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A derivativefree rootfinding algorithm using exponential method and its implementation
BMC Research Notes volume 16, Article number: 276 (2023)
Abstract
Objective
In this paper, we develop a new rootfinding algorithm to solve the given nonlinear equations. The proposed rootfinding algorithm is based on the exponential method. This algorithm is derivativefree and converges fast.
Results
Several numerical examples are presented to illustrate and validation of the proposed methods. Microsoft Excel and Maple implementation of the proposed algorithm is presented with sample computations.
Introduction
Finding an approximate root of nonlinear equations using iterative algorithms plays a significant role in the computational and applied mathematics. The applications of nonlinear equations of the type \(f(x) = 0\) arise in various branches of scientific computing fields. Solving such nonlinear equations is one of the most important problems and frequently appearing in different scientific fields that can be modeled through nonlinear equations. In recent time, several researchers, engineers and scientists focused on solving nonlinear equations numerically as well as analytically. Iterative algorithms play a vital role in finding the solution of such nonlinear problems. In general, the roots of nonlinear or transcendental equations cannot be expressed in closed form or cannot be computed analytically. The rootfinding algorithms provide us to compute approximations to the roots; these approximations are expressed either as small isolating intervals or as floating point numbers. There are various numerical algorithm/methods available in the literature, see for example [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], for more details.
Many new modified/hybrid/multistep iterative algorithms are developed in the last few years, by employing various mathematical algorithms/techniques. Noor et al. discussed the fifthorder second derivativefree algorithm in 2007, see [21], by using the finite difference scheme. GrauSanchez et al. presented a fifthorder ChebyshevHalley type method in 2008, see [22]. Zhanlav et al. proposed a three step fifthorder iterative algorithm in 2010 [23]. Nazeer et al. introduced a novel second derivativefree Householder’s method having fifthorder convergence by using finitedifference scheme [24] in 2016. Recently, in 2021, Amir et al. developed an efficient and derivativefree algorithm for determining an approximate solution of the given nonlinear scalar equations by applying forward and finitedifference schemes similar to Traub’s method, see [25]. In this paper, we propose a new root finding algorithm, which is derivativefree, using exponential method. To propose the algorithm with derivativefree, we employ the forward difference scheme and finite difference scheme. This gives computationally low cost. Microsoft Excel and Maple implementations of the proposed algorithm are presented. Maple and Excel implementations with sample computations for differential and transcendental equations are available in the literature, see for example [18, 19, 26, 27] and there are various techniques for different type of applications, see [20, 28,29,30,31,32,33,34,35], the references cited therein.
Preliminaries
In this paper, we consider the nonlinear equation of the type
Iterations techniques are a common approach widely used in various numerical algorithms/methods. It is a hope that an iteration in the general form of x_{n+1} = g(x_{n}) will eventually converge to the true solution α of the problem (1) at the limit when n → ∞. The concern is whether this iteration will converge, and, if so, the rate of convergence. Specifically we use the following expression to represent how quickly the error e_{n} = α − x_{n} converges to zero. Let e_{n} = α − x_{n} and e_{n+1} = α − x_{n+1} for n ≥ 0 be the errors at nth and (n + 1)th iterations respectively. If two positive constants µ and p exist, and
then the sequence is said to converge to α. Here p ≥ 1 is called the order of convergence; the constant µ is the rate of convergence or asymptotic error constant. This expression may be better understood when it is interpreted as e_{n+1}= µe_{n}^{p} when n → ∞. Obviously, the larger p and the smaller µ, the more quickly the sequence converges.
Theorem 1
[16, 36] Suppose that \(g \in C^{p} [a,b]\). If g^{(k)}(x) = 0, for k = 0, 1, 2,..., p − 1 and \(g^{(p)} (x) \ne 0\), then the sequence {x_{n}} is of order p.
This paper focuses on developing iterative algorithm having fourthorder of convergence. The following section presents the proposed algorithm using Newton–Raphson method and exponential method without computing the derivative.
Main text (a new iterative algorithm)
We assume that α is an exact root of the Eq. (1) and let a be an initial approximation (sufficiently close) to α. In the exponential method, we can find first approximation root using the following formula. See [5] for more details.
If x_{n+1} is the required root, then the exponential formula can be expressed as, for n = 0, 1, 2, 3,...,
which has more than secondorder convergence.
Suppose y_{n} = x_{n+1}, where x_{n+1} is the Newton–Raphson formula, is predictor and corrector, then Traub [37] created a new twostep iterative algorithm as follows, n = 0, 1, 2, 3,...,
It is shown in [37] that the Traub’s method has fourthorder convergence. Since Newton–Raphson formula repeated twice, the Traub’s method includes four computations to execute the algorithm. Amir et al. extended the Traub’s method to derivativefree algorithm by applying forward and finitedifference schemes on Traub’s method.
In this paper, we propose a new twostep iterative algorithm similar to that of Amir et al., and the proposed algorithm has more than fourthorder convergence. The proposed method is created using the exponential method designed by Thota et al. [5]. Using exponential method, one can obtain an approximate root of a given nonlinear equation using the formula (3). The order of convergence of the exponentiation method is more than two, see [5] for more details. Using exponential method (3), the proposed algorithm consists of the following steps:
One can observe that, this is a twostep iteration method to calculate roots of a given nonlinear equations. Since there are two steps in the algorithm and it required four evaluations for its execution. The biggest disadvantage of the algorithm (4) is computational cost of each iteration which is more. In order to reduce the high computational cost, we replace the first derivative by approximation and this suggests a novel derivativefree algorithm. Hence, it can be applied easily to the given non linear equations where the first derivative is not defined in the domain. We use the forward difference approximation in the predictor to approximate the first derivative as follows, here f (x_{n}) ≥ 0,
Now, we use finite difference approximation in the corrector step (i.e., in step 2) as follows
Substituting the Eqs. (5)–(6) in algorithm (4), we obtain a new efficient and derivativefree iterative algorithm to calculate the approximate solution of a given nonlinear equation as follows
where g(x_{n}) and h(x_{n}, y_{n}) are as given (5)–(6). This is a new iterative algorithm to find a root of transcendental equations in twostep without involvement of any derivative. One of the advantages of the proposed algorithm is existence of root where the first derivative does not exist at some particular points in the domain, and another big advance is the computational complexity. This method has more than fourth order convergence and its convergence analysis is presented in the following section.
Analysis of convergence
In this section, we show in the following theorem that the order of converges of the proposed algorithm is five. Let I ⊂ R be an open interval. To prove this, we follow the proofs of ([2], Theorem 5, Theorem 6) or ([16], Theorem 2, Theorem 3, Theorem 4).
Theorem 2
Let f: I → R. Suppose α ∈ I is a simple root of (1) and θ is a sufficiently small neighborhood of α. Then the iterative formula (7) produces a sequence of iterations {x_{n}: n = 1, 2,...} with order of convergence four.
Proof
Let.
where
Since α is a root of f (x), hence f (α) = 0. One can compute that
Hence the Algorithm (7) has fourthorder convergence, by Theorem 1. \(\square\)
One can also verify that the order of convergence of the proposed algorithm as in the following example.
Example 1
Consider the following equation.
It has a root α = − 1. We show, as discussed in proof of Theorem 2, that the proposed algorithm has fourthorder convergence. Following Theorem 2, we have
where
Now
Hence, by Theorem 2, the algorithm in (7) has fourthorder convergence.
Numerical examples
Example 2
Consider a transcendental equation \(e^{x} + \cos (x)  1 = 0\) with \({\text{x}}\_0 =  2\). Now we can compute a real of the given equation using the proposed algorithm (7) as follows.
Suppose \(f(x) = e^{x} + \cos (x)  1,\) then we have \(g(x_{0} ) = \frac{{f(x_{0} + f(x_{0} ))  f(x_{0} )}}{{f(x_{0} )}} = 0.5246013002,\) \(y_{0} = x_{0} \exp \left( {\frac{{  f(x_{0} )}}{{x_{0} g(x_{0} )}}} \right) =  0.5900190724,\) \(h(x_{0} ,y_{0} ) = \frac{{f(y_{0} )  f(x_{0} )}}{{y_{0}  x_{0} }} = 1.181617637\) and, \(x_{1} = y_{0} \exp \left( {\frac{{  f(y_{0} )}}{{y_{0} h(x_{0} ,y_{0} )}}} \right) =  1.025295284.\)
Similarly, we have the values in iteration 2:
Iteration 3:
One can obtain the function value at x_{3} = − 0.9236326590 as f (− 0.9236326590) = − 5.3608 × 10^{−11}. Hence the required root x = − 0.9236326590 is obtained in 3 iterations using the proposed algorithm.
Example 3
Consider a polynomial equation to find a real root.
with x_{0} = 0.6. Following Example 2 using the proposed algorithm (7), we have Iteration 1:
Other iterations values are
Hence, the required approximate root of the given equation (9) is x = 1.929846242848.
Implementation of the proposed algorithm
Implementation in MS Excel
The proposed method can be computed in Excel easily as follows. The number of iterations n, initial guess \( x_{n} ,f\left( {x_{n} } \right),g\left( {x_{n} } \right),y_{n} ,f\left( {y_{n} } \right),h\left( {x_{n} ,{\text{ }}y_{n} } \right),x_{{n + 1}} \;{\text{and}}\;f\left( {x_{{n + 1}} } \right) \) are placed in Excel cells, for example, in A5, B5, C5, D5, E5, F5, G5, H5 and I5 respectively. Enter the respective values in 6th row, i.e., n = 0, x_{0}, “= f(B6)”, “= (f(B6 + C6) − C6)/C6”, “= B6*EXP((− C6)/(B6 * D6))”, “= f(E6)”, “= (F6 − C6)/(E6 − B6)”, “= E6 * EXP((− F6)/(E6 * G6))” and “= f(H6)” respectively in A6–I6. For sec ond iteration, we need to replace x_{n} by x_{n+1} in B7 using the command “= H6”. The last columns, C6–I6, are drag down to get next iteration value. Finally, drag down the entire 8th row, A7–I7, until the required number of iterations, see Fig. 1. Sample computations using MS Excel are presented in the following section.
Example 4
Consider the Eq. (9) presented in Example 3 for sample computations using MS Excel.
with x_{0} = 0.6. Following the procedure in Section, we have the results as in Fig. 1.
Implementation in Maple
Example 5
Consider the equation e^{x} + cos(x) − 1 = 0 given in Example 2 for sample computations in Maple.
Conclusion
In this paper, we proposed a new rootfinding algorithm to solve the nonlinear equations. The main idea of this algorithm is based on the exponential method. The proposed algorithm doesn’t have any derivative even though exponential method is involved, and moreover it converges fast. Numerical examples are presented to illustrate and validation of the proposed methods. Implementation of the proposed algorithm in Excel and Maple is discussed with sample computations.
Limitations
In this paper, we focused on MS Excel and Maple implementation. However, the proposed algorithms can be implemented in many mathematical software tools such as Mathematica, SCILab, Matlab, etc.
Availability of data and materials
The datasets generated and analyzed during the current study are presented in this manuscript.
Abbreviations
 MS Excel:

Microsoft Excel
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The authors are thankful to the reviewers and editor for providing valuable inputs to improve the quality and present format of manuscript.
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Dr. ST involved in creation of the proposed algorithms and Maple implementation. Dr. MMA is involved in suggestion and verification of the numerical examples in the present paper. Dr. PS is involved in modification and improvement of the numerical examples. Ms. LR is involved in verification of the numerical examples. All authors read and approved the final manuscript.
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Thota, S., Awad, M.M., Shanmugasundaram, P. et al. A derivativefree rootfinding algorithm using exponential method and its implementation. BMC Res Notes 16, 276 (2023). https://doi.org/10.1186/s13104023065541
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DOI: https://doi.org/10.1186/s13104023065541
Keywords
 Iterative algorithms
 Nonlinear equations
 Exponential method
 Algorithm implementation
 Rootfinding algorithms