EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1029-1037 ISSN 1307-5543 – ejpam.com Published by New York Business Global On a parameter-controlled method for multiple zeros Young Hee Geum Department of Mathematics, Dankook University, Cheonan, Korea 330-714 Abstract. For a parameter-controlled Newton-secant method, we investigate the complex dynam- ics with the stability surfaces and parameter spaces using Möbious conjugacy map applied to the form ((z − A)(z − B))m. The basins of attraction of test functions is illustrated according to the color palette. The nonlinear equation related to blood rheology model has been employed to show the basins of attraction. 2020 Mathematics Subject Classifications: 65H05, 65H99 Key Words and Phrases: Conjugacy map, multiple root, nonlinear equation, parameter space, Blood Rheology model 1. Introduction The root-finding problem[10, 11] in diverse field of science, engineering and artificial intelligence is an important task to handle. Researchers are developing new iterative methods to solve the nonlinear equations. With the advent of high-accuracy computer, the numerical methods are improved efficiently and accurately in [3, 7, 8, 12, 14]. A root α of f(x) = 0 is called a multiple root with multiplicitym if f (i)(α) = 0, i = 0, 1, 2, · · · ,m−1 and g(m)(α) ̸= 0. Let f : C → C be an analytic function[1, 13] with an integer multiplicity m ≥ 1. Geum et al. [5] investigated the third order of parameter-controlled Newton-secant scheme given by xn+1 = xn − m(1− tm)f(xn) 2 f ′(xn){f(xn)− f(yn)} , (1) where yn = xn −m(1− t) f(xn)f ′(xn) . The iteration method (1) is expressed as a discrete formula xn+1 = Of (xn), (2) where Of is the iterative method. We have a complex discrete dynamical system[6]. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5010 Email address: conpana@empal.com (Y. H. Geum) https://www.ejpam.com 1029 © 2023 EJPAM All rights reserved. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 1029-1037 1030 zn+1 = Of (zn) = zn − λ f(zn) 2 f ′(zn){f(zn)− f(yn)} , (3) where yn = zn − µ f(zn) f ′(zn) , µ = m(1− t) and λ = m(1− tm). 2. Analysis Applying Möbius conjugacy map M(z) to f(z) = ((z −A)(z −B))m, we have J(z, t) =M ◦Of ◦M−1(z) = z((1 + z)r1r2 − r3δ1) (1 + z)r1r2 − r3δ2 , (4) where M(z) = (z −A)/(z −B), A, B ∈ C ∪ {∞}, A ̸= B, r1 = (t + z)m, r2 = (1 + tz)m, r3 = (1 + z)2m, δ1 = (tm + z) and δ2 = (1 + tmz). Then Of (z, t) is conjugate [2] to J(z, t). In case of m = 1, 2, we have J(z, t) = { z2(t+z) 1+tz , m=1, z(1+t3z+z(1+z)(2+z)+t2(2+z+2z2)+t(−1+z(2+z+z2))) 1+t+(3+t+2t2)z+(1+t)(2+t2)z2+(1+t(−1+2t))z3 , m=2. (5) We locate the fixed points of the iterative method J(z, t). Let J(z, t)−z whose roots are the desired fixed point of J(z, t). z = 0 and z = 1 are the zeros of J(z, t)− z. Since M(z) is a fixed point of J(z, t) for a fixed point z of Of , the explicit form of ϕ(z, t) = J(z, t)− z is given by ϕ(z, t) = (−1 + z)zr3δ3 (1 + z)r1r2 − r3δ2 , (6) and ϕ(z, t) = { z(z−1)ψ1(t) q1(t) , if m=1, z(z−1)ψ2(t) q2(t) , if m=2, (7) where ψ1(z) = z + 1, q1(z) = 1 + tz, ψ2(z) = (1 + t)(1 + z)3, q2(z) = 2(1 + z)4 + t2z(1 + z)4 − z(t+ z)2(1 + tz)2. By aid of Mathematica [15], we compute the derivative of J(z, t) : J ′(z, t) = T1r3 + T2r1 + (1 + z)2r1 2r2 2 ((1 + z)r1r2 − r3δ2)2 , (8) Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 1029-1037 1031 where T1 = (2z + tm(1 + z2))r3 − mz(t + z)−1+m(−1 + z2)r2δ3 T2 = (−mtz(1 + tz)−1+m(−1 + z2)r3δ3 + r2(−((1 + z(4 + z) + tm(1 + z2))r3) + 2m(−1 + z)z(1 + z)2mδ3)), For m = 1 and m = 2, we have J ′(z, t) = { z(3z+t2z+2t(1+z2)) (1+tz)2 , m=1 z((1+z6)δ0+z((1+z4)δ1+z(δ2+z2δ2+zδ3))) δ4 2 , m=2. (9) δ0 =2(1 + t2 + 2t3), δ1 = (9 + t(10 + t(16 + t(6 + 7t)))), δ2 =2(1 + t)(4 + t2)(3 + t+ 2t2), δ3 =(35 + t(38 + t(47 + t(24 + t(13 + t(2 + t)))))), δ4 =1 + t+ (3 + t+ 2t2)z + (1 + t)(2 + t2)z2 + (1 + t(−1 + 2t))z3. The critical point of the parameter-controlled method refers to a point where the derivative of a function is zero, that is J ′(z, t) = 0. The points z = ∞ and z = 0 are critical points associated with (z − A)(z − B). The critical points that are not any roots of the polynomial (z − A)(z − B) are called to be free critical points. As the result of following Algorithm 1, the stability surfaces are displayed in Figure 1 and 2. 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, solve qi(k) = 0 using Mathematica [15]. (4) Compute mi = |J ′(k, t)|. (5) Save (Re(v), Im(v),mi) and choose the next value in B (6) Repeat steps (2)-(5) until desired result is done. (7) Set i = i+ 1 and if i ≤ w, then repeat steps (2)-(6) (8) If i = w, then stop the process. 3. Experiment According to Algorithm 2, the numerical parameter spaces for m = 1 and m = 2 are constructed in Figure 3 and 4. 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, t). The tolerance of 10−4 after up to 1000 iterations is assigned. Algorithm 2 (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. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 1029-1037 1032 (a) |ℜ(t)| ≤ 50, |ℑ(t)| ≤ 50 (b) −3 ≤ ℜ(t) ≤ 1, |ℑ(t)| ≤ 1 (c) −7 ≤ ℜ(t) ≤ −2, − 3 ≤ ℑ(t) ≤ 3 (d) |ℜ(t)| ≤ 500, |ℑ(t)| ≤ 500 Figure 1: Stability surfaces for m = 1. (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. In Figure 5, the basins of attraction associated with m = 1, 2 are appeared. As the last example, we choose the equation in blood rheology model[4] z = ( x8 441 + 8x5 63 − 2857144357x4 50000000000 + 16x2 9 − 906122449x 250000000 + 3 10 )4 , to carry out the experiments. In Figure 6, the basins of attraction for this equation are shown. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 1029-1037 1033 (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 2: Stability surfaces for m = 2. 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. Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 1029-1037 1034 2.6 2.8 3. 3.2 3.4 3.6 3.8 4. 4.2 4.4 4.6 -1. -0.8 -0.6 -0.4 -0.2 0. 0.2 0.4 0.6 2.95 2.952 2.954 2.956 2.958 2.96 2.962 2.964 2.966 2.968 2.97 2.972 2.974 2.976 2.978 2.98 -0.015 -0.013 -0.011 -0.009 -0.007 -0.005 -0.003 -0.001 0.001 0.003 0.005 0.007 0.009 0.011 0.013 (a) 2.6 ≤ ℜ(t) ≤ 4.6, |ℑ(t)| ≤ 1 (b) 2.95 ≤ ℜ(t) ≤ 2.98, |ℑ(t)| ≤ 0.015 -1.5 -1.4 -1.3 -1.2 -1.1 -1. -0.9 -0.8 -0.4 -0.3 -0.2 -0.1 0. 0.1 0.2 4.2 4.22 4.24 4.26 4.28 4.3 4.32 4.34 4.36 4.38 4.4 -0.1 -0.08 -0.06 -0.04 -0.02 0. 0.02 0.04 0.06 0.08 (c) − 1.5 ≤ ℜ(t) ≤ −0.8, |ℑ(t)| ≤ 0.4 (d) 4.2 ≤ ℜ(t) ≤ 4.4, |ℑ(t)| ≤ 0.1 Figure 3: Parameter spaces for m = 1 . Y. H. Geum / Eur. J. Pure Appl. Math, 17 (2) (2024), 1029-1037 1035 0.9 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2.8 2.9 3. 3.1 3.2 3.3 -0.25 -0.15 -0.05 0.05 0.15 (a) 2 ≤ ℜ(t)| ≤ 3, |ℑ(t)| ≤ 0.9 (b) 2.28 ≤ ℜ(t) ≤ 2.3, |ℑ(t)| ≤ 0.25 2.42 2.52 2.62 2.72 -0.15 -0.05 0.05 0.15 -1.7-1.65-1.6-1.55-1.5-1.45-1.4-1.35-1.3-1.25-1.2-1.15-1.1-1.05-1.-0.95-0.9-0.85-0.8-0.75-0.7 -0.5 -0.48 -0.46 -0.44 -0.42 -0.4 -0.38 -0.36 -0.34 -0.32 -0.3 -0.28 -0.26 -0.24 -0.22 -0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0. 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 (c) −1.5 ≤ ℜ(t) ≤ 0.9, |ℑ(t)| ≤ 0.15 (d) 0.3 ≤ ℜ(t) ≤ 1.1, |ℑ(t)| ≤ 0.5 Figure 4: Parameter spaces for m = 2 . -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 (a) p1(z) = (z2 − 1)2 (b) p2(z) = (x3 + 1)3(z) (c) p3(z) = (x3 − 1)3 Figure 5: Basins of attraction of test functions REFERENCES 1036 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 Figure 6: Basins of attraction of Blood Rheology model 4. Conclusion We have studied the complex dynamics using Möbius conjugacy map applied to f(z) = ((z − A)(z − B))m with the known multiplicity m. To obtain the useful information on better initial values of any numerical method, we need to experiment the basins of attractions. As a next work, we focus on developing the higher order methods and illustrating the dynamical behavior to find out the chaotic geometry of their dynamics [6]. For test functions, Euler-Cauchy method is best in the sense that the boundaries of the basins have no lobes in Neta B. [9]. Frow this viewpoint, I will improve the efficient and accurate method for nonlinear equations. 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 Ahlfors. Complex Analysis. 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