EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 710-720 ISSN 1307-5543 – ejpam.com Published by New York Business Global On a sixth-order solver for multiple roots of nonlinear equations Young Hee Geum Department of Mathematics, Dankook University, Cheonan, Korea 330-714 Abstract. A number of iterative schemes with high convergence order to solve nonlinear equations are presented in the literature. In this paper, a sixth-order multiple-zero finder has been developed and the dynamics of selected iterative schemes with uniparametric polynomial weight function are investigated using Möbius conjugacy map applied to the form ((z − A)(z − B))m. The complex dynamics on the Riemann sphere by analyzing the parameter spaces associated with the free critical points are studied, and the numerical experiments are carried out. 2020 Mathematics Subject Classifications: 65H05, 65H99 Key Words and Phrases: Conjugacy map, multiple root, nonlinear equation, parameter space 1. Introduction The root-finding problems[1, 2] occur in various fields of artificial intelligence, science, biology and engineering. Finding the zeros of nonlinear equation [3–5, 16] is the process of finding the value of a variable that satisfies a given equation. A zero α of h(x) = 0 is called a multiple root [14] with multiplicity m if h(i)(α) = 0, i = 0, 1, 2, · · · ,m − 1 and h(m)(α) ̸= 0. It is known that the Newton’s method is the most fundamental method for solving the equations, given by xn+1 = xn −m f(xn) f ′(xn) , n = 0, 1, 2, · · · . (1) Researchers [2, 7–9] are interested in solving roots of nonlinear equations[10, 13, 15, 17, 18] and investigating the dynamical analysis by exploring the relevant parameter plane and basins of attraction [12, 18, 19]. We study the dynamics of a class of sixth-order multiple-zero solvers developed by Geum-Kim-Neta[11] below. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5029 Email address: conpana@empal.com (Geum, Y.H.) https://www.ejpam.com 710 © 2024 EJPAM All rights reserved. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 711  yn = xn −m · h(xn), h(xn) = f(xn) f ′(xn) , wn = xn −m ·Gf (s1) · h(xn), s1 = ( f(yn)f(xn) ) 1 m , xn+1 = xn −m ·Kf (s1, s2) · h(xn), s2 = (f(wn) f(xn) ) 1 m , (2) where Gf : C → C is a analytic function in a small neighborhood of the origin 0 and Kf : C2 → C is holomorphic in a small neighborhood of (0, 0).{ Gf (s1) = 1+(a1−a2−1)s1+a1s12 1+(a1−a2−2)s1+a2s12 , Kf (s1, s2) = 1+(a1−a2−1)s1+a1s12 1+(a1−a2−2)s1+a2s12+[(1−a1+a2)s1−1]s2 , (3) where a1 and a2 are free parameter to be chosen. For brevity of analysis, we select a1 = 1 and consider one parameter a2 = κ(∈ C) here. The numerical method in (2) is written as xn+1 = If (xn, a1, a2), where If (xn, a1, a2) = xn −m ·Kf (s1, s2) · h(xn) is a fixed point operator. The process of solving the nonlinear equation of f(z) = 0 is regarded as a sequence of images of initial value x0 under If below: {x0, If (x0), I2f (x0), · · · Inf (x0), · · · } We investigate the conjugacy map and stability surfaces of the selected iterative scheme in Section 2. The algorithm, related theorems and the parameter planes are shown in Section 3. Finally, conclusions are stated in the last section. 2. Conjugacy map and analysis Via Möbius conjugacy map [6] T (z) = (z −A)/(z −B) when applied to a polynomial f(z) = ((z −A)(z −B))m, If is conjugated to J(z, κ) satisfying J(z,A,B, κ) = H(z,A,B, κ) D(z,A,B, κ) , (4) where z,A,B ∈ C ⋃ {∞}, A ̸= B and H and D are polynomials whose coefficients are dependent upon parameters A,B and κ. Using Mathematica [20] computation with T−1(z) = (Bz − A)/(z − 1), we obtain J(z, κ) as follows J(z, κ) = − z6(3 + 3z + z2 − κ− zκ)H1(z) (−1− 3z − 3z2 + zκ+ z2κ)D1(z) , (5) where H1(z) = (2+7z+8z2+4z3+z4−2zκ−2z2κ−z3κ) and D1(z) = (1+4z+8z2+ 7z3+2z4−zκ−2z2κ−2z3κ). We find out that J is dependent only on κ but independent of parameter A and B. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 712 We figure out that the fixed points of the iterative scheme J(z, κ). Let ϕ(z, κ) = z−J(z, κ) where roots are the desired fixed points of J(z, κ). After a lengthy computation, we find that z = 0 and z = 1 are the zeros of ϕ(z, κ) and we have the following expression of ϕ(z, κ): ϕ(z, κ) = (−1 + z)z(1 + z + z2)2W (z) w(z) , (6) W (z) =1 + z6 + z2(−4 + κ)2 + z4(−4 + κ)2 − 2z(−3 + κ)− 2z5(−3 + κ) + z3(22− 13κ+ 2κ2), w(z) =− 1 + 2z6(−3 + κ) + z(−7 + 2κ) + z4(−47 + 27κ− 4κ2) + z3(−43 + 23κ− 3κ2) + z5(−27 + 15κ− 2κ2)− z2(23− 10κ+ κ2). Theorem 1. (1) If κ = 0, then ϕ(z, κ) = −(−1 + z)z(1 + z)(1 + z + z2)2(1 + 4z + 7z2 + 4z3 + z4) 1 + 6z + 17z2 + 26z3 + 21z4 + 6z5 , and the strange fixed points z are given by z = ±1, z = −0.5± 0.866025i, z = −1.62481± 1.30024i, z = −0.375189± 0.300243i. (2) If κ = 2, then ϕ(z, κ) = −z(1 + z + z2)2(−1 + z − z2 + z3) 1 + z + 2z2 , and the strange fixed points z are given by z = ±i, z = 1, z = −0.5± 0.866025i. (3) If κ = 7 2 , then ϕ(z, κ) = z(1 + z + z2)2(4− 4z + z2 + 4z3 + z4 − 4z5 + 4z6) 4 + 4z + 5z2 + 2z3 + 8z4 + 4z5 , and the strange fixed points z are given by z = −0.5± 0.866025i, z = 0.75± 0.661438i, z = −0.84307± 0.537803i, z = 0.59307± 0.805151i. (4) If κ = 22 5 , then ϕ(z, κ) = z(1 + z + z2)2(25− 70z + 4z2 + 88z3 + 4z4 − 70z5 + 25z6) 25− 20z − 61z2 − 64z3 + 77z4 + 70z5 , and the strange fixed points z are given by z = −0.5±0.866025i, z = 0.536675, z = 1.86, z = 0.67, z = 1.48, z = −0.874796± 0.475352i. (5) Let κ /∈ {0, 2, 72 , 22 5 }. Then W (1z ) = z−6W (z). Proof. (1)-(4) Suppose W (z) = 0 and w(z) = 0 for z. Eliminating κ from the two polynomials, we have the relation F (z) = (z + 1)(z2 + z + 1) = 0. Substituting all the roots of F into W (z) = 0 and w(z) = 0, we get the relations for κ and solving them for κ, we have κ = 0, 2. The remaining part is straightforward. If (z − 1) is a divisor of w(z), Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 713 then w(1) = −154 + 79κ− 10κ2 = 0, yielding κ = 7 2 , 22 5 . Then remaining proof is trivial. (5) By direct computation, we have W (1/z) = z−6w(z). □ (a) |ℜ(t)| ≤ 5000, |ℑ(t)| ≤ 5000 (b) |ℜ(t)| ≤ 5, |ℑ(t)| ≤ 5 (c) 1 ≤ ℜ(t) ≤ 3, − 3 ≤ ℑ(t) ≤ −1 (d) |ℜ(t)| ≤ 5000, |ℑ(t)| ≤ 50001 (e) |ℜ(t)| ≤ 5000, |ℑ(t)| ≤ 5000 (f) 0 ≤ ℜ(t) ≤ 500, − 500 ≤ ℑ(t) ≤ 0 Figure 1: Stability surfaces . To find the stability of fixed points, we compute the derivative of J(z, κ) as follows: J ′(z, κ) = z5 ·Q(z) q(z)2 , (7) where Q(z) =a0 + a2z + a3z 2 + a4z 3 + 2a5z 4 + a7z 5 + 2a6z 6 + a7z 7 + 2a5z 8 + a4z 9 + a3z 10 + a2z 11 + a0z 12, q(z) =(−1 + 2z4 + z3(7− 2κ)− z(−4 + κ)− 2z2(−4 + κ)(−1 + z(−3 + κ) + z2(−3 + κ)), a0 =− 12(−3 + κ), a2 = 399− 235κ+ 34κ2, a3 = 2062− 1594κ+ 400κ2 − 32κ3, Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 714 a4 =6569− 5935κ+ 1908κ2 − 249κ3 + 10κ4, a5 = 7166− 7099κ+ 2602κ2 − 415κ3 + 24κ4, a6 =13061− 13800κ+ 5508κ2 − 986κ3 + 67κ4, a7 =22523− 23435κ+ 9170κ2 − 1600κ3 + 105κ4. Theorem 2. (1) If κ = −1, then J ′(z, κ) = z5(2 + z)2Q1(z) (1 + 2z)4(1 + 3z + 4z2 + z3)2 , where Q1(z) = 12 + 107z + 388z2 + 785z3 + 980z4 + 785z5 + 388z6 + 107z7 + 12z8. (2) If κ = 0, then J ′(z, κ) = z5Q2(z) (1 + 3z + 3z2)2(1 + 3z + 5z2 + 2z3)2 , where Q2(z) = 36+327z+1372z2+3498z3+5964z4+7097z5+5964z6+3498z7+1372z8+ 327z9 + 36z10. (3) If κ = 2, then J ′(z, κ) = z5(12 + 17z + 30z2 + 17z3 + 12z4) (1 + z + 2z2)2 . (4) If κ = 7 2 , then J ′(z, κ) = −z5Q4(z) (2 + z)2(2 + z + 2z2 + 4z4)2 , where Q4(z) = (96 + 304z + 336z2 + 460z3 + 508z4 + 723z5 + 508z6 + 460z7 + 336z8 + 304z9 + 96z10). (5) If κ = 22 5 , then J ′(z, κ) = z5Q5(z) (−5 + 7z + 7z2)2(−5− 3z + z2 + 10z3)2 , where Q5(z) = 10500 + 6475z − 39120z2 − 41690z3 + 19252z4 + 79689z5 + 19252z6 − 41690z7 − 39120z8 + 6475z9 + 10500z10. (6) Let κ /∈ {−1, 0, 2, 72 , 22 5 }. Then Q(1z ) = z12Q(z). Proof. (1)-(5) Suppose that Q(z) = 0, q(z) = 0 for z. By eliminating κ from Q(z) = 0, and q(z) = 0, we get the relation: T (z) = (z − 1)(z + 1)(1 + 2z)(1 + z + z2)(−1 − 4z − Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 715 6z2 − 2z3 + 4z4 + 8z5 + 4z6) = 0. Substituting all the roots of T (z) into Q(z) = 0 and q(z) = 0, we find κ = −1, 0, 2, 72 , 22 5 . □ The stability surfaces are shown by illustrative conical surfaces in Figure 1. 3. Experiment According to Algorithm 1, the numerical parameter spaces are constructed in Figure 2. The systematic color palette in Table 1 is utilized to paint a value according to the orbital period of the point z of J(z, κ). The tolerance of 10−4 after up to 1000 iterations is assigned [? ]. Algorithm 1 (1) Set i = 1 (2) Choose a region B ∈ C and select a point v = (Re(v), Im(v)) in B (3) For the v, find the free critical point. (4) Compute the orbit of J(z, t) within the maximal iterative number. (5) If the orbit converges to one cycle within the given error, then color the point v ac- cording to the color palette in Table 1. (6) Choose the next value in B (7) Repeat steps (2)-(6) until desired result is obtained. (8) Set i = i+ 1 and if i ≤ w, then repeat steps (2)-(8) (9) If i = w, then stop the process. Table 1: Color palette for a n-periodic orbit with n ∈ N ∪ {0} n Cn n = 1 C1 =  magenta, for fixed point ∞ cyan, for fixed point 0 yellow, for fixed point 1 red, for other strange fixed point , 2 ≤ n ≤ 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. n = 0∗ or n > 69 Cn = black. ∗: n = 0 : the orbit is non-periodic but bounded. Theorem 3. Let ψ : C2 → C be defined by ψ(κ, z) = l0(z) + l1(z)κ + l2(z)κ 2, where li(z)(0 ≤ i ≤ 2) are complex polynomials with real coefficients. Suppose ψ(κ, z) = 0. Let z̄ Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 716 be a complex conjugate of z. (1) κ(z̄) = κ(z). (2) If z(κ) is a zero of ψ, then so is z̄(κ̄). Proof. (1) Solving ψ(κ, z) = 0 for κ, we have κ(z) = −l1(z)± √ l1(z)2 − 4l1(z)l2(z) 2l1(z) . (8) Since li(z), (0 ≤ i ≤ 2) are complex polynomials with real coefficients, we have li(z̄) = li(z). From 8, we get κ(z̄) = −l1(z̄)± √ l1(z̄)2 − 4l1(z̄)l2(z̄) 2l1(z̄) = −l1(z)± √ l1(z) 2 − 4l1(z)l2(z) 2l1(z) , ψ(κ, z) = l0(z) + l1(z)κ̄+ l2(z)z̄ 2 = 0. Therefore κ(z) = κ(z̄). (2) Let z(κ) be a zero of ψ(κ, z). Then ψ(κ, z) =0 = ψ(κ, z) =l0(z) + l1(z)κ+ l2(z)κ2 =l1(z) + l1(z)κ̄+ l2(z)κ̄ 2 =l0(z̄) + l1(z̄)κ̄+ l2(z̄)κ̄ 2 =ψ(κ̄, z̄), implying z̄(κ̄) is a zero of ψ(κ, z). □ Let P = {η ∈ C : a critical orbit of z converges to a number wp ∈ C}. It is called the parameter space. There are finite periods in the orbit if the number wp is a finite constant. Otherwise, the orbit is not periodic however it is bounded or goes to infinity. Theorem 4. Let z(κ) be a free critical point of J(z, κ). Then the parameter space is symmetric with respect to its horizontal axis. Proof. Let z(κ) is a root of ψ(κ, z). Then z̄(κ̄) is a root of Q(z) at κ̄. From conjugated map J(z, κ), we obtain |J(z, κ)| =|J(z(κ), κ)| = |J(z(κ), κ)| =|J(z(κ), κ̄)| = |J(z̄(κ̄), κ̄)|. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 717 Then the parameter space with J(z, κ) is symmetric with respect to its horizontal axis. □ The parameter spaces P are illustrated in Figures 2. A point ϵ ∈ P is painted using the color palette 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), red or yellow is not a better choice of t. The complicated patterns are found and for n ∈ N−{1}, n-periodic orbit is budding at period-1 component and 4-periodic component is budding at period-2 component. 4. Conclusion We have investigated the complex dynamical analysis on the Riemann sphere by draw- ing the parameter spaces associated with the free critical points for the uniparametric family of sixth-order multiple-root finders. Such research from a viewpoint of complex dynamics may limit us from treating the real dynamical phenomenon for real nonlinear equations. However, this research for investigating the relevant complex dynamics lies is finding the dynamical behavior of a family of iterative schemes via Möbius conjugacy map by showing the parameter spaces. As the next work, we visualize different types of higher order numerical methods by improving the current work. In addition, we draw the convergent region and the basins of attraction of the developed multiple-root finder in more detail. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 710-720 718 300. 320. 340. 360. 380. 400. 420. 440. 460. 480. 500. 520. 540. 560. 580. 600. 620. 640. -300. -280. -260. -240. -220. -200. -180. -160. -140. -120. -100. -80. -60. -40. -20. 0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200. 220. 240. (a) 2.6 ≤ ℜ(t) ≤ 4.6, |ℑ(t)| ≤ 1 3.7 3.701 3.702 3.703 3.704 3.705 3.706 3.707 3.708 3.709 3.71 3.711 3.712 3.713 3.714 3.715 3.716 3.717 3.718 3.719 3.72 3.721 3.722 3.723 3.724 3.725 3.726 3.727 3.728 3.729 3.73 -0.015 -0.014 -0.013 -0.012 -0.011 -0.01 -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0. 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 (a) 2.6 ≤ ℜ(t) ≤ 4.6, |ℑ(t)| ≤ 1 -15750 -15700 -15650 -15600 -15550 -15500 -15450 -15400 -15350 -15300 -15250 -15200 -15150 -15100 -15050 -15000 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 (a) 2.6 ≤ ℜ(t) ≤ 4.6, |ℑ(t)| ≤ 1 Figure 2: Parameter spaces . REFERENCES 719 Acknowledgements The author is supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant(Project Number:NRF-2021R1A2C1012922 ) Conflicts of Interest : The author declare no conflict of interest. References [1] L.V. Ahlfors. Complex Analysis. McGraw-Hill Book Inc., 1979. [2] S. Amat, S. Busquier, and S. Plaza. Review of some iterative root-finding methods from a dynmaical point of view. Scientia, 10:3–35, 2004. [3] A.F. Beardon. Iteration of Rational Funtions. Springer-Verlag, New York, 1991. [4] R. Behl, A. Cordero, S. Mosta, and J. Torregrosa. On developing fourth-order optimal families of methods for mulitiple roots and their dynamics. Applied Mathematics and Computation, 265:520–532, 2015. [5] R. Behl, A. Cordero, S.S. Motsa, and J. R. Torregrosa. On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathe- matics and Computation, 265:520–532, 2015. [6] P. Blanchard. The dynamics of newton’s mehtod. Pro. Symposia in Appl. Math., 49:139–154, 1994. [7] F. Chicharro, A. Cordero, J. Gutiérrez, and J. Torregrosa. Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computa- tion., 219:7023–7035, 2013. [8] C. Chun and B. Neta. Basins of attraction for zhou-chen-song fourth order family of methods for multiple root. Math. Comput. Simulat., 109:74–91, 2015. [9] A. Cordeor, J. Garćıa-Maimó, J.R. Torregrosa, M.P. Vassileva, and P. Vindel. Chaos in king’s iterative family. Appl. Math. Lett., 119:842–848, 2016. [10] R.L. Devaney. Complex dynamical systems: the mathematics behind the mandelbrot and julia sets. Pro. Symposia in Appl. Math., 49:1–29, 1994. [11] Y.H. Geum, Y.I. Kim, and B. Neta. A sixth-order family of three-point modified newton-like multiple-root finders and thier dynamics behind their extraneous fixed points. Appl. Math. Comput., 283:120–140, 2016. [12] D. Gulick. Encounters with Chaos. McGraw-Hill Inc., 192. REFERENCES 720 [13] L. Hörmander. Introduction to complex analysis in several variables. North-Holland Publishing Company, 1973. [14] H.T. Kung and J.F. Traub. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach., 21:643–651, 1974. [15] S. Lipschutz. Theory and Problems of General Topology, Schaum’s Outline Seires. McGraw-Hill Inc., 1965. [16] Á.A. Magre nán, A. Cordeor, J.M. Gutiérrez, and J.R. Torregrosa. Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane. Mathematics and Computers in Simulation, 105:49–61, 2014. [17] H. Peitgen and P. Richter. The Beauty of Fractals. Springer-Verlag, 1986. [18] B.V. Shabat. Introduction to Complex Analysis PART 2, Functions of several Vari- ables. American Math. society, 1992. [19] J.F. Traub. Iterative Methods for the solution of Equations. Chelsea Publishing Company, 1982. [20] S. Wolfram. The Mathematica Book 5th ed. Wolfram Media, 2003.