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Quadratically Convergent Algorithm for Computing Real Root of Non-Linear

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Quadratically Convergent Algorithm for Computing Real Root of Non-Linear Thota and Srivastav BMC Res Notes (2018) 11:909 https://doi.org/10.1186/s13104-018-4008-z BMC Research Notes RESEARCH NOTE Open Access Quadratically convergent algorithm for computing real root of non‑linear transcendental equations Srinivasarao Thota1* and Vivek Kumar Srivastav2 Abstract Objectives: The present paper describes a new algorithm to fnd a root of non-linear transcendental equations. It is found that Regula-Falsi method always gives guaranteed result but slow convergence. However, Newton–Raphson method does not give guaranteed result but faster than Regula-Falsi method. Therefore, the present paper used these two ideas and developed a new algorithm which has better convergence than Regula-Falsi and guaranteed result. One of the major issue in Newton–Raphson method is, it fails when frst derivative is zero or approximately zero. Results: The proposed method implemented the failure condition of Newton–Raphson method with better conver- gence. Error calculation has been discussed for certain real life examples using Bisection, Regula-Falsi, Newton–Raph- son method and new proposed method. The computed results show that the new proposed quadratically conver- gent method provides better convergence than other methods. Keywords: Root of transcendental equations, Regula-Falsi method, Newton–Raphson method, Quadratic convergence Mathematics Subject Classifcation: 65Hxx, 65H04 Introduction In Regula-Falsi method, two initial guesses are taken in Most of the real life-problems are non-linear in nature such a way that the corresponding function values have therefore it is a challenging task for the mathematician opposite signs. Ten these two points are connected and engineer to fnd the exact solution of such problems through the straight line and next approximation is the [1, 2]. In this reference, a number of methods have been point where this line intersect the x-axis. Tis method proposed/implemented in the last two decades [1, 3–8]. gives guaranteed result but slow convergence therefore Analytical solutions of such non-linear equations are very several researchers have improved this standard Regula- difcult, therefore only numerical method based iterative Falsi method into diferent hybrid models to speed up the techniques are the way to fnd approximate solution. In convergence [1, 3–5, 7, 10, 11, 15, 16]. Tus previously the literature, there are some numerical methods such as published works have revised/implemented Regula-Falsi Bisection, Secant, Regula-Falsi, Newton–Raphson, Mul- method in several ways to obtain better convergence. lers methods, etc., to calculate an approximate root of However, it is found that modifed form of Regual-Falsi the non-linear transcendental equations. It is well known method becomes more complicated from computational [1, 3–11, 14] that all the iterative methods require one or point of view. Terefore, in the present work Regual-Falsi more initial guesses for the initial approximations. method has been used as its standard form with New- ton–Raphson method and found better convergence. Newton–Raphson method is generally used to improve *Correspondence: [email protected]; [email protected] 1 Department of Applied Mathematics, School of Applied Natural the result obtained by one of the above methods. Tis Sciences, Adama Science and Technology University, Post Box No. 1888, method uses the concept of tangent at the initial approxi- Adama, Ethiopia mation point. Te next approximate root is taken those Full list of author information is available at the end of the article © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creat​iveco​mmons​.org/licen​ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creat​iveco​mmons​.org/ publi​cdoma​in/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Thota and Srivastav BMC Res Notes (2018) 11:909 Page 2 of 6 value where the tangent intersect the x-axis. So this Te generalization of this process is described in the fol- method fails where tangent is parallel to x-axis, i.e. the lowing section. derivative of the function is zero or approximately zero. Te order of convergence of Newton–Raphson method is Formulation of proposed algorithm two, therefore it converges very rapidly than other meth- Recall the Eqs. (1) and (2) in terms of iteration formu- ods (Bisection, Regula-Falsi, etc.). However it does not lae by replacing a, b, x by xn−1, xn+1, xn respectively, as always give guaranteed root. Many scientists and engi- follows neers have been proposed diferent hybrid models on xn−1f (xn+1) − xn+1f (xn−1) Newton–Raphson method [8, 9, 12–14, 17–22]. xn = f (xn+1) − f (xn+1) (3) It is clear from the survey [1–22], that the most of new algorithms are either based on three classical methods and namely Bisection, Regula-Falsi and Newton–Raphson f (x − ) or created by hybrid processes. In the present work, the x = x − n 1 , n n−1 ′ (4) proposed new algorithm is based on standard Regula- f (xn−1) Falsi and Newton–Raphson methods, which provides where n is the iteration number and |f (xn−1)| < |f (xn+1)| . guaranteed results and higher order convergence over Now the average of Eqs. (3) and (4) is Regula-Falsi method. Te new proposed algorithm will 1 xn−1f (xn+1) − xn+1f (xn−1) f (xn−1) work even the frst derivative equals to zero where New- x = + x 1 − n 2 n− ′ ton–Raphson method fails. f (xn+1) − f (xn−1) f (xn−1) Main text (5) After simplifcation of (5), we get Consider a continuous function f(x) between a and b such that f(a) and f(b) having opposite signs, consequently f (xn−1) 1 f (a) · f (b)<0 xn = xn− − ′ . Without loss of generality, assume that 2f (xn−1) f(a) is negative and f(b) is positive, and |f (a)| < |f (b)| , ′ f (xn−1) − f (xn+1) + (xn−1 − xn+1)(f (xn−1)) (6) hence at least one root lies between a and b. From Reg- . f (x 1) − f (x 1) ula-Falsi method, the frst approximate root can be calcu- n− n+ lated by using the formula Te value xn in Eq. (6) gives the iterative formula with |f (x )| < |f (x )| f ′(x ) = 0 af (b) − bf (a) n−1 n+1 . If n−1 , then Eq. (6) gives x = , undefned value, then we have to interchange the values f (b) − f (a) (1) xn−1 and xn+1. and, the frst approximate root by using Newton–Raph- Te following Teorem gives the generalization of son method is above formulation. f (a) x = a − , Lemma 1 Let f(x) be a continuous function and (a, b) be ′ (2) f (a) a sufciently small interval such that f (a)f (b)<0, and ′ ′ f (x) exists on [a, b]. Ten the approximation of a root of where f (a) indicates the derivative of f(x) at x = a. f(x) can be fnd using the iterative formula given in Eq. (6). Now, in the present proposed algorithm, we take the average of the iterations in Eqs. (1) and (2) as our frst x approximate root and follow the conditions given Steps for calculating a root below for further iterations: I. Select two initial approximations xn−1 and xn+1 • Choose two values a and b where the root exists as in such that product of the corresponding function Regula-Falsi method. values must be negative, i.e. f (xn−1)f (xn+1)<0. • Select the value such that the corresponding func- II. Now calculate xn using the formula given in Eq. (6). tion is closer to zero as a and the other one as b, i.e. Check f (xn) = 0 , if so, then xn is required root and |f (a)| < |f (b)|. ′ process stop. Otherwise we check the following • If frst derivative at a is zero (i.e., f (a) = 0 ) then possible conditions. interchange the values of a and b i.e., interchange (a, b) to (b, a). (i) For f (xn)f (xn−1)<0 , suppose |f (xn−1)| < |f (xn)| then xn replace by xn−1 and xn−1 replace by xn. Thota and Srivastav BMC Res Notes (2018) 11:909 Page 3 of 6 ′′ f (xn)f (xn+1)<0 |f (xn)| < f (β) (ii) For , suppose Putting 2f ′(β) = A (constant), then |f (xn+1)| then xn+1 replace by xn. ′ III. If f (xn−1) ≈ 0 then interchange xn−1 and xn+1. 2 2 en+2 = A enen+1 + en = Aenen+1 + Aen. (8) IV. Repeat steps I, II and III until we get required approximate solution. p p We have, |en+2|=c|en+1| , c > 0 ; |en+1|=c|en| ; and −1/p 1/p |en|=c |en+1| . Substituting in Eq. (8) and after Te implementation of the proposed algorithm in Mat- simplifcation, we get lab is also discussed (See, Additional fle 1). Tis algo- rithm would help to implement the manual calculations p 1+1/p 2 in commercial packages such as Maple, Mathematica, c1en+1 = c2en+1 + c3en SCILab, Singular,etc. ′ p+1 2 ′ = c2en + c3en, where c2 = c1c | |p ≤ ∗| |p+1 + ∗| |2 ∗ = ′ −1 ∗ = −1 Order of convergence en+1 c1 en c2 en , where c1 c2c1 , c2 c3c1 Te order of converges of any iterative method is defned = as If p 1 , then p |en+1|≤c|en| , (7) |e |≤c∗|e |2, where p is the order of convergence and c is a positive n+1 n ∗ ∗ ∗ fnite constant.
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