EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1902-1912
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A family of optimal cubic-order multiple-root solvers
and their dynamcis

Hwajoon Kim1, Arjun Kumar Rathie2, Young Hee Geum3,∗

1 Department of IT Engineering, Kyungdong University, Yangju 11458, Korea,
2 Department of Mathematics, Vedant College of Engineering and Technology, Rajasthan
Technical University, Village: Tulsi, Post: Jakhmund, District: Bundi, Rajasthan State,
India
3 Department of Applied Mathematics, Dankook University, Cheonan, Chungnam, 31116,
Korea

Abstract. The complex dynamical analysis of the cubic-order iterative family is proposed to draw
the fractal images via Möbius conjugacy map applied to a quadratic polynomial (z−A)m(z−B)m.
The resulting dynamics is clearly visualized through various stability surfaces and parameter spaces
using Mathematica.

2020 Mathematics Subject Classifications: 65H05, 65H99

Key Words and Phrases: Multiple-zero solver, cubic-order, conjugacy, parameter space

1. Introduction

Most nonlinear equations are used in computer science, engineering, medicine and
biology. In order to find the solutions for these nonlinear equations, numerical iterative
schemes are sought. According to the form and property of the problem, the iteration
scheme is one of the generally used methods. With the aid of initial values and recurrence
equations, initial guesses are modified continuously until desired accuracy is gotten. In the
past decade, researchers[1, 3, 8, 15, 18] have studied the development of the higher-order
solver to locate the roots of nonlinear equations. Numerous methods were suggested based
on various considerations and theories [4, 11–13]. The optimal cubic-order methods are
designed[7] {

yn = xn −m(1− t) f(xn)
f ′(xn)

,

xn+1 = xn − mf(yn)
tmf ′(xn)

,
(1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4529

Email addresses: math@kduniv.ac.kr (H.J.Kim),
arjunkumarrathie@gmail.com (A.K. Rathie), conpana@empal.com (Y.H.Geum)

https://www.ejpam.com 1902 © 2023 EJPAM All rights reserved.



H.J.Kim, A.K. Rathie, Y.H.Geum / Eur. J. Pure Appl. Math, 16 (3) (2023), 1902-1912 1903

with the multiplicity index m ∈ N of the sought root and the free parameter t ∈ C .
Let R : S → S be an operator with S is the Riemann sphere.The orbit of a point

z1 ∈ S is defined as the set of images of z1 by {z1, R(z1), . . . R
n(z1), . . .}. If R(za) = za,

a point za ∈ S is a fixed point of R. A point zb is called a critical point if R′(zb) = 0.
The following definition and theorem are important to build the conjugacy map[20] and

to visualize the relevant dynamics.

Definition 1. Let f : X → X and g : Y → Y be two analytic functions. We define
that the functions f and g are topologically conjugate if there exists a homeomorphism
h : X → Y such that h ◦ f = g ◦ h, where ◦ denotes function composition. Then the map
h is called a conjugacy [19].

Theorem 1. Let f and g be defined in Definition 1. Then the following hold[9]:
(a) g = h ◦ f ◦ h−1 and gn = h ◦ fn ◦ h−1 .
(b) If f is topologically conjugate to g via h and ν is a fixed point of g, then h−1(ν)

is a fixed point of f . If f and g are invertible, then the topological conjugacy h maps an
orbit of f onto an orbit of g and the order of points is preserved.

2. Conjugacy Maps

A nonlinear equation (1) is reconstructed in a generic form[2, 5, 9, 14] as a discrete
dynamical system

xn+1 = Rf (xn), (2)

where Rf is the iteration function.
We have the following result for discrete system as follows:

zn+1 = Rf (zn) = zn − mf(yn)

tmf ′(zn)
, (3)

where yn = zn −m(1− t) f(zn)
f ′(zn)

.

Using Möbius conjugacy map M(z) = z−A
z−B and its inverse M−1(z) = Bz−A

z−1 with z,
A ̸= B, A, B ∈ C ∪ {∞} [16, 20], Rf in (3) is conjugated to J satisfying

J(z; t) =
z(−r1r2 + r3 (1 + z) tm)

−r1r2z + r3 (1 + z) tm
, (4)

where r1 = (t+ z)m, r2 = (1 + tz)m and r3 = (1 + z)2m.

From (4), two points z = 0 and z = ∞ are fixed points of the conjugate map J(z; t),
regardless of t-values. And z = 1 is a strange fixed point of J (that is not a root of
f(z) = [(z − A)(z − B)]m) from the fact of J(1; t) = 1, regardless of t-values. That is, J
is dependent on t however independent of A and B.

With Mathematica[17], we find J(z; t) as follows:



H.J.Kim, A.K. Rathie, Y.H.Geum / Eur. J. Pure Appl. Math, 16 (3) (2023), 1902-1912 1904

(a) |ℜ(t)| ≤ 10, |ℑ(t)| ≤ 10 (b) 0.7 ≤ |ℜ(t)| ≤ 1, − 0.4 ≤ |ℑ(t)| ≤ 1

(c) −2 ≤ |ℜ(t)| ≤ 10, − 5 ≤ |ℑ(t)| ≤ 5 (d) −2 ≤ |ℜ(t)| ≤ 10, − 5 ≤ |ℑ(t)| ≤ 5

Figure 1: Stability surfaces for m = 1

J(z; t) =

{
z(t(1+z)3−(t+z)(1+tz))
t(1+z)3−z(t+z)(1+tz)

, if m=1
z(t2(1+z)5−(t+z)2(1+tz)2)
t2(1+z)5−z(t+z)2(1+tz)2

, if m=2
(5)

We find the fixed points of the iteration scheme J(z;λ). Let ϕ(z; t) = z−J(z; t), whose
zeros are the sought fixed points of J . We know that z = 0 and z = 1 are the zeros of ϕ.
Hence ϕ(z; t) is expressed as the following form:

ϕ(z; t) =
r1r2(z − 1)z

r1r2z − r3tm(z + 1)
, (6)

To investigate the dynamics behind iterative map (3) applied to a quadratic polynomial
raised to the power of m, f(z) = (z − A)m(z − B)m, we find out the fixed points of J
and their stability. From the fact that M(z) is a fixed point of J for a fixed point z of Rp

with its inverse M−1(z) = zB−A
z−1 , we calculate the explicit form of ϕ(z; t) = z − J(z; t) for

m ∈ {1, 2} below:

ϕ(z; t) =

{
z(z−1)(t+z)(1+tz)

z2+t2z2−t(1+2z+3z2)
, if m=1

z(z−1)(t+z)2(1+tz)2

−t2(1+z)5+z(t+z)2(1+tz)2
, if m=2

(7)

Theorem 2. Let m = 1. Then the following hold:
(a) If t = −1, then ϕ(z; t) = ((−1 + z)3z)/(1 + 2z + 5z2) and the strange fixed points z
are z = 0 and z = 1.



H.J.Kim, A.K. Rathie, Y.H.Geum / Eur. J. Pure Appl. Math, 16 (3) (2023), 1902-1912 1905

(a) −15 ≤ ℜ(t) ≤ 10, |ℑ(t)| ≤ 10 (b) −12 ≤ ℜ(t) ≤ −5, |ℑ(t)| ≤ 5

(c) |ℜ(t)| ≤ 2, |ℑ(t)| ≤ 5 (d) −3.7 ≤ |ℜ(t)| ≤ 3, |ℑ(t)| ≤ 5

Figure 2: Stability surfaces for m = 2

(b) If t = 0, then ϕ(z; t) = 1− z and the strange fixed point is z = 1.
(c) If t = 1, then ϕ(z; t) = z(1− z) and the strange fixed points are z = 0 and z = 1.
(d) Let φ = (t + z)(1 + tz) with t ̸∈ {−1, 0, 1}. Then φ(1/z) = z−2φ(z) holds for z ̸= 0.
Hence, if z ̸= 0 is a root of φ(z; t), then 1/z is also a root of φ(z; t).

Proof. After an accurate computation and careful algebraic treatments with the aid
of Mathematica, (a), (b) and (c) follow . For the proof of (d) follows from the fact that
φ(1/z) = (t+ 1/z)(1 + t/z) = ((t+ z)(1 + tz))/z2 = z−2φ(z).

Let z ̸∈ {0, 1} be a root of ϕ(z; t) for m = 1, 2. Suppose the numerator and denomina-
tor of ϕ(z; t) have no common factors for some suitable t-values. Then the roots of ϕ(z; t)
are explicitly found.
Differentiating J in (4), we require

J ′(z; t) =
r3t

m(−r1
−1+m

m r2t+ r3t
m + k1z + k2z

2 + r1r2k3z
3)

(r1r2z − r3tm(1 + z))2
, (8)

where k1 = −mr1r2
−1+m

m t+2r3t
m+r1

−1+m
m r2(−1+m(−1+2t)), k2 = r3t

m−r1
−1+m

m r2(2m(−1+

t) + t) and k3 = −(1 +m)r1
−1/m +mr2

− 1
m t.

Computing J ′(z; t) for m = 1 and m = 2, we have

J ′(z; t) =

{ −2tz((1+z)2)(1−3t+t2+(−1−t2)z+(1−3t+t2)z2)
(t+2tz−(1+(−3+t)t)z2)2

, if m=1
−t2z(1+z)4(a+bz+cz2+bz3+az4)

(−t2−4t2z+(2t−10t2+2t3)z2+(1−6t2+t4)z3+(2t−5t2+2t3)z4)2
, if m=2

(9)



H.J.Kim, A.K. Rathie, Y.H.Geum / Eur. J. Pure Appl. Math, 16 (3) (2023), 1902-1912 1906

where a = 4t−10t2+4t3, b = 3−8t+2t2−8t3+3t4, b = −4+16t−36t2+16t3−4t4 and
c = 3− 8t+ 2t2 − 8t3 + 3t4, to study the stability of the fixed points which are in Figures
1-2.

The critical points of the iterative scheme are given by the roots of the derivative of J ,
J ′(z, t) = 0. The points z = 0 and z = ∞ are critical points related with the roots a and
b of the quadratic polynomial (z − a)(z − b). When m = 1, the critical points are z = 0,
z = ∞ and z = ±1. When m = 2, 4 roots ξ can be found numerically for a given t.

3. Dynamical analysis and numerical results

This section describes the complex dynamics involved in the parameter space. The
following theorem is used to find useful properties of symmetry in the parameter space.

Theorem 3. Let z(t) be a free critical point of J(z; t) dependent upon parameter t. Then
that parameter space is symmetric about its horizontal axis. [10]

Theorem 4. Let z be a critical point. Then the following holds [10]:
(a) J ′(z; t) = J ′(1/z; t).
(b) If z ̸= 0 is a critical point, then so is 1/z.

When m = 2, the orbit behavior of two branches cp1(t) = ξ1 and cp2(t) = ξ2 of the free
critical points under the action of J(z; t). The orbit of two branches cp3(t) =

1
cp1(t)

and

cp4(t) =
1

cp2(t)
is similarly described.

Let P = {t ∈ C : a critical orbit of z under J(z; t) converges to a number νp ∈ C} be
the parameter space. If the number νp is a finite constant, there is finite periods in the
orbit. Otherwise, the orbit is not periodic however bounded or goes to infinity.

Table 1: Coloring scheme for a q-periodic orbit with q ∈ N ∪ {0}

q Cq

q = 1 C1 =


magenta, for fixed point ∞
cyan, for fixed point 0

yellow, for fixed point 1

red, for other strange fixed point ,

2 ≤ q ≤ 68 C2 = orange, C3 = light green, C4 = dark red, C5 = dark blue, C6 = dark green, C7 = dark yellow,

C8 = floral white, C9 = light pink, C10 = khaki, C11 = dark orange, C12 = turquoise, C13 = lavender,

C14 = thistle, C15 = plum, C16 = orchid, C17 = medium orchid, C18 = blue violet, C19 = dark orchid,

C20 = purple, C21 = power blue, C22 = sky blue, C23 = deep sky blue, C24 = dodger blue, C25 = royal blue,

C26 = medium spring green, C27 = spring green, C28 = medium sea green, C29 = sea green, C30 = forest green,

C31 = olive drab, C32 = bisque, C33 = moccasin, C34 = light salmon, C35 = salmon, C36 = light coral,

C37 = Indian red, C38 = brown, C39 = fire brick, C40 = peach puff, C41 = wheat, C42 = sandy brown,

C43 = tomato, C44 = orange red, C45 = chocolate, C46 = pink, C47 = pale violet red, C48 = deep pink,

C49 = violet red, C50 = gainsboro, C51 = light gray, C52 = dark gray, C53 = gray, C54 = charteruse,

C55 = electric indigo, C56 = electric lime, C57 = lime, C58 = silver, C59 = teal, C60 = pale turquoise,

C61 = sandy brown, C62 = honeydew, C63 = misty rose, C64 = lemon chiffon, C65 = lavender blush,

C66 = gold, C67 = crimson, C68 = tan.

q = 0∗ or q > 69 Cq = black.

∗: q = 0 implies that the orbit is non-periodic but bounded.



H.J.Kim, A.K. Rathie, Y.H.Geum / Eur. J. Pure Appl. Math, 16 (3) (2023), 1902-1912 1907

-500 -400 -300 -200 -100 0 100 200 300 400 500

-500

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0

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5. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.

-1.

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(a) |ℜ(t)| ≤ 500, |ℑ(t)| ≤ 500 (b) 5 ≤ ℜ(t) ≤ 7, |ℑ(t)| ≤ 1

5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7. 7.1 7.2 7.3

-2.

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1.1

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1.6

1.7

1.8

1.9

0.175

-

(c) |ℜ(t)| ≤ 2, |ℑ(t)| ≤ 5 (d) −3.7 ≤ |ℜ(t)| ≤ 3, |ℑ(t)| ≤ 5

Figure 3: Parameter spaces associated with free critical points cp1 for m = 1 .

-3.-2.6-2.2-1.8-1.4-1.-0.6-0.20.20.61.1.41.82.22.63.3.43.84.24.65.5.45.86.26.67.7.47.88.28.69.9.49.810.210.611.11.411.812.212.613.13.413.814.214.615.

-9.
-8.8
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3.2
3.4
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4.2
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5.2
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6.2
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7.2
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8.2
8.4
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6.8 6.9 7. 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8. 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.

-4.

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3.8

0.065 0.165 0.265

-0.3

-0.2

-0.1

5.55112×10-17

0.1

0.2

(a) −3 ≤ ℜ(t) ≤ 15, |ℑ(t)| ≤ 10 (b) 6.8 ≤ ℜ(t) ≤ 9, |ℑ(t)| ≤ 4 (c) 0.065 ≤ ℜ(t) ≤ 0.35, |ℑ(t)| ≤ 0.3

Figure 4: Parameter spaces associated with free critical points cp2 for m = 1 .

We describe a systematic method coloring a point t ∈ P depending on the period of
the orbit of z under J(z; t) for t ∈ P. Then the point t is drawn in corresponding color
Ck if t induces a k-periodic orbit with k ∈ N∪{0} under J(z; t). We accept the desired k-
periodic convergence of an orbit associated with P after a maximum of 1000-2000 iteration
number[17] and with a tolerance of 10−6. We use color Cq according to the color palette
shown in Table 1.

In Figures 3–6, we have shown the parameter spaces P related with cpj(t), (1 ≤ j ≤ 2).
A point t ∈ P is painted according to the coloring scheme shown in Table 1. In terms
of numerical phenomena, every point of the parameter space P whose color is none of
cyan(root z = a), magenta(root z = b), yellow or red is not a better choice of t. Let Pi



H.J.Kim, A.K. Rathie, Y.H.Geum / Eur. J. Pure Appl. Math, 16 (3) (2023), 1902-1912 1908

-2000. -1800. -1600. -1400. -1200. -1000. -800. -600. -400. -200. 0. 200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000.

-1000.

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0.

200.

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600.

800.

-11. -10. -9. -8. -7. -6. -5. -4. -3. -2. -1. 0. 1. 2. 3. 4. 5. 6. 7.

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0.

1.

2.

3.

4.

5.

6.

7.

8.

9.

(a) |Re(t)| ≤ 2000, |ℑ(t)| ≤ 1000 (b) −11 ≤ ℜ(t) ≤ 7, |ℑ(t)| ≤ 10

3.43 3.4313.4323.4333.4343.4353.4363.4373.4383.439 3.44 3.4413.4423.4433.4443.4453.4463.4473.4483.449 3.45 3.4513.4523.4533.4543.4553.4563.4573.4583.459 3.46

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0.285 0.286 0.287 0.288 0.289 0.29 0.291 0.292 0.293 0.294 0.295

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0.

0.001

0.002

0.003

0.004

(c) 0.1 ≤ ℜ(t) ≤ 0.3, |ℑ(t)| ≤ 0.1 (d) 0.285 ≤ ℜ(t) ≤ 0.295, |ℑ(t)| ≤ 0.005

Figure 5: Parameter spaces associated with free critical points cp1 for m = 2 .

denote the parameter space related with branch cpi for 1 ≤ i ≤ 4. We find the complicated
but beautiful pattern with the behavior that from n( ̸= 1) ∈ N -periodic orbit is budding
at period-1 component and 6-periodic component is budding at period-3 component.

Based on the theoretical result, we compare the proposed scheme (1) with the Dong’s
third-order method[6] as follows:

xn+1 = zn −
(
1− 1√

m

)1−m f(zn)

f ′(xn)
, zn = xn −

√
m

f(xn)

f ′(xn)
.

To plot the complex dynamics of the proposed method (g1) and Dong’s scheme (d1)
with the basins of attraction, we take the test functions having multiple roots with mul-
tiplicity m = 4, 6. In this statistical data for the basin of attraction, abbreviations cpu,
tcon, avg and tdiv denote the value of CPU time for convergence, the value of total con-
vergent points, the value of average iteration number for convergence and the value of
divergent points. As the first example, we select the polynomial p1(z) = (z3 − z)4 with
roots z = 0,±1 of multiplicity m = 4. The method g1 is better in view of cpu and avg. As
the next instance, the polynomial p2(z) = (z2 − 3z +5)6 has the roots z = 1.5± 1.65831i.
The method g1 is better in view of avg and d1 is better in view of tcon. As can be seen
in Figure 7, the picture (c) has shown some black point. The results are listed in Table 2
and Figure 7 .



H.J.Kim, A.K. Rathie, Y.H.Geum / Eur. J. Pure Appl. Math, 16 (3) (2023), 1902-1912 1909

0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27

-0.06

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0.

0.01

0.02

0.03

0.04

0.05

-2. -1.8 -1.6 -1.4 -1.2 -1. -0.8 -0.6 -0.4 -0.2 0.

-1.

-0.8

-0.6

-0.4

-0.2

0.

0.2

0.4

0.6

(a) 0.17 ≤ ℜ(t) ≤ 0.27, |ℑ(t)| ≤ 0.06 (b) −2 ≤ ℜ(t) ≤ 0, |ℑ(t)| ≤ 1.0

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3

-0.1

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0.09

3.74 3.75 3.76 3.77 3.78 3.79 3.8 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89

-0.08

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0.

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0.04

0.05

0.06

0.07

(c) 0.1 ≤ ℜ(t) ≤ 0.3, |ℑ(t)| ≤ 0.1 (d) 3.74 ≤ ℜ(t) ≤ 3.89, |ℑ(t)| ≤ 0.08

Figure 6: Parameter spaces associated with free critical points cp2 for m = 2 .

4. Conclusion

Given the multiplicity m, the complex dynamics were described by means of an Möbius
conjugacy map applied to a polynomial of the form f(z) = (z − A)m(z − B)m with the
stability analysis of strange fixed points.

Futures studies deal with the visualization of different types of numerical methods
accurately by improving the current research. In addition, we will investigate the pa-
rameter space and the basins of attraction of the developed multiple-root finder in detail.
We will observe the beautiful fractal that occurs in numerical methods from a variety of
perspectives.

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of

this article.

Acknowledgements

We would like to express our sincere gratitude to anonymous referees for their valuable
suggestions and comments improving the quality of the current paper.



REFERENCES 1910

Table 2: Typical Example

pm Method cpu tcon avg tdiv

p1 g1 54.406 360,000 5.83743 0
d1 58.407 360,000 6.32902 0

p2 g1 64.312 360,000 5.63653 1156
d1 75.516 360,000 9.84352 0

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

(a) g1,m = 4 (b) d1,m4

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

(c) g1,m = 6 (d) d1,m = 6

Figure 7: Basin of attraction

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