R. Thukral
Padé Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire, LS17 5JS, England
Correspondence to: R. Thukral, Padé Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire, LS17 5JS, England.
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Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
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Abstract
The aim of this paper is to present further improvement of the Simpsontype methods for finding zeros of a nonlinear equation. The new Simpsontype methods are shown to converge of the same order four and five, but with better precision of the zeros. In terms of computational cost, the new iterative methods require four evaluations of functions per iteration and therefore the new methods have an efficiency index better than the classical Simpson method. It is proved that the new methods have a convergence of order four and five. We examine the effectiveness of the new Simpsontype methods by approximating the simple root of a given nonlinear equation. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods and thus verifies the theoretical results.
Keywords:
Simpsontype methods, Newton method, Simple root, Nonlinear equation, Rootfinding, Order of convergence
Cite this paper: R. Thukral, Further Improvement of Simpsontype Methods for Solving Nonlinear Equations, American Journal of Computational and Applied Mathematics , Vol. 9 No. 3, 2019, pp. 5761. doi: 10.5923/j.ajcam.20190903.02.
1. Introduction
In this paper, we present an alternative new fourth and fifthorder iterative methods to find a simple root of the nonlinear equation. It is well known that the techniques to solve nonlinear equations have many applications in Mathematics and applied science. Two wellknown techniques, namely the classical Newton method and the classical Simpson method, for their simplicity with convergence order of two and three respectively [114], are applied to construct the new Simpsontype methods. It already has been established that the new Simpsontype method requires same amount of evaluations of the function as the classical Simpson method and it has been established that the new Simpsontype iterative methods have a better efficiency index than the classical Simpson method [1113]. The prime motive for the development of the new Simpsontype methods was to establish an alternative scheme and demonstrate exceptional convergence speed of the proposed methods.The remaining sections of the paper are organized as follows. Some basic definitions and construction of the new Simpsontype methods are discussed in section 2. In section 3, we prove the order of convergence of the new Simpsontype methods. Finally, in section 4, numerical comparisons are made to demonstrate the performance of the presented methods.
2. Construction of the New Iterative Methods
2.1. Preliminaries
In order to establish the order of convergence of the new Simpsontype methods iterative method, we use the following definitions [3, 9, 11, 14].Definition 1 Let be a realvalued function with a root and let be a sequence of real numbers that converge towards The order of convergence p is given by  (1) 
where and is the asymptotic error constant.Definition 2 Let be the error in the kth iteration, then the relation  (2) 
is the error equation. If the error equation exists, then p is the order of convergence of the iterative method. Definition 3 Let r be the number of function evaluations of the method. The efficiency of the method is measured by the concept of efficiency index and defined as  (3) 
where p is the order of convergence of the method [3].Definition 4 Suppose that and are three successive iterations closer to the root of (1). Then the computational order of convergence may be approximated by  (4) 
where [11].Before we define the new fourth and fifthorder Simpsontype methods, we state essentially the classical thirdorder Simpson method and the recently introduced fourth and fifthorder Simpsontype methods [12, 13].
2.2. The Classical Simpson Thirdorder Method
Since this method is well established [1, 2, 46, 8, 1013], we shall state the essential expressions used in order to calculate the root of the given nonlinear equation. Hence the Simpson thirdorder method is given as  (5) 
where  (6) 
 (7) 
is the initial approximation and provided that the denominator of (5) is not equal to zero.
2.3. The Simpsontype Fourthorder Methods
In this section we state the recently introduced fourthorder Simpsontype method [12] to find simple root of a nonlinear equation. The improvement of the classical Simpson method was made by introducing a new factor in (5). Hence, the improved fourthorder Simpsontype method is given by  (8) 
 (9) 
This is actually a simplified version of fourthorder Simpsontype method given in [12].
2.4. New Simpsontype Fourthorder Methods
In this section we construct an alternative fourthorder Simpsontype method for finding simple root of a nonlinear equation. The improvement of the classical Simpson method was made by introducing a new factor in denominator of (5). Hence, the improved fourthorder Simpsontype method is given by  (10) 
where  (11) 
2.5. The Simpsontype Fifthorder Method
The fifthorder Simpsontype method for finding simple root of a nonlinear equation has been presented in [13]. The order of convergence of the classical Simpson method and the fourthorder Simpsontype method is increased by introducing two different type of factors in (5). Hence, the fifthorder Simpsontype method is given by  (12) 
where is given by (9) and  (13) 
Here also we present a simplified version of fifthorder Simpsontype method given in [13].
2.6. New Simpsontype Fifthorder Method
Here we present an alternative to fifthorder Simpsontype method (12) for finding simple root of a nonlinear equation. Here also, we improve the performance of (12) by introducing two new different type of factors in the denominator of (5). Hence, the new fifthorder Simpsontype method is given by  (14) 
where is given by (11) and  (15) 
3. Convergence Analysis
In this section, we prove the order of convergence of the new fourth and fifthorder Simpsontype methods, given by (10) and (14) respectively.Theorem 1Let be a simple zero of a sufficiently differentiable function for an open interval D. If the initial guess is sufficiently close to then the order of convergence of the new Simpsontype methods defined by (10) and (14) is four and five, respectively.Proof Let be a simple root of , i.e. and , and the error is expressed as  (16) 
Using the Taylor series expansion and taking into account , we have  (17) 
 (18) 
where  (19) 
Dividing (17) by (18), we get  (20) 
and hence, we have  (21) 
The expansion of about is given as  (22) 
Since from (7) we have  (23) 
Taylor expansion of about is  (24) 
The denominator of (5) is given as  (25) 
Dividing the numerator by denominator of (5), we get  (26) 
It is well established that the error equation of (5) is  (27) 
The improvement factor introduced in fourthorder method (8) is  (28) 
and the error equation yields  (29) 
Furthermore, the factor introduced in the fifthorder (12) is  (30) 
and the error equation yields  (31) 
For the purpose of this paper, a new different factor is introduced in denominator of (5) and a new fourthorder Simpsontype method is obtained (10). The new factor is given as  (32) 
and the error equation yields  (33) 
The new fourthorder Simpsontype method is improved by introducing another factor in (10) and we obtain a new fifthorder Simpsontype method (14). The new factor is given by  (34) 
and the error equation yields  (35) 
The error equations (33) and (35) establishes the new fourthorder Simpsontype method and the new fifthorder Simpsontype method defined by (10) and (14) respectively.
4. Application of the New Simpsontype Iterative Methods
In this section, numerical results on some test functions are compared for the new Simpsontype methods (10) and (14) with the established Simpsontype methods. It is apparent that the efficiency index of the new methods (10) and (14) is similar to (8) and (12) respectively. The efficiency index is calculated by the formula (3). Hence, the efficiency index of the new Simpsontype methods given by (14) and (12) is whereas the efficiency index of the fourthorder Simpsontype methods given by (10) and (8) is and the efficiency index of the classical Simpson thirdorder method is given by (5) is To demonstrate the performance of the new iterative methods, we display the difference between the simple root and the approximation for test functions with initial guess . In fact, is calculated by using the same total number of function evaluations for all Simpsontype methods.Numerical example 1We will demonstrate the performance of the Simpsontype methods for the following nonlinear equation  (36) 
the exact value of the simple root of (36) is In Table 1 the errors obtained by the Simpsontype methods described, is based on the initial value . We observe that all the Simpsontype methods are converging to the expected order.Table 1. Errors occurring in the estimates of the root of (36) by the methods described 
 

Numerical example 2We will demonstrate the performance of the Simpsontype methods for the following nonlinear equation  (37) 
the exact value of the simple root of (37) is In Table 2 the errors obtained by the Simpsontype methods described, is based on the initial value . We observe that the new Simpsontype methods are converging to the expected order.Table 2. Errors occurring in the estimates of the root of (37) by the methods described 
 

5. Remarks and Conclusions
In this paper, we have demonstrated the performance of the new Simpsontype methods, namely the Simpsontype fifthorder method and the Simpsontype fourthorder method. The prime motive of the development of the new Simpsontype methods was to establish an alternative iterative method than the classical Simpson thirdorder iterative method and recently introduced the Simpsontype fourth and fifthorder iterative methods. We have examined the effectiveness of the new Simpsontype methods by showing the accuracy of the simple root of a nonlinear equation. It is observed from tables that the proposed methods (10) and (14) have a better performance as compared with recently introduced Simpsontype methods given (8) and (12) respectively. Also, it is wellestablished that the efficiency index of the new fifthorder methods is much better than the fourthorder Simpsontype methods and the classical Simpson thirdorder method. Numerical comparisons are made to demonstrate the performance of the derived method. Finally, empirically we have found that the approximate solution of the new fourth and fifthorder Simpsontype methods are to be substantially more accurate than the established fourth and fifthorder Simpsontype methods.
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