρ), c2=3(9+4ρ − -7-6ρ-ρ2)/2(2+ρ), d1=(17+8ρ+-7-6ρ-ρ2)/2(2+ρ), and d2=3(9+4ρ + -7-6ρ-ρ2)/2(2+ρ).
Proof.
Since z is a root of Ψi(z;ρ) for 1≤i≤2, so is 1/z from the result of Theorem 9 (a). For the proof of (a), Ψ1(z;ρ) can be written as a product of two factors: (31)Ψ1z;ρ=z2+cz+1z2+dz+1.By expanding the right side of the above equation and comparing the coefficients of the first- and second-order terms, we find two relations: (32)c+d=11+7ρ31+ρ,2+cd=54+34ρ91+ρ,which gives the desired values of c and d. Then the four roots can be found explicitly from (z2+cz+1)=0 or (z2+dz+1)=0. Similarly for the proof of (b), Ψ2(z;ρ) can be written as a product of two factors each of which is further decomposed into two factors:(33)Ψ2z;ρ=1+c1z+c2z2+c1z3+z41+d1z+d2z2+d1z3+z4=z2+s1z+1z2+s2z+1z2+u1z+1z2+u2z+1.By the same argument as used in the proof of (a), the desired result follows. This completes the proof.
We are now ready to determine the stability of the fixed points. To do so, it is necessary to compute the derivative of J from (24): (34)J′z;ρ=4z3Q1zw1z2,ifm=1,2z3Q2zw2z2,ifm=2, where(35)Q1z=36z18+23ρ+7ρ2+36z318+23ρ+7ρ2+921+34ρ+13ρ2+9z421+34ρ+13ρ2+2z2459+546ρ+175ρ2,w1z=91+ρ+12z2+ρ+z221+13ρ,Q2z=840+38ρ+9ρ2+8z1240+38ρ+9ρ2+z4200+3850ρ+878ρ2+z114200+3850ρ+878ρ2+z224880+22056ρ+4861ρ2+z1024880+22056ρ+4861ρ2+2z343848+37745ρ+8088ρ2+2z943848+37745ρ+8088ρ2+6z555820+46235ρ+9581ρ2+6z755820+46235ρ+9581ρ2+2z4102448+86152ρ+18081ρ2+2z8102448+86152ρ+18081ρ2+z6393440+324176ρ+66891ρ2,w2z=42+ρ+10z512+5ρ+4z15+7ρ+z620+9ρ+6z352+21ρ+z2188+81ρ+z4280+111ρ.We first check the existence of ρ-values for common factors (divisors) of Qi(z) and wi(z). Besides, wi(z) will be checked if it has divisors z, z2, z3. The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.
Theorem 13.
Let m=1 in (34). Then the following hold:
If ρ=-3, then J′(z;ρ)=4z3.
If ρ=-1, then J′(z;ρ)=2z2(9z2+22z+9)/2z+32.
If ρ=-3/5, then J′(z;ρ)=12z3(11z2+34z+11)/11z+32.
If ρ=-1/3, then J′(z;ρ)=4z3(3z+5)(15z2+26z+15)/5z+33.
If ρ=-5/3, then J′(z;ρ)=4z3(3z-1)(3z2-14z+3)/z-33.
If ρ=-27/17, then J′(z;ρ)=-12z3(5+94z+5z2)/z+152.
Let ρ∉{-3,-1,-3/5,-1/3,-5/3,-27/17}. Let z be a fixed point of J(z;ρ) satisfying ϕ(z;ρ)=0. Then J′(z;ρ)=J′(1/z;ρ) holds for z≠0.
Proof.
The proofs of (a)–(f) immediately follow from the same argument as used in the proofs of Theorem 10. Eliminating ρ from the two polynomials Qi(z) and wi(z) plays a key role in obtaining the relation G(z)=(-3+z)(-1+z)z(1+z)(3+5z)(3+2z+3z2)=0, whose roots enable us to deduce some desired ρ-values. Additional requirement that z,z2,z3 are candidates for common divisors of Qi(z) and wi(z) gives only ρ=-1. For the proof of (g), via direct computation with the aid of Mathematica symbolic capability, we find J′(z;ρ)-J′(1/z;ρ)=-(4Q1zz+1Δ1/q1(z)q1(1/z)2z8)ϕ(z;ρ)=0, where Δ1=9(1+ρ)+3z(5+ρ) + 2z2(3+5ρ)+3z3(5+ρ)+9z4(1+ρ). This completes the proof.
Theorem 14.
Let m=2 in (34). Then the following hold:
If ρ=-2, then J′(z;ρ)=2z2(3+35z+112z2+159z3+112z4+35z5+3z6)/2+7z+7z2+z32.
If ρ=-4, then J′(z;ρ)=z3(8+17z+8z2)/1+2z2.
If ρ=-988/409, then J′(z;ρ)=z3σ1/-170-951z-1735z2-955z3+258z4+178z52, where σ1=121040+727134z+624992z2-5092795z3-17328190z4-24412987z5-17328190z6 − 5092795z7+624992z8+727134z9+121040z10.
Let ρ∉{-2,-4,-988/409}. Let z be a fixed point of J(z;ρ) satisfying ϕ(z;ρ)=0. Then J′(z;ρ)=J′(1/z; ρ) holds for z≠0.
Proof.
The proofs of (a)–(c) immediately follow from the same argument as used in the proofs of Theorem 13. For the proof of (d), via direct computation with the aid of Mathematica symbolic capability, we find J′(z;ρ)-J′(1/z;ρ)=-(2Q2zz+1Δ2/q2(z)q2(1/z)2z16)ϕ(z;ρ)=0, where Δ2=4(2+ρ)+4z8(2+ρ) + 4z(13+6ρ)+4z7(13+6ρ)+2z3(98+39ρ)+2z5(98+39ρ)+z2(136+57ρ)+z6(136+57ρ)+z4(204+83ρ). This completes the proof.
Table 2 summarizes the stability results for the strange fixed points ζ of J for special ρ-values with m∈{1,2}.
Stability check from |J′(ζ;ρ)| of strange fixed points ζ for special ρ-values with 1≤m≤2.
m
ρ
ζ
Number of ζ
|J′(ζ;ρ)|: t∗
1
-3/5
-1
1
(-11±85)/6
4
9/4: r
24/7: r
76/9: r
-3
1
±(1+3)/2
3
4: r
4: r
-1
1
(-5±i11)/6
3
16/5: r
14/3: r
-1/3
1
-1.70469±0.885458i
-0.461974±0.239960i
5
7/2: r
4.76095: r
4.76095: r
-5/3
1
-0.712486±0.701686i
0.545819±0.837902i
5
8: r
5.12085: r
6.37915: r
-27/17
-0.723969±0.689831i
0.690636±0.723202i
4
5.09653: r
4.90792: r
2
−2
1
-1.80197±0.880307i
-0.448023±0.218869i
5
54/17: r
6.49038: r
6.49038: r
−4
-1(triple)
1
4
1: p
11/3: r
-988/409
-2.35462
-1.68698±0.790067i
0.765729±0.643163i
-0.486146±0.227678i
−0.424696
8
834.411: r
6.04594: r
4.98848: r
6.04594: r
834.411: r
∗|J′(ζ;ρ)|: t implies that ζ is attractive, parabolic, and repulsive, if t=a(|J′|<1), t=p(|J′|=1), and t=r(|J′|>1), respectively.
We are ready to discuss the stability of the fixed points described in Theorems 10 and 11 in terms of parameter ρ.
Theorem 15.
Let m=1 and ρ∉{-3,-1,-3/5,-1/3,-5/3,-27/17}. Then the following hold:
The strange fixed point z=1 becomes an attractor, a parabolic (indifferent, neutral) point, and a repulser, respectively, when 3+2ρ/27+17ρ<1/32, 3+2ρ/27+17ρ=1/32, and 3+2ρ/27+17ρ>1/32.
The strange fixed point z=1 is a superattractor if ρ=-3/2.
Proof.
(a) From the case of m=1 in (34), we find J′(1;ρ)=32(3+2ρ/27+17ρ). Solving J′1;ρ=323+2ρ/27+17ρ=1 for ρ, we obtain an ellipse x+5685/38072/32/12692+y2/(32/32060952)=1 in the cross-sectional ρ-parameter plane for z=1 to be a parabolic point, where x=Re(ρ), y=Im(ρ). (b) Solving J′1;ρ=0 easily yields ρ=-3/2.
Theorem 16.
Let m=2 and ρ∉{-2,-4,-988/409}. Then the following hold:
The strange fixed point z=1 is a parabolic (neutral, indifferent) point, respectively, when 64+27ρ/988+409ρ<1/54, 64+27ρ/988+409ρ=1/54, and 64+27ρ/988+409ρ > 1/54.
The strange fixed point z=1 is a superattractor if ρ=-64/27.
Proof.
From the case of m=2 in (34), we find J′(1; ρ) = 5464+27ρ/988+409ρ. Solving J′1;ρ=5464+27ρ/988+409ρ=1 for ρ, we obtain an ellipse x+4634756/19584832/27000/19584832+y2/54005/8981779301472=1 in the cross-sectional ρ-parameter plane for z=1 to be a parabolic point, where x=Re(ρ), y=Im(ρ). (b) Solving J′1;ρ=0 easily yields ρ=-64/27.
We now proceed to discuss the stability of the strange fixed points ζ for conjugate map J(z;ρ) with m∈{1,2} using J′(ζ;ρ). As a consequence of Theorems 13 (g) and 14 (d) together with Corollary 12, the stability can be stated at most five strange fixed points including z=1. Then the stability of these fixed points can be best described by illustrative conical surfaces shown in Figures 1-2. The top row of each figure refers to a stability surface for strange fixed point z=1. The stability surfaces for the remaining fixed points zi(1≤i≤8) are displayed in order from top to bottom and from left to right in each case of m=1 and m=2. The underlying theory is clearly verified via cross-sectional views of the stability surfaces with ρ-parameter domains.
Stability surfaces of the strange fixed points J(z;ρ) for m=1.
Stability surfaces of the strange fixed points J(z;ρ) for m=2.
4. Extraneous Fixed Points
In this section, we will consider different complex dynamics behind the extraneous fixed points to be defined now. The fixed points of Rf are zeros of f(x) under consideration. The iteration function Rf, however, might possess other fixed points that are not zeros of f. Such fixed points different from zeros of f are called the extraneous fixed points [31, 32] of the iteration function Rf. Extraneous fixed points may form attractive, indifferent, repulsive cycles or periodic orbits to display chaotic dynamics behind the basin of attraction under investigation. The existence of such extraneous fixed points would affect the global iteration dynamics, which was demonstrated via König functions by Vrscay and Gilbert [32]. Particularly the presence of attractive cycles induced by the extraneous fixed points of Rf may alter the basin of attractions due to the trapped sequence xn. Even in the case of repulsive or indifferent fixed points, an initial value x0 chosen near a desired root may converge to another unwanted remote root. Indeed, these aspects of the Schröder functions [32, 33] were observed in an application to the family of functions fkx=xk-1,k≥2 for simple-root finders. By taking a particular member of the family into account for multiple-root finders, we are further interested in the dynamics applied to prototype quadratic polynomial z2-1 raised to the power m, being the multiplicity of zero α under consideration. Such dynamical aspects motivate our investigation of the extraneous fixed points that may affect the basins of attraction for the proposed methods (5).
The structure of Hf(xn) in (20) clearly characterizes a variety of iterative methods. The zero α of f(x) is obviously a fixed point of Rf. The points ξ≠α for which Hf(ξ)=0 are extraneous fixed points of Rf.
Let H(z) represent Hf(z) when f(z) is a finite-order rational function of z. Then it would be of great interest for us to investigate the complex dynamics of the rational iterative map Rp of the form [26](36)zn+1=Rpzn=zn-pznp′znHpzn,in connection with the basins of attraction for a variety of polynomials p(zn). Clearly, Rp(z) represents classical Newton’s method with weight function Hp(z) and may possess its fixed points as zeros of p(z) or extraneous fixed points associated with H(z).
We now turn to complex dynamics [28–30] behind the basins of attraction of iterative map (36) applied to a prototype quadratic polynomial raised to the power of m. We are interested also in the investigation of unified dynamics associated with these extraneous fixed points. To this end, we apply a simple quadratic polynomial raised to the power of multiplicity m, that is, f(z)=z2-1m to Hf(xn), simple-root cases of which were introduced by Cayley [34] and Vrscay and Gilbert [32] in dynamical studies of the Schröder and König functions for a family of functions fk(z)=zk-1, k∈N, to minimize perturbations of the Julia set boundaries.
Hence in this section we will discuss the complex dynamics of (36) associated with its extraneous fixed points. To this end, we first write H(z) associated with Hf applied to f(z)=z2-1m in the form of(37)Hz=ω+FζDζ,where ζ=z2 and ω=a+c+γ(1-dλ) with γ=2m/(m+2) is a constant independent of z;(38)Fζ=b+d2m4m-2ζ2m+ρb+d2m2m-1δ1m-1δ2ζm-λγc-dρδ12m-2δ22,Dζ=δ1m-1δ22m2m-1ζm+δ1m-1δ2ρ are polynomials having no common factors with(39)δ1=2m-γ2ζ-γ2,δ2=2m-γζ+γ.Hence, the roots of F(ζ)+ω·D(ζ)=0 may indeed express the desired extraneous fixed points of H, provided that D(ζ)≠0. In this paper, we limit ourselves to considering a simple form of H(z) by selecting λ=-m(m+2)/τγκ, ρ=-m(m+2)/τκ, and d=-m2(m+4)3κ/(m+2)2τ with τ=m2+2m-4 and κ=m/m+2m so that we have a=c=0 and b=-mm3+6m2+8m-8κ/2m+22. Consequently, F(ζ) and D(ζ) reduce to the following:(40)Fζ=m4τ-m+43δ12m-2δ22κτ2+22m-2m2m-3m+24δ1m-1δ2ζmτ-24m-3m4m-5m+23ζ2mκ,Dζ=δ1m-1δ22m2m-1ζm-mm+2τκδ1m-1δ2.
In order to compare the dynamics behavior of (36) behind the extraneous fixed points, let us now investigate the corresponding H(z) of the existing three optimal methods, Kan, Sol, and Li, introduced in Section 1. By similarly following the development procedure of H(z) as shown in (37), we find with ζ=z2(41)Hz=m-3m-2δ12m-2δ22+22m-1m2m-22m+73m-2δ1m-1δ2ζmκ-24m-2m4m-4η1ζ2mκ22δ12m-2δ22-22m-1m2m-22m+7δ1m-1δ2ζmκ+24m-2m-1m+8m4m-4ζ2mκ2forKan,-2-2mm-2m2-2mδ12m-2δ22+m-2m+2δ1m-1δ2ζmκ-22m-2m2m-3η2ζ2mκ22κζmδ1m-1δ2-22m-1m2m-3m2+2m-4ζmκforSol,m-m-2δ1m-1δ2+22m-1m2mζmκ2δ1m-1δ2-2m2m-1ζmκforLi,whereη1=3m3+19m2+16m+16 and η2=m4+2m3-4m2-8m-16.
Since H(z) in (40) or (41) defines a high-order rational function as the multiplicity m increases, it is convenient to study the typical cases of m∈{2,3,4,5} for locating the corresponding extraneous fixed points ξ by solving H(ξ)=0 for ξ=ζ1/2. In fact, Table 3 lists the extraneous fixed points ξ and their stability from the value of Rp′ξ applied to a prototype polynomial p(z)=z2-1m, respectively, for values of 2≤m≤5. As can be seen in the table, all extraneous fixed points of the listed methods are found to be repulsive. Observe that methods Sol and Li do not possess the extraneous fixed points when m=2. In addition, critical points of the proposed methods applied to a polynomial f(z)=z2-1m are found and displayed in Table 4 for values of 2≤m≤5.
Extraneous fixed points ξ=ζ1/2 with stability check for selected cases with 2≤m≤5.
In the latter part of Section 6, complex dynamics behind the extraneous fixed points will be discussed along with chaotic behavior of rational iterative maps (36) when applied to various polynomials p(z), based on visual description of their basins of attraction along with comparison of their dynamic properties and characteristics.
5. Numerical Experiment
We have conducted numerical experiments with a number of test functions using Mathematica Version 7 to confirm the optimal fourth-order convergence. We have assigned 200 significant digits to the minimum number of precision digits and prescribed error bound of ε=10-150 throughout the current experiment. The initial values x0 are selected close to the sought zero α for guaranteed convergence to the desired root. All computations have been performed by Mathematica Version 7 with AMD Kaveri 7850 CPU having 3700 Mhz of clock core speed under Windows 7 operating system.
Definition 17 (computational convergence order).
Assume that theoretical asymptotic error constant η=limn→∞en/en-1p and convergence order p≥1 are known. Define pn=log|en/η|/logen-1 as the computational convergence order. Note that limn→∞pn=p.
Typical methods Y1,Y2,…,Y6 have been applied to the test functions F_{1}–F_{6} below:(42)F1x=1+e-2xx2-x+74,m=4,α=1-33i2,i=-1,F2x=4+3sinx-2x26,m=6,α≈-0.9,F3x=cosπ2x+e1-x2-15,m=5,α=1,F4x=ex2+x+1x2-72logx2-6,m=3,α=7,F5x=cosx2+x-xlogx2-π+3+18x2+2-π,m=9,α=π-2,F6x=log4x3-6cosx3-7+π2logx3-7,m=8,α=71/3.As seen in Table 5, the order of convergence is four and the values of computational asymptotic error constant well approach theoretical value η.
Convergence for test functions F1(x)–F6(x) with selected methods Y1–Y6.
MT
F
n
xn
|F(xn)|
|xn-α|
|en/en-14|
η
pn
0
0.490000000000000-2.57000000000000i
0.000678292
0.0298039
1
0.500000115882539-2.59807622867990i
1.67×1025
1.17×10-7
0.1484996662
0.1356337586
3.97420
Y1
F1
2
0.500000000000000-2.59807621135332i
3.78×10-112
2.55×10-29
0.1356337326
4.00000
3
0.500000000000000-2.59807621135332i
9.98×10-459
5.79×10-116
0.1356337586
4.00000
4
0.500000000000000-2.59807621135332i
0.0×10-795
0.0×10-119
0
−0.890000000000000
3.66×10-7
0.0154886
1
−0.905488640138161
1.61×10-47
2.90×10-9
0.05047432426
0.04879773270
3.99189
Y2
F2
2
−0.905488637233314
4.73×10-200
3.47×10-36
0.04879773239
4.00000
3
−0.905488637233314
3.48×10-855
7.11×10-144
0.04879773270
4.00000
4
−0.905488637233314
0.0×10-1196
0.0×10-200
0
1.19000000000000
0.102779
0.190000
1
0.999178734348023
2.17×1013
0.0008×10-13
0.6301867327
0.2584060085
3.46320
Y3
F3
2
0.999999999999883
1.27×10-62
1.17×10-53
0.2572719538
4.00062
3
1.00000000000000
1.55×10-259
4.84×10-53
0.2584060085
4.00000
4
1.00000000000000
0.0×10-1000
0.0×10-200
0
2.65000000000000
0.486088
0.00424869
1
2.64575131054487
8.73×10-22
5.19×10-10
1.594967181
1.992474266
4.04075
Y4
F4
2
2.64575131106459
1.91×10-104
1.45×10-37
1.992474316
4.00000
3
2.64575131106459
4.38×10-435
8.89×10-148
1.992474266
4.00000
4
2.64575131106459
0.0×10-593
0.0×10-199
0
1.19000000000000
1.41×1012
0.0215466
1
1.06845276583309
2.37×10-53
6.27×10-7
2.911081724
2.437575882
3.95374
Y5
F5
2
1.06845339326982
2.47×10-217
3.77×10-25
2.4375758821
4.00000
3
1.06845339326982
2.89×10-873
4.96×10-98
2.437575882
4.00000
4
1.06845339326982
0.0×10-1799
0.0×10-199
0
1.89000000000000
0.0000253422
0.0229312
1
1.91292827592121
1.07×10-36
2.90×10-6
10.51276543
14.87454509
4.09193
Y6
F6
2
1.91293118277239
3.41×10-160
1.06×10-21
14.87411967
4.00000
3
1.91293118277239
3.46×10-654
1.89×10-83
14.87454509
4.00000
4
1.91293118277239
0.0×10-1597
0.0×10-199
The following test functions f1,f2,…,f7 listed below further confirm the convergence behavior of our proposed methods (5). Consider(43)f1x=x9+4x7-5,α=-0.92+0.39i,m=1,x0=-0.91+0.41i,f2x=ex6-x3-7-1x6-x3-7,α=1+2921/3=1.47,m=2,x0=1.46,f3x=ex2+1x2-72secxlogx2-6,α=-7=-2.64575,m=3,x0=-2.59,f4x=5-x+x24cotx3+7,α=1-19i2,m=4,x0=0.49-2.25i,f5x=x5-9x4+3x-27ex2+xsinπx9logx2-6x-263,α=9,m=5,x0=9.12,f6x=x2+54logx3-3x2+5x-142,α=-5i,m=6,x0=-2.18i.Table 6 shows the comparison of xn-α among listed methods Kan, Sol, and Li and Y1–Y6 described in Section 1. In Table 6, the least errors within the prescribed error bound are highlighted in bold face. Although we are limited to the selected current experiments, within three iterations, a strict comparison shows that method Y4 displays slightly better convergence for test functions f1 and f3, and method Sol displays slightly better convergence for test function f4, while method Kan displays slightly better convergence for test functions f5 and f6. In addition, both methods Sol and Li show similar convergence for test function f2. If we closely view the definition of the asymptotic error constant, we will find that the local convergence is dependent on the function f(x), an initial guess x0, the multiplicity m, and zero α. Consequently, for a given set of test functions, one method is hardly expected to always show better performance than the others.
Comparison of |xn-α| for high-order iterative methods.
f(x)
x0
|xn-α|
Kan
Sol
Li
Y1
Y2
Y3
Y4
Y5
Y6
-0.91+0.41i
|x1-α|
3.02e-5∗
4.05e-5
7.1e-6
1.67e-5
1.67e-5
2.17e-5
2.52e-7
2.92e-5
2.17e-5
f1
|x2-α|
5.49e-17
2.13e-16
3.69e-20
2.66e-18
2.66e-18
9.63e-18
1.19e-27
4.2e-17
9.63e-18
|x3-α|
6.00e-64
1.64e-61
2.71e-77
1.71e-69
1.71e-69
3.73e-67
6.11e-109
1.8e-64
3.73e-67
1.46
|x1-α|
3.3e-5
1.99e-5
1.99e-5
2.88e-5
2.88e-5
3.17e-5
8.65e-5
7.36e-5
3.43e-5
f2
|x2-α|
1.55e-15
1.21e-16
1.21e-16
7.92e-16
7.92e-16
1.27e-15
3.67e-11
9.57e-15
1.90e-15
|x3-α|
7.65e-57
1.67e-61
1.67e-61
4.52e-58
4.52e-58
3.38e-57
2.8e-30
2.73e-54
1.79e-56
-2.61
|x1-α|
1.09e-5
1.12e-5
1.11e-5
1.09e-5
1.09e-5
1.09e-5
1.07e-5
1.09e-5
1.09e-5
f3
|x2-α|
6.94e-20
8.08e-20
7.46e-20
6.94e-20
6.94e-20
6.83e-20
6.08e-20
6.76e-20
6.83e-20
|x3-α|
1.10e-76
2.15e-76
1.51e-76
1.10e-76
1.10e-76
1.03e-76
6.17e-77
9.84e-77
1.03e-76
0.49-2.25i
|x1-α|
1.09e-5
3.78e-6
4.11e-6
4.29e-6
4.29e-6
4.32e-6
4.44e-6
4.33e-6
4.32e-6
f4
|x2-α|
6.94e-20
6.67e-24
1.01e-23
1.26e-23
1.26e-23
1.30e-23
1.50e-23
1.32e-23
1.30e-23
|x3-α|
1.10e-76
6.46e-95
3.82e-94
9.42e-94
9.42e-94
1.07e-93
1.99e-93
1.17e-93
1.07e-93
9.03
|x1-α|
1.09e-5
4.50e-5
4.47e-5
4.45e-5
4.45e-5
4.45e-5
4.45e-5
4.45e-5
4.45e-5
f5
|x2-α|
6.94e-20
3.27e-16
3.17e-16
3.13e-16
3.13e-16
3.13e-16
3.11e-16
3.13e-16
3.13e-16
|x3-α|
1.10e-76
9.08e-61
8.06e-61
7.70e-61
7.70e-61
7.65e-61
7.50e-61
7.63e-61
7.65e-61
−2.19i
|x1-α|
1.09e-5
1.72e-4
1.53e-4
1.56e-4
1.56e-4
1.56e-4
1.58e-4
1.57e-4
1.56e-4
f6
|x2-α|
6.94e-20
6.35e-15
5.99e-15
7.02e-15
7.02e-15
7.18e-15
7.73e-15
7.27e-15
7.18e-15
|x3-α|
1.10e-76
1.18e-56
1.37e-56
2.88e-56
2.88e-56
3.17e-56
4.39e-56
3.35e-56
3.17e-56
∗3.02e-5=3.02×10-5.
It is important to properly select initial values guaranteeing the convergence of iterative schemes. For ensured convergence of iterative map (5), it requires good initial values close to zero α. It is not easy to determine how close the initial values are to zero α, since initial values are generally dependent upon computational precision, error bound, and the given function f(x) under consideration. One efficient way of selecting stable initial guesses is to directly use visual basins of attraction [27, 35, 36]. Since the area of convergence can be seen on the basins of attraction, it would be a reasonable measure of convergence behavior. One would say that a method having a larger area of convergence implies a more stable method. Obviously a quantitative analysis becomes an essential tool for measuring the size of area of convergence. In the next section, we will illustrate the basins of attraction of the listed methods when applied to a variety of polynomials with multiple zeros and discuss underlying relevant dynamics.
6. Basins of Attraction
Throughout the current dynamics experiment, for effective constructing basins of attraction, we have employed a tolerance ε=10-4 for convergence within a maximum of 40 iterations. To illustrate the desired basins of attraction, we first take a 600×600 uniform grid point in a 6×6 square region centered at the origin of the complex plane, which contains all roots of test functions selected. We then paint the initial points on the basins of attraction, with a diversity of colors ranging from bright ones to dark ones based on the iteration number for convergence. In Figures 3–8, the black points are regarded as the points for which the corresponding iteration scheme starting from an initial point does not converge to any root of the test polynomial under consideration. We have applied all the methods mentioned in Section 1 to a variety of polynomials having multiple roots with multiplicity of m=2,3,4,5,6. In Tables 7–12, abbreviations CPU, TCON, AVG, and TDIV denote CPU time measured in units of seconds for convergence, the number of total convergent points, the number of average iteration for convergence, and the number of divergent points, respectively. At this point, we now begin by presenting various examples to display the desired basins of attraction.
Typical Example 1 with P1(z)=(z3-z)2,m=2.
Method
CPU
TCON
AVG
TDIV
Kan
205.469
360,000
6.20867
0
Sol
111.884
351,220
5.88726
8780
Zhou
76.331
360,000
7.34681
0
Y1
184.503
360,000
6.03626
0
Y2
195.422
360,000
6.22663
0
Y3
200.024
360,000
6.34617
0
Y4
118.436
360,000
6.35492
0
Y5
308.039
360,000
9.56036
0
Y6
488.376
360,000
13.786
0
Typical Example 1 with P2(z) = (z3+5z-7)3, m=3.
Method
CPU
TCON
AVG
TDIV
Kan
442.013
360,000
5.73999
0
Sol
296.418
359,898
5.57455
102
Zhou
84.599
359,996
6.72502
4
Y1
357.679
360,000
5.44998
0
Y2
397.303
360,000
5.84558
0
Y3
411.795
360,000
6.03896
0
Y4
423.153
360,000
6.01186
0
Y5
206.343
360,000
5.35253
0
Y6
973.993
360,000
13.2845
0
Typical Example 1 with P3(z)=(z2+3z+5)4,m=4.
Method
CPU
TCON
AVG
TDIV
Kan
201.023
359,594
5.54699
406
Sol
145.612
360,000
5.36095
0
Zhou
54.226
360,000
6.00988
0
Y1
176.749
360,000
5.32411
0
Y2
187.17
360,000
5.58189
0
Y3
189.401
359,998
5.66574
2
Y4
198.277
359,725
5.73911
275
Y5
189.385
359,996
7.25579
4
Y6
384.09
359,994
10.9976
6
Typical Example 1 with P4(z)=(z2-z)6,m=6.
Method
CPU
TCON
AVG
TDIV
Kan
331.018
359,994
5.76332
6
Sol
235.64
359,998
5.56602
2
Zhou
55.615
360,000
6.45457
0
Y1
251.661
360,000
5.70224
0
Y2
252.815
360,000
5.78377
0
Y3
253.236
360,000
5.79558
0
Y4
252.316
360,000
5.78629
0
Y5
267.401
360,000
8.74116
0
Y6
416.866
359,891
8.59493
109
Typical Example 1 with P5(z)=(z2-1)2,m=2.
Method
CPU
TCON
AVG
TDIV
Kan
60.231
360,000
5.28004
0
Sol
48.375
350,408
5.00397
9592
Zhou
39.952
360,000
5.92232
0
Y1
61.948
360,000
5.31739
0
Y2
46.723
360,000
5.38721
0
Y3
60.481
360,000
5.07764
0
Y4
62.557
360,000
5.31739
0
Y5
464.556
360,000
13.8001
0
Y6
158.309
360,000
13.8001
0
Typical Example 1 with P6(z)=(z2-1)5,m=5.
Method
CPU
TCON
AVG
TDIV
Kan
101.743
360,000
5.45698
0
Sol
75.38
360,000
5.26937
0
Zhou
37.472
360,000
5.92232
0
Y1
119.169
360,000
5.59266
0
Y2
119.84
360,000
5.58321
0
Y3
96.877
360,000
5.33871
0
Y4
100.932
360,000
5.59266
0
Y5
150.634
360,000
8.93996
0
Y6
164.113
359,891
8.93996
0
(a) Kan, (b) Sol, (c) Zhou, (d) Y1, (e) Y2, (f) Y3, (g) Y4, (h) Y5, and (i) Y6, for the roots of the polynomial P1(z)=(z3-z)2.
(a) Kan, (b) Sol, (c) Zhou, (d) Y1, (e) Y2, (f) Y3, (g) Y4, (h) Y5, and (i) Y6, for the roots of the polynomial P2(z)=(z3+5z-7)3.
(a) Kan, (b) Sol, (c) Zhou, (d) Y1, (e) Y2, (f) Y3, (g) Y4, (h) Y5, and (i) Y6, for the roots of the polynomial P3(z)=(z2+3z+5)4.
(a) Kan, (b) Sol, (c) Zhou, (d) Y1, (e) Y2, (f) Y3, (g) Y4, (h) Y5, and (i) Y6, for the roots of the polynomial P4(z)=(z2-z)6.
(a) Kan, (b) Sol, (c) Zhou, (d) Y1, (e) Y2, (f) Y3, (g) Y4, (h) Y5, and (i) Y6, for the roots of the polynomial P4(z)=(z2-1)2.
(a) Kan, (b) Sol, (c) Zhou, (d) Y1, (e) Y2, (f) Y3, (g) Y4, (h) Y5, and (i) Y6, for the roots of the polynomial P4(z)=(z2-1)5.
As a first example, we have taken the following polynomial: (44)P1z=z3-z2 whose roots z=0, ±1 are all real with multiplicity m=2. Based on Table 7 and Figure 3, we find that Y1 has shown best AVG and TDIV, followed by Kan and Y2. As can be seen in Figure 3, Sol has shown considerable amount of black points. These points causing divergence behavior were observed from the last column of Table 7. The best result for CPU is given by Zhou and the worst one is given by Y6.
Our next sample has triple roots. The polynomial (45)P2z=z3+5z-73has three roots z=-0.559719-2.43718i,-0.559719+2.43718i,1.11944 of multiplicity 3. The statistical results are listed in Table 8 and relevant basins of attraction are illustrated in Figure 4. The method Y5 has performed best AVG and TDIV. As can be seen in Figure 4, Sol has shown considerable amount of black points, while Zhou has shown a few black ones. The best result for CPU is given by Y5 and the worst one is given by Y6.
As a third example, we have taken the following polynomial whose roots are all of multiplicity four: (46)P3z=z2+3z+54whose roots are z=-1.5-1.65831i,-1.5+1.65831i. The statistical results are presented in Table 9 and relevant basins of attraction are illustrated in Figure 5. The method Y1 has shown best AVG and TDIV. As can be seen in Figure 5, Kan and Y4 have shown considerable amount of black points, while Y5 and Y6 have shown a few black ones. The best result for CPU is given by Zhou and the worst one is given by Y2.
In the fourth example, we have taken the polynomial having two roots of unity (47)P4