EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2323-2347 ISSN 1307-5543 – ejpam.com Published by New York Business Global A general family of fifth-order iterative methods for solving nonlinear equations Ali Zein Department of Applied Mathematics, Palestine Polytechnic University, Hebron, Palestine Abstract. A family of fifth-order iterative methods is proposed for solving nonlinear equations using the weight function technique. This family offers flexibility through its structure and the choice of weight functions, resulting in a wide range of new specific schemes. It is demonstrated that this proposed family includes several well-known and recent methods as special cases. In addition, several new particular methods are designed to achieve better results than existing methods of the same type. Convergence analysis is conducted, and numerical examples in both real and complex domains are provided for several specific schemes within the proposed family. Comparisons between the existing methods within this family and the newly introduced methods generally indicate improved performance among the new members. Notably, the study of complex dynamics and basins of attraction reveals that our new specific schemes have broader sets of initial points that lead to convergence. 2020 Mathematics Subject Classifications: 65H05, 65B99 Key Words and Phrases: Fifth-order method, Weight function, Basins of attraction, Efficiency index, Nonlinear equations, Iterative methods 1. Introduction Solving nonlinear equations is a very important problem in various fields of science and technology [2]. It is mostly impossible to find exact or analytical solutions for these equations. Therefore, iterative methods are used to find approximate solutions for them. In the literature, numerous iterative methods have been developed for approximating a root of a nonlinear equation f(x) = 0. Newton’s method is the most popular and widely used iterative scheme defined by xn+1 = xn − f(xn) f ′(xn) , n = 0, 1, . . . , where x0 is the initial approximation to a root α. It is well known that this method is quadratically convergent to simple roots. A root α is simple if f(α) = 0 but f ′(α) ̸= 0. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4949 Email address: alizein@ppu.edu (A. Zein) https://www.ejpam.com 2323 © 2023 EJPAM All rights reserved. A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2324 In the last two decades, many higher order iterative methods have been derived and analyzed for solving nonlinear equations, see [1–4, 6–12, 14–23] and the references therein. In particular, many fifth-order iterative methods are found in the literature, see [6, 7, 9– 12, 14, 18, 20]. Commonly, the order of convergence and the efficiency index are important in rating iterative methods. The efficiency index E.I. is defined as E.I. = p 1 d , where p is the order of convergence and d is the total number of new function evaluations per iteration. According to the Kung and Traub conjecture [13] a multi-point method which uses d evaluations could achieve, at most, a convergence order p = 2d−1. Methods that reach this bound are known as optimal. According to the Kung -Traub conjecture, a fifth-order scheme requires a minimum of four function evaluations per iteration. To achieve fifth-order convergence, several authors have modified classical methods such as Newton’s method, Chebyshev’s method, and Chebyshev-Halley’s method [7, 11, 12, 18]. But these methods require the evaluation of the second derivative, which can be expensive. Recently, several fifth-order methods have been developed by using the weight function technique [10, 14, 20]. In this paper, a two-step family of fifth-order iterative methods is presented. Conver- gence analysis is established, and a general error equation is derived for the family. The methods in this family require four new function evaluations per iteration, which is the min- imum number of new function evaluations per iteration for fifth-order iterative methods, in the sense of Kung -Traub conjecture. Thus, the efficiency index is E.I. = 5 1 4 = 1.495. The presented family has the flexibility of choice in its formation and in selecting suitable weight functions, offering a large number of particular schemes. It is shown that several well-known and recent schemes are considered as special cases of the family. Namely, these methods include the Ham and Chun method [9], the Fang et al. method [6], the Sivakumar and Jayaraman method [20], the Liu et al. method [14], and the recent method of Khirallah and Alkhomsan [10]. Furthermore, the flexibility of choice within the presented family enables us to establish several specific schemes that exhibit better stability compared to existing methods of the same type. Indeed, four new particular schemes are deduced from the general family. Numerical results and the basins of attraction show better performance for the new schemes compared with the existing schemes mentioned above. This paper is organized as follows: Section 2 provides preliminaries. Section 3 presents the construction and convergence analysis of the new family of fifth-order methods. In Section 4, some special cases of the proposed family are established. Also, some well-known schemes are listed as particular cases of the proposed family. Section 5 is devoted to study the stability of particular methods by using the basins of attraction technique. In section 6, numerical results for particular methods in real domain are given and comparisons are made. 2. Preliminaries Definition 1. [5] Suppose {xn} is a sequence that converges to a limit α. If there exist A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2325 an integer constant p ≥ 1 and a non-zero constant C such that lim n→∞ xn+1 − α (xn − α)p = C, then p is called the order of convergence and C is called the asymptotic error constant. Let en = xn − α is the error in the nth iteration. The equation en+1 = Cep n + O(ep+1 n ) is called the error equation for the method, where p is the order of convergence, see [23]. Definition 2. [23] Let α be a root of f(x) = 0 and suppose that xn−1, xn, and xn+1 are three consecutive iterations closer to the root α. Then the computational order of convergence can be computed by using the formula p ≈ ln |(xn+1 − α)/(xn − α)| ln |(xn − α)/(xn−1 − α)| . 3. Construction of the new family and convergence analysis Assume that α is a simple root of f(x) = 0, where f : I ⊆ R → R is a sufficiently differentiable function. The new family of fifth-order iterative methods consists of two steps as follows: yn = xn − f(xn) f ′(xn) , (1) xn+1 = yn − G(ηn) f(yn) Af ′(xn) + Bf ′(yn) , (2) where A and B are parameters, and G is a weight function in terms of ηn = 1 − f ′(yn) f ′(xn) . (3) The first step of this family involves the classical Newton’s method. In the second step, the parameters A and B are chosen arbitrarily. Then, the weight function G is designed to achieve fifth-order convergence, as demonstrated in Theorem 1. Note that, the symbolic computations in all proofs in this paper are performed by using Maple software. Remark 1. In this paper, we remark that: (i) The notation yn given in (1) represents the Newton step. (ii) The conventional notation cj = f (j)(α) j!f ′(α) , j = 2, 3, ... is employed. A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2326 Theorem 1. Let α ∈ I be a simple root of a sufficiently differentiable function f : I ⊆ R → R for an open interval I. If x0 is sufficiently close to α, and the weight function G(η) satisfies G(0) = A + B, G′(0) = A, G′′(0) = 5A + B 2 , (4) then the iterative family (2) converges to α with order of convergence five and satisfies the error equation en+1 = (( 14A + 4B − 4 3G′′′(0) ) c4 2 A + B − c2 2c3 ) e5 n + O(e6 n). (5) Provided A + B ̸= 0 and |G′′′(0)| < ∞. Proof. Expanding f(xn) and f ′(xn) about α by using Taylor’s series, we get f(xn) = f ′(α)[en + c2e2 n + c3e3 n + c4e4 n + c5e5 n] + O(e6 n) (6) and f ′(xn) = f ′(α)[1 + 2c2en + 3c3e2 n + 4c4e3 n + 5c5e4 n + 6c6e5 n] + O(e6 n). (7) From (6) and (7), we get f(xn) f ′(xn) = en − c2e2 n + (2c2 2 − 2c3)e3 n + (−4c3 2 + 7c2c3 − 3c4)e4 n + (8c4 2 − 20c2 2c3 + 10c2c4 + 6c2 3 − 4c5)e5 n + O(e6 n). (8) Subtracting α from both sides of (1), then using en = xn − α and inserting (8), we have yn − α = en − f(xn) f ′(xn) = c2e2 n − (2c2 2 − 2c3)e3 n − (−4c3 2 + 7c2c3 − 3c4)e4 n − (8c4 2 − 20c2 2c3 + 10c2c4 + 6c2 3 − 4c5)e5 n + O(e6 n). (9) Expanding f(yn) and f ′(yn) about α and using (9) we obtain f(yn) = f ′(α)[c2e2 n + (−2c2 2 + 2c3)e3 n + (5c3 2 − 7c2c3 + 3c4)e4 n − 2(6c4 2 − 12c2 2c3 + 5c2c4 + 3c2 3 − 2c5)e5 n] + O(e6 n) (10) and f ′(yn) = f ′(α)[1 + 2c2 2e2 n − (4c3 2 − 4c2c3)e3 n + (8c4 2 − 11c2 2c3 + 6c2c4)e4 n − 2c2(8c4 2 − 14c2 2c3 + 10c2c4 − 4c5)e5 n] + O(e6 n). (11) From (7), (10), and (11), we have A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2327 f(yn) Af ′(xn) + Bf ′(yn) = c2e2 n A + B + [ (−4c2 2 + 2c3)A − 2(c2 2 − c3)B ] e3 n (A + B)2 + [ (13c3 2 − 14c2c3 + 3c4)A2 + (12c3 2 − 21c3c2 + 6c4)AB + (3c3 2 − 7c3c2 + 3c4)B2 ] e4 n (A + B)3 + [ (−38c4 2 + 64c2 2c3 − 20c2c4 − 12c2 3 + 4c5)A3 − (48c4 2 − 124c2 2c3 + 50c2c4 + 30c2 3 − 12c5)BA2 − (22c4 2 − 76c2 2c3 + 40c2c4 + 24c2 3 − 12c5)B2A − (4c4 2 − 16c2 2c3 + 10c2c4 + 6c2 3 − 4c5)B3 ] e5 n (A + B)4 + O(e6 n). (12) Using (7) and (11) we can write the weight function variable η given in (3) as ηn = 1 − f ′(yn) f ′(xn) = 2c2en + (−6c2 2 + 3c3)e2 n + (16c3 2 − 16c2c3 + 4c4)e3 n +(−40c4 2 + 61c2 2c3 − 22c2c4 − 9c2 3 + 5c5)e4 n +(96c5 2 − 198c3 2c3 + 88c2 2c4 + 66c2c2 3 − 28c2c5 − 24c3c4 + 6c6)e5 n + O(e6 n). (13) Then, expanding the weight function G(ηn) around zero, we get G(ηn) = G(0) + G′(0)ηn + G′′(0) 2 η2 n + G′′′(0) 6 η3 n + . . . = G(0) + 2G′(0)c2en + [ (−6c2 2 + 3c3)G′(0) + 2G′′(0)c2 2 ] e2 n + [ (48G′(0) − 36G′′(0) + 4G′′′(0))c3 2 3 − (16G′(0) − 6G′′(0))c2c3 + 4G′(0)c4 ] e3 n + · · · . (14) Because the least order in equation (12) is two, we expand G(ηn) to the third order. Finally, according to (9), (12) and (14), the error equation of the scheme (2) is en+1 = yn − α − G(ηn) f(yn) Af ′(xn) + Bf ′(yn) = [A + B − G(0)]c2 e2 n A + B + [ (−2c2 2 + 2c3)A2 + ( (−4c2 2 + 4c3)B + (4G(0) − 2G′(0))c2 2 − 2G(0)c3 ) A − 2 ( (c2 2 − c3)B + (−G(0) + G′(0))c2 2 + G(0)c3 ) B ] e3 n (A + B)2 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2328 + [( 4A3 + (12B − 13G(0) + 14G′(0) − 2G′′(0))A2 + (12B − 12G(0) + 24G′(0) − 4G′′(0))AB + (4B − 3G(0) + 10G′(0) − 2G′′(0))B2 ) c3 2 − 7(A + B)(A2 + (2B − 2G(0) + G′(0))A + B(B − G(0) + G′(0)))c2c3 + 3c4(A + B)2(A + B − G(0)) ] e4 n (A + B)3 + ξe5 n + O(e6 n), (15) where ξ = ξ (A, B, c2, ..., c5, G(0), G′(0), G′′(0), G′′′(0)) is the coefficient of e5 n. To obtain a fifth-order convergence, we need the coefficients of e2 n, e3 n and e4 n in (15) to be zero. It is clear that the first condition in (4) makes the coefficient of e2 n zero. To simplify the calculations, we substitute G(0) = A + B in (15), to get en+1 = 2c2 2[A − G′(0)] e3 n A + B − c2 [( 9A2 + ( 4B − 14G′(0) + 2G′′(0) ) A − (B + 10G′(0) − 2G′′(0))B ) c2 2 − (7c3(A − G′(0))(A + B)) ] e4 n (A + B)2 + γe5 n + O(e6 n), (16) where γ = γ(A, B, c2, ..., c5, G′(0), G′′(0), G′′′(0)). Obviously, the second condition in (4) eliminates the third-order error. Thus, applying G′(0) = A in (16) leads to en+1 = c3 2(5A + B − 2G′′(0)) e4 n A + B + µe5 n + O(e6 n), (17) where µ = µ(A, B, c2, ..., c5, G′′(0), G′′′(0)). By applying the third condition in (4), the fourth-order error vanishes in the error equation (17). That is, using G′′(0) = 5A + B 2 results in en+1 = (( 14A + 4B − 4 3G′′′(0) ) c4 2 A + B − c2 2c3 ) e5 n + O(e6 n). This ends the proof of the theorem. The case A+B = 0 is excluded from Theorem 1. In this case, without loss of generality, the family of methods (2) can be written as xn+1 = yn − G(ηn) f(yn) f ′(xn) − f ′(yn) . (18) For the scheme (18), we have the following convergence result. Theorem 2. Let α ∈ I be a simple root of a sufficiently differentiable function f : I ⊆ R → R for an open interval I. If x0 is sufficiently close to α, and the weight function G(η) satisfies G(0) = 0, G′(0) = 1, G′′(0) = 2, G′′′(0) = 15 2 , (19) A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2329 then the iterative scheme (18) converges to α with order of convergence five and satisfies the error equation en+1 = (1 3 ( 42 − G(4)(0) ) c4 2 − c2 2c3 ) e5 n + O(e6 n). (20) Provided |G(4)(0)| < ∞. Proof. Following the proof of Theorem 1, according to (7), (10) and (11), we have f(yn) f ′(xn) − f ′(yn) = en 2 + −2c2 2 + c3 4c2 e2 n + · · · . (21) Since the least order in (21) is one, we expand G(ηn) to the fourth order, i.e. G(ηn) = G(0) + G′(0)ηn + G′′(0)η2 n 2 + G′′′(0)η3 n 6 + G(4)(0)η4 n 24 + · · · . (22) Now, combining (9), (13), (21) and (22), we get the error equation for the scheme (18) en+1 = yn − α − G(ηn) f(yn) f ′(xn) − f ′(yn) = −G(0)en 2 + [ (2G(0) − 4G′(0) + 4)c2 2 − G(0)c3 ] e2 n 4c2 + · · · . (23) To make the coefficients of en and e2 n zero, we obtain the first two conditions in (19), i.e. G(0) = 0 and G′(0) = 1. Substituting these values in the error equation (23), we get en+1 = −c2 2(G′′(0)−2)e3 n +7c2 [( G′′(0) − 2G′′′(0) 21 − 9 7 ) c2 2 − c3(G′′(0) − 2) 2 ] e4 n + · · · . It is clear that setting G′′(0) = 2 eliminates the third-order error. Subsequently, the fourth-order error is eliminated when G′′′(0) = 15 2 . Therefore, we establish the third and fourth conditions in (19), leading to the following form of the error equation en+1 = (1 3 ( 42 − G(4)(0) ) c4 2 − c2 2c3 ) e5 n + O(e6 n). Thus, the proof is complete. 4. Particular cases of the proposed family Many fifth-order iterative methods can be derived from the family (2) because of the arbitrary nature of A and B. Furthermore, several weight functions can satisfy (4) for a specific choice of A and B. Here, some well-known methods are listed as particular cases of the family (2): A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2330 (i) Let A = −1 and B = 5, then the weight function G(η) satisfies the condition (4) if: G(0) = 4, G′(0) = −1, G′′(0) = 0. Taking G(η) = 4 − η, the scheme (2) becomes xn+1 = yn − [ f ′(yn) + 3f ′(xn) 5f ′(yn) − f ′(xn) ] f(yn) f ′(xn) , (24) which is the method proposed by Ham and Chun (HCM) [9]. (ii) If A = 1 and B = 0, then (4) holds if G(η) satisfies G(0) = 1, G′(0) = 1, G′′(0) = 5 2 . (25) Taking G(η) = 5 + 3(1 − η)2 1 + 7(1 − η)2 , the scheme (2) takes the form xn+1 = yn − [ 5f ′2(xn) + 3f ′2(yn) f ′2(xn) + 7f ′2(yn) ] f(yn) f ′(xn) , (26) which is the method of Fang et al. (FLM) [6]. Another choice for the weight function satisfying (25) is G(η) = 1 + η + 5 4η2 − 1 6η3, leading to the following scheme xn+1 = yn − [ 2 − f ′(yn) f ′(xn) + 5 4 ( 1 − f ′(yn) f ′(xn) )2 − 1 6 ( 1 − f ′(yn) f ′(xn) )3] f(yn) f ′(xn) , (27) this is a special iterative scheme of the second family of fifth-order iterative methods introduced by Liu et al. (LM) [14]. (iii) Suppose A = 0 and B = 1, by (4) G(0) = 1, G′(0) = 0, G′′(0) = 1 2 . We can choose G(η) = 1 + 1 4η2, then the corresponding method is given as xn+1 = yn − [ 5 4 − f ′(yn) 2f ′(xn) + f ′2(yn) 4f ′2(xn) ] f(yn) f ′(yn) , (28) which is the method of Sivakumar and Jayaraman (PJM) [20]. A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2331 (iv) Let A = 1 and B = 1, then G(η) satisfies (4) if G(0) = 2, G′(0) = 1, G′′(0) = 3. Assuming the weight function as G(η) = 4 − 4η 2 − 3η , we get the method of Khirallah and Alkhomsan (AMK1) [10] xn+1 = yn + [ 4f ′(yn) f ′(xn) − 3f ′(yn) ] f(yn) f ′(xn) + f ′(yn) . (29) Next, we present four new particular methods of the family (2) by considering alter- native choices for A and B as follows: 1. If A = −2 and B = 7, the weight function G(η) satisfies the condition (4) if G(0) = 5, G′(0) = −2, G′′(0) = −3 2 . Taking G(η) = 40 − 31η 8 − 3η , the scheme (2) takes the form xn+1 = yn − [9f ′(xn) + 31f ′(yn) 5f ′(xn) + 3f ′(yn) ] f(yn) 7f ′(yn) − 2f ′(xn) . (30) We denote this new scheme as AZ1. 2. Let A = −1 and B = 4, the condition (4) holds if the weight function G(η) satisfies G(0) = 3, G′(0) = −1, G′′(0) = −1 2 . Assuming G(η) = 12 − 7η 4 − η , then the resulting scheme (AZ2) is given as xn+1 = yn − [5f ′(xn) + 7f ′(yn) 3f ′(xn) + f ′(yn) ] f(yn) 4f ′(yn) − f ′(xn) . (31) 3. Let A = 10 and B = 3, then the weight function G(η) satisfies the condition (4) if G(0) = 13, G′(0) = 10, G′′(0) = 53 2 . Using G(η) = 1040 − 578η 80 − 106η leads to the following scheme (AZ3) xn+1 = yn − [231f ′(xn) + 289f ′(yn) −13f ′(xn) + 53f ′(yn) ] f(yn) 10f ′(xn) + 3f ′(yn) . (32) A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2332 4. If A = 1 and B = −1, i.e. A + B = 0, then the weight function G(η) is selected to satisfy the condition (19) in Theorem 2 to obtain fifth-order schemes of type (18). Several forms for G(η) can be designed to satisfy (19). For example, assuming G(η) = −4η η3 + η2 + 4η − 4 , we obtain the following scheme (AZ4) xn+1 = yn + 4f ′2(xn)f(yn) (f ′(xn) − f ′(yn))2 (2f ′(xn) − f ′(yn)) − 4f ′2(xn)f ′(yn) . (33) The error equations for the methods mentioned above are derived using the general error equation (5) when A+B ̸= 0 and the error equation (20) when A+B = 0. As shown, error equation (5) involves the evaluation of G′′′(0), while error equation (20) requires the evaluation of G(4)(0). Table 1 presents the mentioned methods along with their respective error equations Table 1: Error equations for particular methods within the proposed family of iterative methods (2). Method Equation Error equation HCM (24) en+1 = (3 2c4 2 − c2 2c3 ) e5 n + O(e6 n) FLM (26) en+1 = (7 2c4 2 − c2 2c3 ) e5 n + O(e6 n) LM (27) en+1 = (46 3 c4 2 − c2 2c3 ) e5 n + O(e6 n) PJM (28) en+1 = ( 4c4 2 − c2 2c3 ) e5 n + O(e6 n) AMK1 (29) en+1 = −c2 2c3e5 n + O(e6 n) AZ1 (30) en+1 = ( 9 20c4 2 − c2 2c3 ) e5 n + O(e6 n) AZ2 (31) en+1 = (5 6c4 2 − c2 2c3 ) e5 n + O(e6 n) AZ3 (32) en+1 = (231 260c4 2 − c2 2c3 ) e5 n + O(e6 n) AZ4 (33) en+1 = −c2 2c3e5 n + O(e6 n) A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2333 5. Basins of attraction Any iterative method can converge if the initial estimate is chosen appropriately, re- gardless of the order of convergence. Therefore, applying an iterative method to some test functions for one or more initial estimates does not fully illustrate the stability of the method. To address this, many authors have considered complex dynamics and basins of attraction for testing the stability and reliability of iterative methods, see [2, 3, 8, 10]. Indeed, basins of attraction shows how an iterative method converges based on different initial estimates in a specified region. This technique generates a picture in complex plane that provides a visual comparison for different methods. To plot the basins of attraction, we specify a rectangular region, denoted as D, in the complex plane that contains all the roots of the considered complex polynomial g(z). Different colors are assigned to different roots at a point z0 ∈ D based on the corresponding roots to which the method converges when starting from z0. The brightness of the colors depends on the number of iterations required for convergence; a brighter color indicates fewer iterations, while a darker color indicates that the method requires more iterations. Points in the region D are assigned the black color if the method does not converge according to the convergence criteria. In this section, we assume a tolerance of 10−3 with a maximum of 20 iterations. Table 2: Complex polynomials and their roots. Function Roots g1(z) = z2 − 1 ±1 g2(z) = z3 − 1 1, − 0.5 ± 0.866025i g3(z) = z3 + z + i −0.56228 − 0.662359i, 1.32472i, 0.56228 − 0.662359i g4(z) = z4 − 1 ±1, ± i g5(z) = z4 − z + i −0.759845 + 0.592595i, − 0.532605 − 1.08829i, 0.181924 + 0.732098i, 1.11052 − 0.236405i g6(z) = z5 − 1 1, − 0.809017 ± 0.587785i, 0.309017 ± 0.951057i g7(z) = z6 − 1 ±1, − 0.5 ± 0.866025i, 0.5 ± 0.866025i g8(z) = z11 − 1 1, − 0.959493 ± 0.281733i, −0.654861 ± 0.75575i, −0.142315 ± 0.989821i, 0.415415 ± 0.909632i, 0.841254 ± 0.540641i g9(z) = z6 + 10z3 − 8 −2.20663, 0.906359, −0.45318 ± 0.78493i, 1.10332 ± 1.911i g10(z) = z(z2 + 1)(z2 − 4) 0, ± i, ±2 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2334 In Table 2, ten complex polynomials are listed with their roots as test cases. We use the region D = [−2, 2] × [−2, 2] for polynomials g1, g2, ..., g8, and D = [−3, 3] × [−3, 3] for g9 and g10. A grid of 320 × 320 points is used in all experiments. We compare the results of our new particular methods from the proposed family, namely AZ1 (30), AZ2 (31), AZ3 (32), and AZ4 (33), with those of well-known methods: HCM (24), FLM (26), LM (27), PJM (28), and AMK1 (29). To provide a summary, these methods are listed in Table 3. Table 3: Particular methods within the family (2). The first step is yn = xn − f(xn) f ′(xn) in all methods. Method Second step of the method HCM xn+1 = yn − [ f ′(yn) + 3f ′(xn) 5f ′(yn) − f ′(xn) ] f(yn) f ′(xn) FLM xn+1 = yn − [ 5f ′2(xn) + 3f ′2(yn) f ′2(xn) + 7f ′2(yn) ] f(yn) f ′(xn) LM xn+1 = yn − [ 2 − f ′(yn) f ′(xn) + 5 4 ( 1 − f ′(yn) f ′(xn) )2 − 1 6 ( 1 − f ′(yn) f ′(xn) )3] f(yn) f ′(xn) PJM xn+1 = yn − [ 5 4 − f ′(yn) 2f ′(xn) + f ′2(yn) 4f ′2(xn) ] f(yn) f ′(yn) AMK1 xn+1 = yn + [ 4f ′(yn) f ′(xn) − 3f ′(yn) ] f(yn) f ′(xn) + f ′(yn) AZ1 xn+1 = yn − [9f ′(xn) + 31f ′(yn) 5f ′(xn) + 3f ′(yn) ] f(yn) 7f ′(yn) − 2f ′(xn) AZ2 xn+1 = yn − [5f ′(xn) + 7f ′(yn) 3f ′(xn) + f ′(yn) ] f(yn) 4f ′(yn) − f ′(xn) AZ3 xn+1 = yn − [231f ′(xn) + 289f ′(yn) −13f ′(xn) + 53f ′(yn) ] f(yn) 10f ′(xn) + 3f ′(yn) AZ4 xn+1 = yn + 4f ′2(xn)f(yn) (f ′(xn) − f ′(yn))2 (2f ′(xn) − f ′(yn)) − 4f ′2(xn)f ′(yn) A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2335 In Table 4, the number of black points is given for all test cases with the average value across all cases for each method. In Figures 1, 2, 3, and 4, we illustrate the basins of attraction for the functions g2, g4, g5, and g6, respectively. It is clear that our four new methods and the AMK1 method exhibit good dynamical behavior, i.e they are more stable than other methods. Based on the number of black points for the given test functions, we can rank the best methods as follows: AZ1, followed by AMK1, then AZ2 and AZ3 being nearly equal, and then AZ4. Table 4: Number of black points for iterative methods for each test case and average across all cases. MD g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 Average HCM 320 415 108 3688 54 6997 11002 18087 1714 104 4249 FLM 488 1035 450 9296 282 13627 18556 31282 5572 272 8086 LM 612 10929 4969 27936 10469 29856 32916 41961 22971 5742 18836 PJM 326 2928 1066 14844 1343 16903 20024 32524 8446 928 9933 AMK1 320 0 49 640 0 141 726 6541 8 0 842 AZ1 320 0 0 640 0 77 740 6017 0 0 779 AZ2 320 0 6 656 0 200 982 6543 10 0 872 AZ3 320 0 8 656 0 163 930 6659 0 0 874 AZ4 320 0 29 648 0 197 926 7161 11 0 929 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2336 (a) HCM (b) FLM (c) LM (d) PJM (e) AMK1 (f) AZ1 (g) AZ2 (h) AZ3 (i) AZ4 Figure 1: Basins of attraction associated with g2(z) = z3 − 1 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2337 (a) HCM (b) FLM (c) LM (d) PJM (e) AMK1 (f) AZ1 (g) AZ2 (h) AZ3 (i) AZ4 Figure 2: Basins of attraction associated with g4(z) = z4 − 1 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2338 (a) HCM (b) FLM (c) LM (d) PJM (e) AMK1 (f) AZ1 (g) AZ2 (h) AZ3 (i) AZ4 Figure 3: Basins of attraction associated with g5(z) = z4 − z + i A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2339 (a) HCM (b) FLM (c) LM (d) PJM (e) AMK1 (f) AZ1 (g) AZ2 (h) AZ3 (i) AZ4 Figure 4: Basins of attraction associated with g6(z) = z5 − 1 6. Numerical results In this section, we consider numerical examples to confirm the order of convergence and demonstrate the efficiency of the newly constructed methods: AZ1 (30), AZ2 (31), A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2340 AZ3 (32), and AZ4 (33). We compare these new methods with the well-known methods: HCM (24), FLM (26), LM (27), PJM (28), and AMK1 (29). All these methods, as shown, belong to the proposed family (2). All computations are performed using Maple 2021 with a precision of 2000 significant digits. The applied stopping criterion is defined as |xn−xn−1| < 10−50. The computational order of convergence ρ is approximated by using the formula [4] ρ = ln |(xn+1 − xn)/(xn − xn−1)| ln |(xn − xn−1)/(xn−1 − xn−2)| . Table 5 lists eight of the most frequently used test functions in research. Their simple roots are provided up to 25 decimal places. Tables 6, 7, 8, and 9 display the following metrics: the number of iterations (N) at which the stopping criterion is satisfied, the computational order of convergence (ρ), the error |xN −xN−1|, |f(xN )|, and the processing time in seconds. The processing time is determined as the mean of 10,000 executions to obtain reasonably accurate values. Table 5: The test functions and their simple roots. Function Root f1(x) = x3 + 4x2 − 10 1.365230013414096845760807 f2(x) = sin2 x − x2 + 1 1.404491648215341226035087 f3(x) = x2 − ex − 3x + 2 0.257530285439860760455367 f4(x) = ln(x2 + x + 2) − x + 1 4.152590736757158274996989 f5(x) = xex2 − sin2 x + 3 cos x + 5 −1.207647827130918927009417 f6(x) = esin x − x + 1 2.630664147927903633975327 f7(x) = x5 + x − 10000 6.308777129972689094767572 f8(x) = √ x2 + 2x + 5 − 2 sin x − x2 + 3 2.331967655883964010308044 Clearly, the computational order of convergence for all considered methods confirms the theoretical analysis. The numerical results categorize the methods into two groups: • The methods AZ1, AZ2, AZ3, AZ4 and AMK1. • The methods HCM, FLM, LM, PJM. In general, for the given test functions, the first category yields better results than the second category, with fewer iterations in some test cases and higher accuracy in most cases. Indeed, the results for the methods in the first category are comparable. It appears that for some cases, AMK1 is the best method, while AZ1 is the best choice for other cases. Based on the processing time, it seems that the methods HCM, AMK1, AZ1, AZ2, and AZ3 require less execution time compared to the other methods. A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2341 Table 6: Numerical results for test functions f1(x) and f2(x). Function Method x0 N |xN − xN−1| |f(xN )| ρ Time f1 HCM 0.8 4 9.15 × 10−51 7.63 × 10−251 5.00 0.0019 FLM 5 1.80 × 10−183 5.92 × 10−914 5.00 0.0031 LM 28 1.89 × 10−101 3.46 × 10−503 4.99 0.0203 PJM 5 2.49 × 10−137 3.44 × 10−683 5.00 0.0030 AMK1 4 1.34 × 10−68 1.02 × 10−340 4.99 0.0021 AZ1 4 3.34 × 10−86 7.82 × 10−429 5.00 0.0025 AZ2 4 7.14 × 10−70 1.03 × 10−346 5.00 0.0022 AZ3 4 1.93 × 10−67 1.62 × 10−334 5.00 0.0021 AZ4 5 1.02 × 10−222 2.63 × 10−1111 5.00 0.0032 f1 HCM 2.0 4 6.00 × 10−74 9.22 × 10−367 4.99 0.0019 FLM 4 3.12 × 10−63 9.16 × 10−313 4.99 0.0024 LM 5 1.41 × 10−248 7.94 × 10−1239 5.00 0.0040 PJM 4 4.60 × 10−63 7.39 × 10−312 4.99 0.0024 AMK1 4 1.97 × 10−111 7.12 × 10−555 5.00 0.0021 AZ1 4 2.72 × 10−90 2.80 × 10−449 5.00 0.0025 AZ2 4 3.76 × 10−81 4.18 × 10−403 4.99 0.0022 AZ3 4 2.63 × 10−80 7.62 × 10−399 4.99 0.0022 AZ4 4 1.31 × 10−88 9.11 × 10−441 4.99 0.0029 f2 HCM 1.0 5 2.64 × 10−195 1.63 × 10−973 5.00 0.0036 FLM 5 1.65 × 10−128 3.83 × 10−639 5.00 0.0045 LM 6 1.17 × 10−135 3.10 × 10−674 5.00 0.0063 PJM 5 1.26 × 10−66 1.16 × 10−329 5.00 0.0051 AMK1 4 7.61 × 10−63 3.42 × 10−312 4.99 0.0032 AZ1 4 1.11 × 10−83 4.82 × 10−416 4.99 0.0033 AZ2 4 1.03 × 10−58 7.55 × 10−291 5.00 0.0030 AZ3 4 1.71 × 10−56 1.01 × 10−279 5.00 0.0032 AZ4 5 4.28 × 10−168 1.92 × 10−838 5.00 0.0054 f2 HCM 3.0 5 9.57 × 10−195 1.02 × 10−970 5.00 0.0039 FLM 5 3.00 × 10−167 7.69 × 10−833 5.00 0.0039 LM 5 2.01 × 10−128 4.63 × 10−638 4.99 0.0050 PJM 5 7.36 × 10−167 7.77 × 10−831 5.00 0.0042 AMK1 5 7.32 × 10−250 2.81 × 10−1247 5.00 0.0044 AZ1 5 1.41 × 10−226 1.61 × 10−1130 5.00 0.0046 AZ2 5 2.87 × 10−210 1.25 × 10−1048 5.00 0.0039 AZ3 5 1.41 × 10−208 3.95 × 10−1040 5.00 0.0046 AZ4 5 1.24 × 10−249 3.88 × 10−1246 5.00 0.0044 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2342 Table 7: Numerical results for test functions f3(x) and f4(x). Function Method x0 N |xN − xN−1| |f(xN )| ρ Time f3 HCM −1.0 4 5.87 × 10−104 1.01 × 10−519 4.99 0.0021 FLM 4 2.26 × 10−95 5.12 × 10−477 5.00 0.0024 LM 4 1.35 × 10−74 1.12 × 10−372 5.00 0.0026 PJM 4 1.40 × 10−95 3.91 × 10−478 4.99 0.0027 AMK1 4 2.91 × 10−88 3.93 × 10−441 4.99 0.0022 AZ1 4 8.63 × 10−92 8.38 × 10−459 4.99 0.0020 AZ2 4 2.55 × 10−95 1.77 × 10−476 4.99 0.0022 AZ3 4 6.94 × 10−96 2.62 × 10−479 4.99 0.0024 AZ4 4 2.41 × 10−91 1.52 × 10−456 4.99 0.0024 f3 HCM 1.5 4 8.53 × 10−80 6.53 × 10−399 5.00 0.0021 FLM 4 3.57 × 10−82 5.09 × 10−411 5.00 0.0023 LM 4 1.11 × 10−84 4.32 × 10−423 5.00 0.0028 PJM 4 9.64 × 10−83 6.08 × 10−414 4.99 0.0023 AMK1 4 2.46 × 10−78 1.70 × 10−391 5.00 0.0022 AZ1 4 8.94 × 10−79 1.00 × 10−393 5.00 0.0023 AZ2 4 3.80 × 10−79 1.29 × 10−395 5.00 0.0022 AZ3 4 3.36 × 10−79 6.93 × 10−396 5.00 0.0024 AZ4 4 1.65 × 10−78 2.28 × 10−392 5.00 0.0028 f4 HCM 2.0 4 1.80 × 10−67 5.05 × 10−339 5.00 0.0023 FLM 4 6.50 × 10−59 4.92 × 10−296 5.00 0.0027 LM 5 4.57 × 10−145 2.73 × 10−726 5.00 0.0036 PJM 4 2.31 × 10−54 3.05 × 10−273 5.00 0.0031 AMK1 4 1.15 × 10−81 2.90 × 10−410 5.00 0.0024 AZ1 4 9.02 × 10−77 1.08 × 10−385 5.00 0.0022 AZ2 4 3.68 × 10−73 1.43 × 10−367 5.00 0.0023 AZ3 4 1.14 × 10−72 4.21 × 10−365 5.00 0.0026 AZ4 4 5.79 × 10−78 9.44 × 10−392 4.99 0.0026 f4 HCM 6.0 4 2.42 × 10−118 2.20 × 10−593 4.99 0.0024 FLM 4 3.09 × 10−113 1.19 × 10−567 4.99 0.0027 LM 4 3.81 × 10−101 1.10 × 10−506 4.99 0.0029 PJM 4 1.37 × 10−112 2.20 × 10−564 4.99 0.0031 AMK1 4 2.25 × 10−124 8.39 × 10−624 4.99 0.0025 AZ1 4 3.82 × 10−122 1.48 × 10−612 4.99 0.0021 AZ2 4 1.42 × 10−120 1.22 × 10−604 4.99 0.0024 AZ3 4 2.28 × 10−120 1.34 × 10−603 4.99 0.0027 AZ4 4 1.81 × 10−123 2.84 × 10−619 4.99 0.0026 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2343 Table 8: Numerical results for test functions f5(x) and f6(x). Function Method x0 N |xN − xN−1| |f(xN )| ρ Time f5 HCM -2.5 6 3.11 × 10−61 1.84 × 10−301 4.99 0.0082 FLM 7 2.88 × 10−102 5.38 × 10−506 4.99 0.0105 LM 8 3.12 × 10−207 4.42 × 10−1030 5.00 0.0139 PJM 7 1.22 × 10−120 8.61 × 10−598 4.99 0.0107 AMK1 5 1.63 × 10−140 1.07 × 10−697 5.00 0.0067 AZ1 6 2.25 × 10−200 2.58 × 10−997 5.00 0.0094 AZ2 6 4.97 × 10−127 1.70 × 10−631 4.99 0.0093 AZ3 6 7.25 × 10−123 2.25 × 10−612 5.00 0.0098 AZ4 6 8.47 × 10−103 3.99 × 10−509 5.00 0.0099 f5 HCM -1.0 4 4.08 × 10−67 7.16 × 10−331 5.00 0.0051 FLM 5 2.75 × 10−237 4.27 × 10−1181 5.00 0.0069 LM 6 4.81 × 10−111 3.86 × 10−549 5.00 0.0091 PJM 5 9.30 × 10−195 2.24 × 10−968 5.00 0.0069 AMK1 4 2.71 × 10−64 1.34 × 10−316 4.99 0.0048 AZ1 4 5.71 × 10−71 2.74 × 10−350 4.99 0.0054 AZ2 4 3.33 × 10−87 2.28 × 10−432 4.99 0.0056 AZ3 4 1.26 × 10−100 3.61 × 10−501 5.00 0.0058 AZ4 5 3.78 × 10−241 7.04 × 10−1201 5.00 0.0086 f6 HCM 1.5 5 3.22 × 10−54 1.27 × 10−270 4.98 0.0025 FLM 6 3.34 × 10−249 1.65 × 10−1245 5.00 0.0042 LM 9 1.44 × 10−92 3.66 × 10−462 4.99 0.0085 PJM 6 1.04 × 10−235 5.06 × 10−1178 5.00 0.0045 AMK1 5 1.40 × 10−57 1.81 × 10−287 4.98 0.0025 AZ1 5 9.92 × 10−57 3.33 × 10−283 4.98 0.0030 AZ2 5 6.55 × 10−56 4.25 × 10−279 4.98 0.0028 AZ3 5 8.76 × 10−56 1.82 × 10−278 4.98 0.0030 AZ4 5 9.62 × 10−60 2.80 × 10−298 4.99 0.0035 f6 HCM 3.5 4 5.22 × 10−71 1.41 × 10−354 5.00 0.0021 FLM 4 2.32 × 10−67 2.70 × 10−336 5.00 0.0023 LM 5 2.05 × 10−243 2.15 × 10−1216 5.00 0.0039 PJM 4 2.34 × 10−65 2.83 × 10−326 5.00 0.0030 AMK1 4 1.63 × 10−74 3.87 × 10−372 5.00 0.0022 AZ1 4 1.49 × 10−73 2.54 × 10−367 5.00 0.0022 AZ2 4 1.12 × 10−72 6.12 × 10−363 5.00 0.0020 AZ3 4 1.51 × 10−72 2.76 × 10−362 5.00 0.0025 AZ4 4 1.24 × 10−77 9.83 × 10−388 5.00 0.0028 A. Zein / Eur. J. Pure Appl. Math, 16 (4) (2023), 2323-2347 2344 Table 9: Numerical results for test functions f7(x) and f8(x). Function Method x0 N |xN − xN−1| |f(xN )| ρ Time f7 HCM 4.0 Div - - - - FLM 10 6.66 × 10−63 3.15 × 10−309 4.99 0.0065 LM Div - - - - PJM 34 4.60 × 10−84 5.75 × 10−415 4.99 0.0256 AMK1 5 7.16 × 10−61 7.54 × 10−300 5.00 0.0033 AZ1 5 8.08 × 10−72 1.38 × 10−355 4.99 0.0033 AZ2 5 1.03 × 10−73 3.14 × 10−364 4.99 0.0030 AZ3 5 2.35 × 10−77 2.22 × 10−382 4.99 0.0037 AZ4 6 3.57 × 10−147 2.33 × 10−731 5.00 0.0046 f7 HCM 8.0 5 1.26 × 10−248 2.57 × 10−1238 5.00 0.0033 FLM 5 6.57 × 10−194 2.94 × 10−964 5.00 0.0034 LM 5 2.73 × 10−142 1.80 × 10−705 5.00 0.0037 PJM 5 2.18 × 10−195 1.39 × 10−971 5.00 0.0038 AMK1 4 2.79 × 10−65 6.80 × 10−322 5.00 0.0024 AZ1 4 3.41 × 10−72 1.85 × 10−357 4.99 0.0024 AZ2 4 2.40 × 10−58 2.13 × 10−287 4.99 0.0027 AZ3 4 3.07 × 10−57 8.51 × 10−282 4.99 0.0028 AZ4 4 4.47 × 10−61 7.16 × 10−301 4.99 0.0028 f8 HCM 1.0 4 2.20 × 10−151 1.03 × 10−756 5.00 0.0092 FLM 4 4.58 × 10−152 3.07 × 10−760 5.00 0.0085 LM 4 3.00 × 10−152 3.17 × 10−761 5.00 0.0095 PJM 4 2.86 × 10−152 2.67 × 10−761 5.00 0.0090 AMK1 4 5.61 × 10−151 1.31 × 10−754 5.00 0.0084 AZ1 4 4.31 × 10−151 3.34 × 10−755 5.00 0.0099 AZ2 4 3.40 × 10−151 9.86 × 10−756 5.00 0.0092 AZ3 4 3.29 × 10−151 8.24 × 10−756 5.00 0.0092 AZ4 4 5.60 × 10−151 1.30 × 10−754 5.00 0.0092 f8 HCM 4.0 4 1.30 × 10−84 7.37 × 10−423 4.99 0.0083 FLM 4 1.28 × 10−52 5.28 × 10−263 5.00 0.0079 LM 5 3.12 × 10−209 3.87 × 10−1046 5.00 0.0114 PJM 4 1.90 × 10−54 3.43 × 10−272 5.00 0.0082 AMK1 5 3.60 × 10−245 1.42 × 10−1225 5.00 0.0111 AZ1 4 6.97 × 10−55 3.69 × 10−274 4.99 0.0093 AZ2 4 6.29 × 10−61 2.12 × 10−304 4.99 0.0092 AZ3 4 5.93 × 10−62 1.58 × 10−309 4.99 0.0095 AZ4 4 1.43 × 10−57 1.42 × 10−287 4.99 0.0095 REFERENCES 2345 7. Conclusion We have used the weight function procedure to develop a two-step family of fifth-order iterative methods for solving nonlinear equations. This family has a wide range of choices due to its formation and the ability to select various weight functions. This flexibility allows us to design specific methods with improved stability and accuracy. Furthermore, we show that the proposed family includes several well-known and recent fifth-order iterative methods as special cases. We provide a detailed convergence analysis and assess the methods’ stability using basins of attraction. The basins of attraction in the complex plane, along with numerical results in the real domain, reveal that the newly constructed methods AZ1, AZ2, AZ3, and AZ4, as well as the recent method AMK1 [10], exhibit superior stability and accuracy when compared to other known fifth-order methods within the same family. Acknowledgements The author would like to thank the anonymous reviewers for their comments and suggestions. References [1] H. A. Abro and M. M. Shaikh. A new time-efficient and convergent nonlinear solver, applied mathematics and computation. Applied Mathematics and Compu- tation, 335:516–536, 2019. [2] F. I. Chicharro, A. Cordero, N. Garrido, and J. R. Torregrosa. Wide stability in a new family of optimal fourth-order iterative methods. Computational Mathematics and Mathematical Physics, 1:e1023, 2019. [3] A. Cordero, M. M.-Martinez, and J. R. Torregrosa. Chaos and stability in a new iterative family for solving nonlinear equations. Algorithms, 14:1–24, 2021. [4] A. Cordero and J. R. Torregrosa. Variants of newton’s method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190:686–698, 2007. [5] J. F. Epperson. An Introduction to Numerical Methods and Analysis. John Wiley & Sons, New Jersey, 2013. [6] L. Fang, L. Sun, and G. P. He. An efficient newton-type method with fifth-order convergence for solving nonlinear equations. Computational and Applied Mathematics, 27:269–274, 2008. [7] M. Grau and J. L. Diaz-Barrero. An improvement of the euler-chebyshev iterative method. Journal of Mathematical Analysis and Applications, 315:1–7, 2006. REFERENCES 2346 [8] M. A. Hafiz and M.Q. Khirallah. An optimal fourth order method for solving nonlinear equations. Journal of mathematics and Computer Science, 23:86–97, 2021. [9] Y. M. Ham and C. B. Chun. A fifth-order iterative method for solving nonlinear equations. Applied Mathematics and Computation, 194:287–290, 2007. [10] M. Q. Khirallah and A. M. Alkhomsan. A new fifth-order iterative method for solving non-linear equations using weight function technique and the basins of attraction. Journal of Mathematics and Computer Science, 28:281–293, 2023. [11] J. Kou and Y. Li. The improvements of chebyshev-halley methods with fifth-order convergence. Applied Mathematics and Computation, 188:143–147, 2007. [12] J. Kou, Y. Li, and X. Wang. A family of fifth-order iterations composed of newton and third-order methods. Applied Mathematics and Computation, 186:1258–1262, 2007. [13] H. T. Kung and J. F. Traub. Optimal order of one-point and multipoint iteration. Journal of Association of Computer Mathematics, 21:634–651, 1974. [14] C. S. Liu, E. R. El-Zahar, and C.W. Chang. Three novel fifth-order iterative schemes for solving nonlinear equations. Mathematics and Computers in Simulation, 187:282– 293, 2021. [15] K.I. Noor and M.A. Noor. Predictor-corrector halley method for nonlinear equations. Applied Mathematics and Computation, 188:1587–1591, 2007. [16] K.I. Noor, M.A. Noor, and S. Momani. Modified householder iterative method for nonlinear equations. Applied Mathematics and Computation, 190:1534–1539, 2007. [17] M. A. Noor, K. I. Noor, E. Al-Said, and M. Waseem. Some new iterative methods for nonlinear equations. Mathematical Problems in Engineering, 2010:1–12, 2010. [18] M.A. Noor and K.I. Noor. Fifth-order iterative methods for solving nonlinear equa- tions. Applied Mathematics and Computation, 188:406–410, 2007. [19] M. S. Petković, B. Neta, L. D. Petković, and J. Džunić. Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam-Boston-Heidelberg-London-New York, 2013. [20] P. Sivakumar and J. Jayaraman. Some new higher order weighted newton methods for solving nonlinear equation with applications. Mathematical and Computational Applications, 24:21 pages, 2019. [21] O. S. Solaiman and I. Hashim. Two new efficient sixth order iterative methods for solving nonlinear equations. Journal of King Saud University-Science, 31:701–705, 2019. REFERENCES 2347 [22] O. S. Solaiman and I. Hashim. An iterative scheme of arbitrary odd order and its basins of attraction for nonlinear systems. Computers, Materials & Continua, 66:1427–1444, 2021. [23] S. Weerakoon and T.G.I. Fernado. A variant of newton’s method with accerated third-order convergence. Applied Mathematics Letters, 13:87–93, 2000.