Hindawi Publishing Corporation Journal of Nonlinear Dynamics Volume 2015, Article ID 162818, 9 pages http://dx.doi.org/10.1155/2015/162818
Research Article On the Complex Dynamics of Continued and Discrete Cauchy’s Method Mohamed Lamine Sahari Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria Correspondence should be addressed to Mohamed Lamine Sahari;
[email protected] Received 2 May 2015; Accepted 17 September 2015 Academic Editor: Ivo Petras Copyright © 2015 Mohamed Lamine Sahari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let p be a complex polynomial of fixed degree n. In this paper we show that Cauchy’s method may fail to find all zeros of p for any initial guess 𝑧0 lying in the complex plane and we propose several ways to find all zeros of a given polynomial using scaled Cauchy’s methods.
1. Introduction For a large class of computational problems a zero of a polynomial 𝑝 in the complex plane C has to be found. Cauchy’s method [1] is an interesting candidate for the numerical solution. Let 𝑧0 ∈ C be a starting point for a zero of 𝑝. We set 𝑧𝑘+1 = 𝑔𝜆 (𝑧𝑘 ) = 𝑧𝑘 − 𝜆𝑝 (𝑧𝑘 ) ,
𝜆 ∈ R.
(1)
If 𝛼 is a simple root of 𝑝(𝑧) and 𝜆 is selected in such way that the following criterion is satisfied, 1 − 𝜆𝑝 (𝛼) < 1,
(2)
However, someone may wonder about the basin of attraction 𝐴 (𝛼) = {𝑧 ∈ C : 𝑔𝜆𝑘 (𝑧) → 𝛼, 𝑘 → ∞} ,
(5)
of a root 𝛼 of 𝑝 and the relationship with the vector field and trajectories of the differential system: 𝑑𝑧 (𝑡) = −𝜎𝑝 (𝑧 (𝑡)) , 𝑑𝑡
𝑡 ≥ 0, 𝜎 = ±1, 𝑧 (0) = 𝑧0 ,
(6)
known as the continuous Cauchy’s method. It is assumed that discrete Cauchy’s method (1) may be interpreted as Euler step for the differential equation (6).
the Cauchy method has the following properties: (a) 𝛼 is an attractive fixed point of a map 𝑔𝜆 : C → C (C = C ∪ {∞}). (b) For 𝑧0 sufficiently close to 𝛼, the sequence of iterates, 𝑧𝑘 = 𝑔𝜆𝑘 (𝑧0 ) = 𝑔𝜆 (𝑔𝜆𝑘−1 (𝑧0 )) ,
(3)
will converge to the root 𝛼 and 𝑧𝑘+1 − 𝛼 𝑧𝑘 − 𝛼 = 1 − 𝜆𝑝 (𝛼) ; that is, the sequence (3) converges linearly.
(4)
2. Examples and Graphics Let us consider a famous example given by Cayley (1879, [2]). Let 𝑝 (𝑧) = 𝑧3 − 1,
(7)
for 𝛼1 = 1, 𝛼2 = −1/2 + 𝑖(1/2)√3, and 𝛼3 = −1/2 − 𝑖(1/2)√3, the three roots of 𝑝(𝑧), and color a point 𝑧0 ∈ C blue if 𝑘=∞ {𝑧𝑘 }𝑘=∞ 𝑘=0 converges to 𝛼1 , color it green if {𝑧𝑘 }𝑘=0 converges to 𝑘=∞ 𝛼2 , and color it red if {𝑧𝑘 }𝑘=0 converges to 𝛼3 . Any remaining point gets coloured white (e.g., if 𝑧𝑘 𝛼1,2,3 ). Figure 1 shows basins of attraction for (7) with a value of 𝜆, respectively,
2
Journal of Nonlinear Dynamics
(a)
(b)
Figure 1: Basins of attraction for 𝑝(𝑧) = 𝑧3 − 1 with (a) 𝜆 = −0.1 and 𝑧 ∈ [−3.00, 3.00] × [−3.00, 3.00]𝑖. (b) 𝜆 = 0.1 and 𝑧 ∈ [−5.00, 5.00] × [−3.00, 3.00]𝑖.
(a)
(b)
Figure 2: Basins of attraction for 𝑝(𝑧) = 𝑧3 − 1 with (a) 𝜆 = −0.01. (b) 𝜆 = 0.01.
taken −0.1 and 0.1 for the region [−3.0, 3.0] × [−5.0, 5.0] in Figure 1(a) and [−5.0, 5.0]×[−3.0, 3.0] in Figure 1(b). Figure 2 indicates corresponding basins of attraction for the gradient method with 𝜆 = −0.01 and 𝜆 = 0.01 for the region [−2.1, 2.1] × [9.5, 14.5] in Figure 2(a) and [−15.0, −10.0] × [−1.5, 1.5] in Figure 2(b). Note that the boundary of the basins of attraction for each root of 𝑝 exhibits a fractal structure (see Figures 1–5). These figures show also the role played by the parameter 𝜆; 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 𝜆 → 0 in (1).
The answer can be given by differential system (6). Denote Δ𝑡 = 𝜆𝜎, where 𝜎 = sign(𝜆). To see this, let 𝑧𝑘+1 = 𝑧𝑘 − 𝜎Δ𝑡𝑝 (𝑧𝑘 ) .
(8)
𝑧𝑘+1 − 𝑧𝑘 = −𝜎𝑝 (𝑧𝑘 ) . Δ𝑡
(9)
Rearranging yields
Finally, letting Δ𝑡 → 0 (⇒ 𝜆 → 0) we have 𝑑𝑧 (𝑡) = −𝜎𝑝 (𝑧 (𝑡)) . 𝑑𝑡
(10)
Journal of Nonlinear Dynamics
3 2.0
2.0
1.6
1.6
1.2
1.2
0.8
0.8
0.4
0.4
0.0
0.0
−2.0 −1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 −0.4 x −0.8 y
1.2
1.6
2.0
−2.0 −1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 −0.4 x −0.8 y
−1.2
−1.2
−1.6
−1.6
−2.0
−2.0
(a)
1.2
1.6
2.0
(b)
Figure 3: Vector field for continuous Cauchy’s method when 𝑝(𝑧) = 𝑧3 − 1, with 𝜎 = −1 in (a) and 𝜎 = 1 in (b).
We can see in Figure 3(a) that the fixed point 𝛼1 is attractive while 𝛼2,3 are repulsive; in Figure 3(b), the fixed point 𝛼1 is repulsive while 𝛼2,3 is attractive. It is clear that this qualitative behavior change is controlled by the sign of the 𝜎 parameter.
Theorem 4 (case of continuous Cauchy’s method). If 𝑧(𝑡) satisfies (6) for all 𝑡 > 0 with 𝜎 = 1 and 𝛼 is the root of 𝑝(𝑧) when Re(𝑝 (𝛼)) > 0, then
3. Cauchy’s Method as a Dynamical System
for some 𝑧0 in C.
A critical point of a holomorphic map is usually a point, where the derivative vanishes. In particular, a critical points of 𝑔𝜆 = 𝑖𝑑 − 𝜆𝑝 are solutions of 𝑔𝜆 (𝑧) = 1 − 𝜆𝑝 (𝑧) = 0. Thus if 𝑝 is degree 𝑛, then 𝑔𝜆 admits at the most 𝑛−1 different critical points.
Proof. Let the monic with degree 𝑛
Definition 1 (immediate basins). Let 𝜉 be an attracting fixed point of 𝑔𝜆 . The connected component 𝐴∗ (𝜉) of the basins of attraction 𝐴(𝜉) that contains 𝜉 is called its immediate basins.
lim 𝑧 (𝑡) = 𝛼,
𝑡→∞
𝑝 (𝑧) = 𝑧𝑛 + 𝑎𝑛 𝑧𝑛−1 + 𝑎𝑛−1 𝑧𝑛−2 + ⋅ ⋅ ⋅ + 𝑎1 ,
Theorem 3 (case of discrete Cauchy’s method). If 𝛼 is a root of 𝑝(𝑧) with Re(𝑝 (𝛼)) > 0, then Cauchy’s method converges towards this root for some 𝜆 > 0. Proof. We have |1 − 𝜆𝑝 (𝛼)|2 = |𝑝 (𝛼)|2 𝜆2 − 2𝜆Re(𝑝 (𝛼)) + 1 (where Re(𝑧) denotes the real part of complex number 𝑧) and, for 0 < 𝜆 < 2Re(𝑝 (𝛼))/|𝑝 (𝛼)|2 , condition (2) is obtained.
(12)
with 𝑛 simple roots 𝛼1 , 𝛼2 , . . . , 𝛼𝑛 and we have 𝑛 1 1 1 =∑ 𝑛 𝑝 (𝑧) 𝑖=1 (𝑧 − 𝛼𝑖 ) ∏ 𝑗=1, 𝑗=𝑖̸ (𝛼𝑖 − 𝛼𝑗 )
(13)
The following theorem is the main result in the study of basins. Theorem 2 (Fatou, [3]). If 𝜉 is an attracting fixed point of 𝑔𝜆 , then the immediate basins 𝐴∗ (𝜉) contain at least one critical point. We deduce directly from the last theorem that the number of attractive fixed points (attractive roots of 𝑝) of Cauchy’s method is at most equal to the degree of 𝑝. Thus the Cauchy’s method cannot find all roots of the complex polynomial 𝑝(𝑧). In the subsequent two results we will explain this fact.
(11)
𝑛
1 1 , (𝛼 ) (𝑧 − 𝛼 ) 𝑝 𝑖 𝑖 𝑖=1
=∑ 𝑧
∫
𝑧0
𝑛 𝑑𝑧 1 [ln (𝑧 − 𝛼𝑖 ) − ln (𝑧0 − 𝛼𝑖 )] . =∑ 𝑝 (𝑧) 𝑖=1 𝑝 (𝛼𝑖 )
(14)
Assume that 𝑧(𝑡) solves (6). Rearranging terms yields 𝑑𝑧 = −𝑑𝑡, 𝑝 (𝑧)
(15)
and integrating with respect to 𝑡 and in accordance with (14) 𝑛
𝑖=𝑛 1 1/𝑝 (𝛼𝑖 ) ) = ln (𝑧 − 𝛼 ) ln (𝑧 − 𝛼 ∏ 𝑖 𝑖 𝑖=1 𝑝 (𝛼𝑖 ) 𝑖=1
∑
= −𝑡 + ln (𝐶) .
(16)
4
Journal of Nonlinear Dynamics and in the singular points 𝛼1 , 𝛼2 , and 𝛼3 , we have
Finally, exponentiation shows that 𝑖=𝑛
1/𝑝 (𝛼𝑖 )
∏ (𝑧 − 𝛼𝑖 )
= 𝐶𝑒−𝑡 ,
𝐶 ∈ C.
(17)
𝑖=1
𝛽1 (𝛼1 ) = 𝛽2 (𝛼1 ) = 1 − 3𝜆, 3 𝛽1 (𝛼2 ) = 𝛽1 (𝛼3 ) = 1 − 𝜆 (1 − 𝑖√3) , 2 3 𝛽2 (𝛼2 ) = 𝛽2 (𝛼3 ) = 1 − 𝜆 (1 + 𝑖√3) , 2
4. Computer Experiments with Scaled Cauchy’s Methods To obtain the condition (2), we used the scaled complex of (1); that is, 𝑧𝑘+1 = 𝐺𝜆 (𝑧𝑘 ) = 𝑧𝑘 − 𝜆𝜌 (𝑧𝑘 ) 𝑝 (𝑧𝑘 ) , 𝜆 > 0, 𝜌 : C → C.
(18)
𝛽1 (𝛼1 ) = 𝛽2 (𝛼1 ) < 1
2 if 𝜆 ∈ (0, ) , 3
𝛽1 (𝛼2 ) = 𝛽2 (𝛼2 ) = 𝛽1 (𝛼3 ) = 𝛽2 (𝛼3 ) < 1 1 if 𝜆 ∈ (0, ) . 3
The continuous form is given by 𝑑𝑧 (𝑡) = −𝜌 (𝑧 (𝑡)) 𝑝 (𝑧 (𝑡)) , 𝑑𝑡
𝑡 ≥ 0, 𝑧 (0) = 𝑧0 .
(19)
Subsequently, we have shown the basins of attraction and vector field according to three different choices of the function 𝜌. Case 1 (𝜌(𝑧) = sign(Re(𝑝 (𝑧)))). The function 𝜌 : 𝑧 → sign(Re(𝑝 (𝑧))) is not holomorphic and a similar analysis of (2) of Section 1 is not possible in C; then we consider the dynamics (18) with 𝑝(𝑧) = 𝑧3 − 1 in R2 . Consider 3 2 {𝑥𝑘+1 = 𝑥𝑘 − 𝜎 (𝑥𝑘 , 𝑦𝑘 ) (𝑥𝑘 − 3𝑥𝑘 𝑦𝑘 − 1) , 𝐺𝜆 := { 𝑦 = 𝑦𝑘 − 𝜎 (𝑥𝑘 , 𝑦𝑘 ) (3𝑥𝑘2 𝑦𝑘 − 𝑦𝑘3 ) , { 𝑘+1
Thus the fixed point 𝛼1 is attractive for 𝜆 ∈ (0, 2/3), and then 𝛼2 , 𝛼3 are attractive for 𝜆 ∈ (0, 1/3). Figure 4(a) showed the basins of attraction of 𝑧3 − 1 using (20) and 𝜆 = 0.10, and Figure 4(b) represents the vectors field of (19). Case 2 (𝜌(𝑧) = 1/ Re(𝑝 (𝑧))). The dynamics (18) with 𝑝(𝑧) = 𝑧3 − 1 can be formulated in R2 as 𝜆 (𝑥𝑘3 − 3𝑥𝑘 𝑦𝑘2 − 1) { { = 𝑥 − , 𝑥 { 𝑘+1 𝑘 { { 3 (𝑥𝑘2 − 𝑦𝑘2 ) { 𝐺𝜆 := { { { 𝜆 (3𝑥𝑘2 𝑦𝑘 − 𝑦3 ) { { {𝑦𝑘+1 = 𝑦𝑘 − . 3 (𝑥𝑘2 − 𝑦𝑘2 ) {
(20)
with if 𝑥2 − 𝑦2 ≥ 0, {𝜆 𝜎 (𝑥, 𝑦) = { −𝜆 if 𝑥2 − 𝑦2 < 0. {
(21)
(26)
The map 𝐺𝜆 has the following Jacobian
The Jacobian of the transformation 𝐺𝜆 is expressed by 𝐽11 𝐽12 𝐽𝐺𝜆 (𝑥, 𝑦) = ( ), 𝐽21 𝐽22
(25)
𝐽𝐺𝜆 (𝑥, 𝑦) = (
𝐽11 𝐽12 𝐽21 𝐽22
),
(27)
(22) with
with 𝐽11 = 1 − 𝜎 (𝑥, 𝑦) (3𝑥2 − 3𝑦2 ) , 𝐽12 = 6𝜎 (𝑥, 𝑦) 𝑥𝑦, 𝐽21 = 1 − 𝜎 (𝑥, 𝑦) (3𝑥2 − 3𝑦2 ) ,
𝐽11 = 1 − 𝜆 (23) 𝐽12 = 2𝜆
𝐽22 = −6𝜎 (𝑥, 𝑦) 𝑥𝑦. The eigenvalues of the matrix 𝐽𝐺𝜆 (𝑥, 𝑦) are 𝛽1 and 𝛽2 and are given by 𝛽1 (𝑥, 𝑦) = 1 − 3𝜎 (𝑥, 𝑦) 𝑥2 + 3𝜆𝑦𝑘2 + 𝑖6𝜎 (𝑥, 𝑦) 𝑥𝑦, 𝛽2 (𝑥, 𝑦) = 1 − 3𝜎 (𝑥, 𝑦) 𝑥2 + 3𝜆𝑦𝑘2 − 𝑖6𝜎 (𝑥, 𝑦) 𝑥𝑦,
(24)
(𝑥3 − 3𝑥𝑦2 − 1) 3𝑥2 − 3𝑦2 , + 2𝜆 2 3 (𝑥2 − 𝑦2 ) 3 (𝑥2 − 𝑦2 )
(𝑥3 − 3𝑥𝑦2 − 1) 𝑦 𝑥𝑦 − 2𝜆 , 2 𝑥 2 − 𝑦2 3 (𝑥2 − 𝑦2 ) 2
𝐽21 = −2𝜆
𝑥2
3
(3𝑥 𝑦 − 𝑦 ) 𝑥 𝑥𝑦 + 2𝜆 , 2 2 −𝑦 3 (𝑥2 − 𝑦2 )
(3𝑥2 𝑦 − 𝑦3 ) 𝑦 3𝑥2 − 3𝑦2 𝐽22 = 1 − 𝜆 . − 2𝜆 2 3 (𝑥2 − 𝑦2 ) 3 (𝑥2 − 𝑦2 )
(28)
Journal of Nonlinear Dynamics
5 5 4 3 2 1 0 −5
−4
−3
−2
−1
0
x
2 1 −1 y
3
4
5
−2 −3 −4 −5 (a)
(b)
Figure 4: (a) Basins of attraction for 𝑝(𝑧) = 𝑧3 − 1 using (20) and 𝜆 = 0.10. (b) Vector field when 𝑝(𝑧) = 𝑧3 − 1 (𝑧 ∈ [−5.00, 5.00] × [−5.00, 5.00]𝑖).
In a similar way, the eigenvalues of the matrix 𝐽𝐺𝜆 (𝑥, 𝑦) are 𝛽1 (𝑥, 𝑦) and 𝛽2 (𝑥, 𝑦) at the singular points
and thus we have 𝛽1 (𝛼1 ) = 𝛽2 (𝛼1 ) < 1
if 𝜆 ∈ (0, 2) ,
𝛽1 (𝛼2 ) = 𝛽2 (𝛼2 ) = 𝛽1 (𝛼3 ) = 𝛽2 (𝛼3 ) < 1
𝛽1 (𝛼1 ) = 𝛽2 (𝛼1 ) = 1 − 𝜆, 𝛽1 (𝛼2 ) = 𝛽1 (𝛼3 ) = 1 − 𝜆 (1 − 𝑖√3) ,
1 if 𝜆 ∈ (0, ) . 2
(29)
𝛽2 (𝛼2 ) = 𝛽2 (𝛼3 ) = 1 − 𝜆 (1 + 𝑖√3) ,
(30)
Case 3 (𝜌(𝑧) = 𝑝 (𝑧)). In this case
3 2 2 2 2 3 {𝑥𝑘 = 𝑥 − 3𝜆 (𝑥 − 3𝑥𝑦 − 1) (𝑥 − 𝑦 ) − 6𝜆 (3𝑥 𝑦 − 𝑦 ) 𝑥𝑦, 𝐺𝜆 := { 𝑦 = 𝑦 − 3𝜆 (3𝑥2 𝑦 − 𝑦3 ) (𝑥2 − 𝑦2 ) + 6𝜆 (𝑥3 − 3𝑥𝑦2 − 1) 𝑥𝑦. { 𝑘
(31)
𝐽12 = 18𝜆𝑥𝑦 (𝑥2 − 𝑦2 ) + 6𝜆 (𝑥3 − 3𝑥𝑦2 − 1) 𝑦
The Jacobian of the transformation 𝐺𝜆 is given by
− 6𝜆 (3𝑥2 − 3𝑦2 ) 𝑥𝑦 − 6𝜆 (3𝑥2 𝑦 − 𝑦3 ) 𝑥,
𝐽11 𝐽12 𝐽𝐺𝜆 (𝑥, 𝑦) = ( ), 𝐽21 𝐽22
(32)
𝐽21 = −18𝜆𝑥𝑦 (𝑥2 − 𝑦2 ) − 6𝜆 (3𝑥2 𝑦 − 𝑦3 ) 𝑥 + 6𝜆 (3𝑥2 − 3𝑦2 ) 𝑥𝑦
with
+ 6𝜆 (𝑥3 − 3𝑥𝑦2 − 1) 𝑦, 𝐽22 = 1 − 3𝜆 (3𝑥2 − 3𝑦2 ) (𝑥2 − 𝑦2 )
𝐽11 = 1 − 3𝜆 (3𝑥2 − 3𝑦2 ) (𝑥2 − 𝑦2 ) 3
2
2 2
− 6𝜆 (𝑥 − 3𝑥𝑦 − 1) 𝑥 − 36𝜆𝑥 𝑦 − 6𝜆 (3𝑥2 𝑦 − 𝑦3 ) 𝑦,
+ 6𝜆 (3𝑥2 𝑦 − 𝑦3 ) 𝑦 − 36𝜆𝑥2 𝑦2 + 6𝜆 (𝑥3 − 3𝑥𝑦2 − 1) 𝑥. (33)
6
Journal of Nonlinear Dynamics 3 2 y 1 0 −3
−2
−1 x
0
1
2
3
−1 −2 −3
(a)
(b)
Figure 5: (a) Basins of attraction for 𝑝(𝑧) = 𝑧3 − 1 using (26) and 𝜆 = 0.10. (b) Vector field when 𝑝(𝑧) = 𝑧3 − 1 (𝑧 ∈ [−3.00, 3.00] × [−3.00, 3.00]𝑖). 3 2 1 0 −3
−2
−1
1
0
x
2
3
−1 y −2 −3
(a)
(b)
Figure 6: (a) Basins of attraction for 𝑝(𝑧) = 𝑧3 − 1 using (31) and 𝜆 = 0.010. (b) Vector field when 𝑝(𝑧) = 𝑧3 − 1 (𝑧 ∈ [−3.00, 3.00] × [−3.00, 3.00]𝑖).
We notice that the transformation 𝐺𝜆 possesses the following fixed points: 𝛼1 = (1, 0) , 1 1 𝛼2 = (− , √3) , 2 2 1 1 𝛼3 = (− , − √3) , 2 2 𝛼4 = (0, 0) .
(34)
At the points 𝛼𝑖 , (𝑖 = 1, 2, 3) the eigenvalues of 𝐽𝐺𝜆 (𝛼𝑖 ), (𝑖 = 1, 2, 3) are equal to 1 − 9𝜆; in this case the fixed points 𝛼𝑖 (𝑖 = 1, 2, 3) are attractive in the interval (0, 2/9) and 𝛼4 = (0, 0) is a indifferent fixed point (see Figure 6), since 1 0 ). 𝐽𝐺𝜆 (𝛼4 ) = ( 0 1
(35)
In the last case, the method, 𝑧𝑘+1 = 𝑧𝑘 − 𝜆𝑝 (𝑧𝑘 ) 𝑝 (𝑧𝑘 ),
𝜆 > 0,
(36)
Journal of Nonlinear Dynamics
7
Figure 7: Basins of attraction for 𝑧3 − 1 = 0 using Newton’s method in the range [−2.00, 2.00] × [−2.00, 2.00].
6. Further Motivation
is the minimization process of the function given by 𝜙 (𝑧) =
1 2 𝑝 (𝑧) , 2
(37)
called the discrete steepest descent method, since 𝜙 (𝑧) = 𝑝 (𝑧) 𝑝 (𝑧).
(38)
Note that 𝜙 is locally convex and the local convergence of (19) is established (see [4]).
5. 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 𝑧 → 𝜆𝜌(𝑧) that yield the functional condition 1 − 𝜆𝜌 (𝛼) 𝑝 (𝛼) = 0, (39) when 𝑝(𝛼) = 0, and we can take 𝜌 (𝑧) =
1 , 𝑝 (𝑧)
(40)
𝜆 = 1. Substituting (40) into (18), we get familiar Newton’s method; that is, 𝑧𝑘+1 = 𝑧𝑘 −
𝑝 (𝑧𝑘 ) . 𝑝 (𝑧𝑘 )
(41)
Numerical investigations into the basins of attraction of (41) and their boundary for the cubic polynomial 𝑝(𝑧) = 𝑧3 − 1 have been carried out, and pictures of these sets are well known [5–8] (see Figure 7).
The method (1) is defined for any 𝜆 ∈ C. However, we will only be concerned with a small real parameter 𝜆. The study of the 𝜆-parameter plane allows the identification of the singular points other than the fixed points, which are the periodic points. 𝑗
Definition 5. Let 𝑧∗ ∈ C satisfy 𝑔𝜆𝑛 (𝑧∗ ) = 𝑧∗ and 𝑔𝜆 (𝑧∗ ) ≠ (𝑧∗ ) for 𝑗 = 1, . . . , 𝑛 − 1. Then 𝑧∗ is a periodic point of period 𝑛. The set, 𝛾 (𝑧∗ ) = {𝑧∗ , 𝑔𝜆 (𝑧∗ ) , . . . , 𝑔𝜆𝑛−1 (𝑧∗ )} ,
(42)
is called a periodic orbit or a cycle (of period 𝑛). Contrary to the fixed points of 𝑔𝜆 which are roots of the polynomial 𝑝, 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 𝑔𝜆 for 𝜆 ≠ 0 when 𝑝(𝑧) = 𝑧3 − 1 are 𝜒1 = −√3/3√𝜆 and 𝜒2 = √3/3√𝜆. In Figure 8(a) the global behaviour of the orbit of the critical point 𝜒1 is pictured. The horizontal and vertical axes correspond to the real and imaginary part of the complex parameter 𝜆 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 𝑔𝜆 . 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 𝑝(𝑧) =
8
Journal of Nonlinear Dynamics
(a)
(b)
Figure 8: The 𝜆-parameter plane of Cauchy’s method for critical point 𝜒1 = −√3/3√𝜆.
(a)
(b)
Figure 9: The 𝜆-parameter plane of Cauchy’s method for critical point 𝜒2 = √3/3√𝜆.
𝑧3 − 1, the complex structure of the set exhibited in Figures 8-9 requires a more explicit study in forthcoming paper.
References
Conflict of Interests
[1] A. Cauchy, “M´ethodes g´en´erales pour la r´esolution des syst`emes d’´equations simultan´ees,” Comptes Rendus de l’Acad´emie des Sciences, vol. 25, pp. 536–538, 1878.
The author declares that there is no conflict of interests regarding the publication of this paper.
[2] A. Cayley, “Desiderata and suggestions: no. 3. The NewtonFourier imaginary problem,” American Journal of Mathematics, vol. 2, no. 1, p. 97, 1879.
Acknowledgment The authors would like to thank Professor Ilhem Djellit for their help and valuable suggestions.
[3] P. Fatou, “Sur les quations fonctionnelles,” Bulletin de la Soci´et´e Math´ematique de France, vol. 47-48, pp. 161–314, 1919. [4] W. B. Richardson Jr., “Steepest descent using smoothed gradients,” Applied Mathematics and Computation, vol. 112, no. 2-3, pp. 241–254, 2000.
Journal of Nonlinear Dynamics [5] G. Ardelean, “A comparison between iterative methods by using the basins of attraction,” Applied Mathematics and Computation, vol. 218, no. 1, pp. 88–95, 2011. [6] M. L. Sahari and I. Djellit, “Fractal Newton basins,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 28756, 16 pages, 2006. [7] M. L. Sahari and I. Djellit, “Fractalisation du bassin d’attraction dans l’algorithme de Newton Modifi´e,” ESAIM: Proceedings and Surveys, vol. 20, pp. 208–216, 2007. [8] F. Von Hasseler and H. Kiete, “The relaxed Newton’s method for rational function,” Random & Computational Dynamics, vol. 3, pp. 71–92, 1995. [9] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, NY, USA, 1983.
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