EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2419-2430 ISSN 1307-5543 – ejpam.com Published by New York Business Global New Variants of Newton’s Method for Solving Nonlinear Equations Buddhi Prasad Sapkota1,∗, Jivandhar Jnawali1 1 Department of Mathematics, Ratna Rajyalaxmi Campus, Tribhuvan University, Kathmandu, Nepal Abstract. Two Newton-type iterative techniques have been created in this work to locate the true root of univariate nonlinear equations. One of these can be acquired by modifying the double Newton’s method in a straightforward manner, while the other can be gotten by modifying the midpoint Newton’s method. The iterative approach developed by McDougall and Wortherspoon is employed for the change. The study demonstrates that the modified double Newton’s approach outperforms the current one in terms of both convergence order and efficiency index, even though both methods assess the same amount of functions and derivatives every iteration. In comparison to the midpoint Newton’s technique, which has a convergence order of 3, the modified midpoint Newton’s method has a convergence order of 5.25 and requires two extra functions to be evaluated per iteration. In order to evaluate the effectiveness of recently introduced approaches with current methods, some numerical examples are shown in the final section. 2020 Mathematics Subject Classifications: 65H05 Key Words and Phrases: Newton’s method, Numerical method, Nonlinear equation, Conver- gence, Efficiency index 1. Introduction A collection of various nonlinear equations with objective functions and/or nonlinear constraint functions is referred to as a nonlinear equation system [1]. Finding the roots of some nonlinear equation systems requires numerical approaches. The numerical ap- proach is a strategy for figuring out mathematical issues so that they may be resolved using standard operations like add, subtract, multiply, and divide. One of the numerical techniques for resolving a system of nonlinear equations is the Newton method. Some mathematicians are modifying Newton methods to arrive at a novel technique. The Mid- point Newton approach, which can find the equation’s root more quickly [13] is the result ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4951 Email addresses: buddi.sapkota@rrlc.tu.edu.np (B. Sapkota), jivandhar.jnawali@rrlc.tu.edu.np (J. Jnawali) https://www.ejpam.com 2419 © 2023 EJPAM All rights reserved. B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2420 of combining the Newton method and the Midpoint rule. Singh MK [14] developed New- ton’s method based on contra harmonic mean with order of convergence six. He JH extended the frequency-amplitude formulation to solve nonlinear conservative oscillators with general initial conditions which is similar to the Hamiltonian approach. Finding the solutions to nonlinear equations is one of the most important and challenging problems in almost all areas of science and engineering, but solving such equations analytically is almost impossible. In those situations, the construction of efficient numerical methods based on an iterative procedure for finding the roots of such equations becomes a fasci- nating and attractive task. In literature, there are numerous methods available for finding the roots of such equations in different situations. Newton’s method is one of the best- known and most frequently used basic numerical methods to solve nonlinear equations. It’s an iterative formula for obtaining the simple root of the nonlinear equation In similar circumstances, developing effective numerical algorithms based on an iterative process to locate the roots of such equations turns out to be an intriguing and alluring task. There are many techniques described in the literature for locating the roots of these equations in various contexts. One of the most popular and widely used fundamental numerical techniques for solving nonlinear equations is Newton’s method. It is an iterative formula for finding the nonlinear equation’s simple root. xn+1 = xn − f(xn) f ′(xn) . (1) This method is quadratically convergent [2] and its efficiency index is 1.414 [9]. During the last two decades, several higher-order variants of Newton’s methods have appeared. Some of them can be found in [3]-[17]. Here, we will briefly describe those methods that encouraged us to carry out our own work. Now, consider double Newton’s methods yn = xn − f(xn) f ′(xn) , xn+1 = yn − f(yn) f ′(yn) .  (2) This method is fourth-order convergent [11, 15] and its efficiency index is the same as Newton’s method. Many researchers use the numerical integration technique to improve the local order of convergence of Newton’s method. Some of them may be found in [3–8, 10, 16, 17] and references therein. To get the improved convergence order of Newton’s method, Özban in [13] considered Newton’s theorem f(x) = f(xn) + ∫ x xn f ′(t) dt (3) and used the mid-point rule to approximate the integral, that is,∫ x xn f ′(t) dt = (x− xn)f ′ (x+ xn) 2 . (4) B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2421 Then, they explored the variant of Newton’s method given by xn+1 = xn − f(xn) f ′(xn+x∗ n 2 ) , (5) where x∗n = xn − f(xn) f ′(xn) . This is the midpoint Newton’s method and it has been proven that its convergence order is 3. Recently, McDougall andWortherspoon [12] obtained a variant of Newton’s method having the order of convergence 1 + √ 2 with a minor change in Newton’s method. The method is given below: If x0 is the initial approximate solution, we can write x∗0 = x0 − f(x0) f ′(x0) , x1 = x0 − f(x0) f ′[12(x0 + x∗0)] .  (6) For n ≥ 1, we have x∗n = xn − f(xn) f ′[12(xn−1 + x∗n−1)] , xn+1 = xn − f(xn) f ′[12(xn + x∗n)] .  (7) This is a kind of predictor-corrector method. In Newton’s method, the derivatives are calculated at the previously iterated point, but in the methods (6)-(7), the derivatives are calculated at some convenient point. As the supreme goal of this paper, two iterative methods are constructed as a variant of methods (2) and (5) using methods (6)-(7). Also, it has to be shown that both newly introduced methods can easily compete with similar existing methods. 2. Methods with Convergence Analysis At first, we present the iterative method as a variant of method (2) using the view of method (6)-(7) as given below: If x0 is taken as the initial approximate solution, x∗0 = x0 x∗∗0 = x0 − f(x0) f ′ ( x0 + x∗0 2 ) = x0 − f(x0) f ′(x0) x1 = x∗∗0 − f(x∗∗0 ) f ′(x∗∗0 )  (8) B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2422 followed by (for n ≥ 1) x∗n = xn − f(xn) f ′ ( xn−1 + x∗n−1 2 ) x∗∗n = xn − f(xn) f ′ ( xn + x∗n 2 ) xn+1 = x∗∗n − f(x∗∗n ) f ′(x∗∗n )  (9) To analyze it’s convergence, we prove: Theorem 1. Let a sufficiently differentiable function f : D ⊂ R → R has a simple root α in the interval D. If x0 is sufficiently close to α, then to solve the nonlinear equation f(x) = 0, (8)-(9) has a convergence of order 4.45. Proof. Let en = xn − α be the error in the nthiterationxn and denote cj = 1 j! . fj(α) f ′(α) , j = 2, 3, 4.... It is standard to work out that the error equation in the Newton method 1 : en+1 = c2e 2 n, (10) where the terms with higher powers of en are omitted. Let us go on to discuss the convergence of the methods (8)-(9). Let e∗n, e ∗∗ n and en denote errors at iterates x∗n, x ∗∗ n and xn respectively. Then clearly e∗0 = e0 and in the view of (10), from (8), we get the error equation e∗∗0 = c2e 2 0, (11) and e1 = c2(e ∗∗ 0 )2 = c32e 4 0 (12) Here, x∗1 = x1 − f(x1) f ′[12(x0 + x∗0)] = x1 − f(x1) f ′(x0). Taylor series and binomial expansion gives the error equation for x∗1, we have α+ e∗1 = α+ e1 − f(α+ e1) f(α+ e0) e∗1 = e1 − e1 + c2e 2 1 + c3e 3 1 +O(e41) 1 + 2c2e0 + 3c3e20 +O(e30) = e1 − ( e1 + c2e 2 1 + c3e 3 1 +O(e41) ) ( 1− 2c2e0 − 3c3e 2 0 + 4c22e 2 0 +O(e30) ) = 2c2e0e1 = 2c42e 5 0, (13) B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2423 neglecting the higher powers of e0. Next, the error in x∗∗1 can be found as follows: e∗∗1 = e1 − f(α+ e1) f ′ ( α+ e1+e∗1 2 ) = e1 − ( e+ c2e 2 1 + c3e 3 1 )( 1 + 2c2( e1 + e∗1 2 ) + 3c3( e1 + e∗1 2 )2 )−1 = c2e1e ∗ 1 = 2c52e 9 0 (14) by using (12) and (13). Also, from the view of the error equation (10), the error equation for x2 : e2 = c2(e ∗∗ 1 )2 = 22c112 e180 (15) In fact, for n ≥ 2, the errors at x∗n, x ∗∗ n and xn respectively, can be found by the relations e∗n = c2enen−1, (16) e∗∗n = c2ene ∗ n = c22e 2 nen−1, (17) and en+1 = c2e ∗∗ n 2 = c52e 4 ne 2 n−1. (18) To calculate the order of convergence of (8)-(9), the following relation is needed: en+1 = Aepn (19) Thus, en = Aepn−1 or en−1 = A − 1 p e 1 p n . (20) From (16), (17) and (18), we have Aepn = c52e 4 nA −2/pe2/pn . Equating the power of en, we obtain p = 4 + 2 p or, p2 − 4p− 2 = 0 or, p = 4± √ 24 2 . For positive sign, p = 2 + √ 6 ≈ 4.45. Consequently, the convergence order of methods (8)-(9) is 4.45. Alternative Proof: Accumulating the relations from the proof given above, we get the errors at xn, x∗n, and x∗∗n stages as given in the following Table-1: We construct the table B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2424 Table 1: Successive errors. n en e∗n e∗∗n 0 e0 e0 c2e 2 0 1 c32e 4 0 2c42e 5 0 2c52e 9 0 2 22c112 e180 22c142 e220 24c262 e400 3 28c532 e800 210c652 e980 218c1192 e1780 4 238c2392 e3560 246c2932 e4360 284c5332 e7920 5 2168c10672 e15840 ... ... ... ... ... ... as done in [12]. The sequence formed by the power of e0 in the errors at each iteration is 4, 18, 80, 356, 1584, · · · (21) The sequence of their successive ratios is 18 4 , 80 18 , 356 80 , 1584 356 , · · · or 4.5, 4.44, 4.45, 4.45, · · · . If {Ri} denotes the ith term of the sequence (21), then it holds relation Ri = 4Ri−1 + 2Ri−2, i = 2, 3, 4, · · · . If we take the limit Ri Ri−1 = Ri−1 Ri−2 = p Then, we get p2 − 4p− 2 = 0, (22) which gives its positive root p = 2 + √ 6 ≈ 4.45. Consequently, convergence order of the method (8)-(9) is at least 4.45. □ Next, we modify method (5) by using methods (6)-7) as below: If x0 is the initial approximation, then x∗0 = x0, x1 = x∗0 − f(x∗0) f ′[12(x ∗ 0 + z1)] , with z1 = x0 − f(x0) f ′ ( x0+x∗ 0 2 ) .  (23) x∗1 = x1 − f(x1) f ′[12(x1 + z∗1)] , with z∗1 = x1 − f(x1) f ′ ( x0+x∗ 0 2 ) . } (24) B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2425 For n ≥ 1, we get the following iterations: x∗n = xn − f(xn) f ′[12(xn + z∗n)] ,where z∗n = xn − f(xn) f ′ ( xn−1+x∗ n−1 2 ) xn+1 = x∗n − f(x∗n) f ′ ( x∗ n+zn+1 2 ) ,where zn+1 = xn − f(xn) f ′ ( xn+x∗ n 2 ) .  (25) The proof of the convergence results of the above theorem is given below. Theorem 2. Let a sufficiently differentiable function f : D ⊂ R → R has a simple root α in the interval D. If x0 is sufficiently close to α, then the methods (23)-(25) to solve the nonlinear equation f(x) = 0, have convergence order 5.1925. Proof. Let en and e∗n be the errors in xn and x∗n,respectively. Again, suppose ck = fk(α) k!f ′(α) , k = 2, 3, 4.... The error equation of method (5) as obtained by Özban [13] : en+1 = ae3n, (26) where a = c22 − 1 4c3 and the higher powers of en being neglected. In the same line of proof of above result, e∗0 and e1, in x∗0 and x1 in (23)-(25) are given by e∗0 = e0 e1 = ae30, (27) where higher powers of small quantity are neglected. Next, we find e∗1 in x∗1. Taylor and binomial series expansion gives: f(x1) f ′(x0) = f(α+ e1) f ′(α+ e0) = ( e1 + c2e 2 1 + c3e 3 1 +O(e41) )( 1 + 2c2e0 + 3c3e 2 0 +O(e30) )−1 = e1 − 2c2e0e1 +O(e50), and hence z∗1 = x1 − f(x1) f ′(x0) = α+ 2c2e0e1 +O(e50). Also, x1 + z∗1 2 = α+ e1 + 2c2e0e1 + ... 2 , and f ′(x1 + z∗1 2 ) = f ′(α) ( 1 + c2e1 + 2c22e0e1 +O(e50) ) . (28) B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2426 Consequently, using (27) and (28), e∗1 in x∗1 at the equation (24) is given by e∗1 = e1 − ( e1 + c2e 2 1 +O(e31) )( 1 + c2e1 + 2c22e0e1 +O(e50) )−1 = 2c22e0e 2 1 = be70, where b = 2c22a 2. From (25), using e∗1, we now find e2 in x2 = x∗1 − f(x∗1) f ′( x∗ 1+z2 2 ) , where z2 = x1 − f(x1) f ′ (x1+x∗ 1 2 ) . Now, f ′(x1 + x∗1 2 ) = f ′(α+ e1 + e∗1 2 ) = f ′(α) ( 1 + c2e1 + c2e ∗ 1 + 3 4 c3e 2 1 +O(e90) ) in turn, f(x1) f ′ (x1+x∗ 1 2 ) = ( e1 + c2e 2 1 +O(e31) )( 1 + c2e1 + c2e ∗ 1 + 3 4 c3e 2 1 +O(e90) )−1 = e1 + 1 4 c3e 3 1 − c2e1e ∗ 1 + ... and therefore z2 = α− 1 4 c3e 3 1 + c2e1e ∗ 1 + .... Here, x∗1 + z2 2 = α+ 1 2 ( e∗1 − 1 4 c3e 3 1 + c2e1e ∗ 1 + ... ) , and f ′(x∗1 + z2 2 ) = f ′(α) ( 1 + c2e ∗ 1 − 1 4 c2c3e1 3 + c22e1e ∗ 1 + ... ) Thus, the error e2 in x2 is : e2 = ce31e ∗ 1, where c = −1 4c2c3 and higher powers of small quantities are neglected. Consequently, for n ≥ 1, we get: en+1 = ce3ne ∗ n. (29) To compute en+1 explicitly, the values of e∗n are needed. The value of e∗1 is already known. Like above, e∗2 in x∗2 can be found, and its value is given by e∗2 = de1e 2 2, B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2427 where d = c22 and, again, for n ≥ 2, the following formula can be obtained: e∗n = den−1e 2 n. (30) To investigate the convergence order of methods (23)-(25), the following form should be obtained: en+1 = Aepn. (31) Also, en = Aepn−1 or en−1 = A − 1 p e 1 p n . (32) From relations (29), (30) and (31), we have Aepn = en+1 = ce3ne ∗ n = ce3nden−1e 2 n = cde5nA − 1 p e 1 p Equating the power of en, we get p = 5 + 1 p or, p2 − 5p+ 1 = 0 ∴ p = 5± √ 29 2 Hence, the convergence order of method (23)-(25)is 5+ √ 29 2 ≈ 5.1925 Note: The above theorem can also be proved in the same line of alternative proof of Theorem 1. 3. Numerical Examples Here, we will be discussing some numerical examples to exhibit the performance of the newly developed methods. We will compare these methods with Newton’s method (1), McDougall and Wotherspoon (8-(9), midpoint Newton’s method (5), and double New- ton’s method (2). For the numerical computations, MATLAB software is used under the stopping criteria |xn+1 − xn| < 10−15 or |f(xn)| < 10−15. To determine the initial approximation, Intermediate Value Theorem or graphical method can be used. For the comparison, the following function has been used: (i) f1 = x3 + 4x2 − 10 (ii) f2 = x6 − x− 1 (iii) f3 = sin2 x− x2 + 1 (iv) f4 = xex 2 − sin2 x+ 3 cosx+ 5 (v) f5 = cosx− xex + x2 (vi) f6 = ex 2+7x−30 − 1 (vii) f7 = ex sinx+ ln(x2 + 1) (viii) f8 = (x− 2)23 − 1 where, NM,MWM, MNM, DNM denote Newton’s method, McDougall and Wother- spoon method, mid-point Newton’s method and double Newton’s method respectively. B. Sapkota, J. Jnawali / Eur. J. Pure Appl. Math, 16 (4) (2023), 2419-2430 2428 Table 2: Comparison of various methods with distinct initial approximation x0 . Function x0 Number of Iterations Root NM MWM MNM DNM Method Method (8)-(9) (23)-(25) 3 6 5 4 3 3 3 1.365230013414097 f1 1 5 4 3 3 3 3 - -1 24 32 9 12 5 7 - 3 10 9 7 5 5 5 1.34724138401519 f2 1 6 5 4 3 3 3 - 0 7 7 4 4 4 3 -0.778089598678601 10 8 7 5 4 4 4 1.404491648215341 f3 4 6 5 4 3 3 3 - -3 6 5 4 3 3 3 -1.404491648215341 1 8 7 Div 5 11 80 -1.207647827130919 f4 -2 9 8 6 5 5 5 - -1.5 7 6 5 4 4 4 - -1.5 7 5 5 4 4 4 0.639154096332008 f5 5 11 9 7 6 6 5 - 10 17 14 10 9 8 7 - 6 53 46 34 27 26 22 3 f6 5 35 29 22 18 17 15 - 4 19 16 12 10 10 8 - 0.5 6 5 4 3 3 3 0 f7 1 7 6 5 4 4 3 - 2 6 6 5 4 3 3 - 3.5 14 12 9 7 7 6 3 f8 4 21 17 13 11 10 9 - 4.5 26 21 16 13 13 11 - 4. 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