Research Article  Open Access
Grégory Antoni, "An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations", International Journal of Engineering Mathematics, vol. 2016, Article ID 8565821, 12 pages, 2016. https://doi.org/10.1155/2016/8565821
An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations
Abstract
This study concerns the development of a straightforward numerical technique associated with Classical Newton’s Method for providing a more accurate approximate solution of scalar nonlinear equations. The proposed procedure is based on some practical geometric rules and requires the knowledge of the local slope of the curve representing the considered nonlinear function. Therefore, this new technique uses, only as input data, the firstorder derivative of the nonlinear equation in question. The relevance of this numerical procedure is tested, evaluated, and discussed through some examples.
1. Introduction
The resolution of nonlinear problems is an issue frequently encountered in several scientific fields such as mathematics, physics, or many engineering branches, for example, mechanics of solids [1–8]. In most cases, these problems are governed by nonlinear equations not having any analytical solution. In this regard, the introduction of iterative methods is therefore needed in order to provide a numerical approximate solution associated with any type of nonlinear equation [9–23]. Among these iterative algorithms, Classical Newton’s Method (CNM) [24, 25] is one of the most used mainly for the following reasons: (i) the simplicity for numerical implementation in any scientific computation software; (ii) the only knowledge of the firstorder derivative of the considered function; (iii) the quadratic rate of convergence. In this paper, we propose a New Numerical Technique (NNT) based on geometric considerations which enable providing a more accurate approximate solution than that obtained by CNM. The present study is organized as follows: (i) in the first part, Section 2.1, we outline the scientific framework of this study, then, in second part, Section 2.2, we recall CNM including some convergence results, and, finally, in the third part, Section 2.3, we present NNT which uses only the firstorder derivative of the considered nonlinear equation in order to enhance the predictive abilities of CNM; (ii) in Section 3, the numerical relevance of the proposed procedure is addressed, assessed, and discussed on some specific examples.
2. A New Numerical Technique (NNT) Combined with Classical Newton’s Method (CNM)
2.1. Problem Statement
We consider scalarvalued nonlinear equation (with ) in the following form (see Figure 1):where denotes the class of infinitely differentiable functions in domain , is the simple solution (socalled “simple true zero” or “simple root”) on interval , that is, with .
(a)
(b)
2.2. Classical Newton’s Method (CNM)
2.2.1. Iterative Algorithm
For using Classical Newton’s Method (CNM) [24, 25], we consider only the firstorder term in Taylor series expansion associated with function (i.e., the linearization of the considered function):where denotes the firstorder derivative of functionat point.
The equation of tangent straight line passing at point associated with function is (; see Figure 2) as follows:where (resp.,) denotes th (resp., th) iteration associated with variable (with ).
(a)
(b)
Using (3), the linearization of (1) must check the following relation ( and ):
Based on (4),th iterative point provided by CNM is solution () such as
2.2.2. Convergence Results
Considering Taylor series expansion of functionand truncation error () between th iterative approximate solution () and true zero (), that is, with and , we havewithwhere denotes order derivative of function at point with , is factorial, represent the HigherOrder Terms which check: or , and are Landau notations associated with the asymptotic behavior of function , and is the square norm or Euclideannorm associated with the quantity (here, Euclidean distance reduces to absolute value , i.e., ).
Since that , (6)(7) can be written as follows (with):
The firstorder derivative associated with function at point (i.e., ) checks:
Combining (5), (8)(9), and leads to the following:
An iterative numerical scheme has th order of convergence with associated convergence rate to number (i.e., “true zero”) if there exists and [25] such aswhere is the absolutevalue function of variable(such as when , when , and when ) and “” is the limit operator. It is important to emphasize that (i) if and then the convergence is linear (or “firstorder” type, e.g., bisection or false position method); (ii) ifandthen the convergence is sublinear; (iii) ifand (or) then the convergence is superlinear (e.g., secant method); (iv) if(with) then the convergence is quadratic (or “secondorder” type, e.g., CNM); (v) if (with) then the convergence is cubic (or “thirdorder” type, e.g., TMNM; see Section 3.1). th order of convergence means that the number of significant digits isfold at each iteration, for example, in the case of CNM (resp., TMNM), where (resp.,), the number of exact decimals doubles (resp., triples) at each iteration .
Using (10)(11), we can see that CNM has quadratic convergence () with a rate of convergence [24]:
2.3. New Numerical Technique (NNT)
2.3.1. Proposed Iterative Procedure
In this section, we propose a New Numerical Technique (NNT) for improving the accuracy associated with the approximate solution provided by CNM (see Section 2.2, which is used for finding the roots of scalar nonlinear equations. NNT is based on some practical geometric rules and requires only the determination of the local slope of the curve representing the considered nonlinear equation taking into account the firstorder derivation (see [26–28]).
Here, we present the main steps associated with the development of NNT:(i)We consider normal straight line associated with the curve representing nonlinear equation at point (see [26, 27] and Figure 2):(ii)We introduce straight line having direction vector which depends on the sum of direction vectors associated with normal and tangent straight lines passing by point (with ; see Figure 2), that is: with where and are the components of direction vector associated with straight line in any orthonormal basis .(iii)Using CNM, we define th iterative point (with ; see (5) and also Figure 2):(iv)We introduce also straight line in the following form (with ; see Figure 2): with(v)Combining (14)(15) and (17)(18), we adopt the solution of the equation (with ; see Figure 2), that is: According to (16) and (19), th iterative point associated with NNT (see Figure 2) can be rewritten (with ):
The iterative numerical scheme associated with NNT is coupled with CNM and therefore th iterative solution can be written as follows (with, ; see Figure 2):with the different conditions associated with the iterative solution (with ):(i)First condition [A1]: and and .(ii)Second condition [A2]:
2.3.2. Convergence Analysis
Similar to that in Section 2.2.2, we analyze the convergence associated with NNT which is presented in Section 2.2.
Using (9) leads to the following:
In line with (22), we have the following:
On the other hand, we have the following:
Combining (8) and (24) leads to the following:
According to (19) and using (23) and (25), it holds that
In line with (11) and (26), we can see that the rate of convergenceassociated with NNT isand the order of convergence is of lineartype (i.e., ) if and quadratictype (i.e., ) if .
By taking (12) and (27), the rate of convergenceof NNT combined with CNM is
It is important to emphasize that the associated convergence order is lineartype () if with condition A (i.e., [A1] or [A2]) and quadratictype () if with condition (i.e., [A1] or [A2]) and elsewhere with .
3. Numerical Examples
3.1. Some Preliminary Remarks
In this section, we propose to test, evaluate, and analyze the iterative numerical procedure presented in Section 2.3 on some particular examples. This New Numerical Technique (NNT) is coupled with Classical Newton’s Method (CNM) in order to provide a more accurate approximate solution associated with scalar nonlinear equations. The numerical predictions obtained by combining both NNT and CNM are compared with those provided by Thirdorder Modified Newton’s Method (TMNM) [22, 26]. All numerical results presented here have been made with Matlab software (see [25, 29–32]).
For stopping the iterative process associated with each considered algorithm, we adopt three coupled types of Convergence Criterion (CC):where denotes the maximum number of iterations and (resp.) is the tolerance parameter associated with the residue (resp., approximation) error criterion. Here, the considered values for each CC are , , and .
The iterative numerical scheme associated with TMNM (see [22, 26]) iswhere denotes the secondorder derivative of function at point . It should be noted that order of convergence is cubic (i.e., ) and rate of convergence is (see [22, 26])
3.2. Examples
We consider the following scalar nonlinear equations:where , , and represent exponential, natural logarithm, and cosinus functions.
The approximate numerical solution of roots associated with functions , , and are, respectively,
3.3. Results and Discussion
All the numerical results associated with scalar nonlinear functions to are presented in Tables 1–6 and also in Figures 3–20. For each scalar nonlinear function (with ), we test two guest points for starting iterative procedure: (i) in the case of function , we adopt the firstguest point (resp., secondguest point ) and we show different approximate solutions (with ) and also the evolution of both residue error and approximation error provided by Classical Newton’s Method (CNM; see Section 2.2), CNM coupled to the proposed New Numerical Technique (NNT) with both conditions [A1] and [A2] (CNM + NNT; see Section 2.3), and Thirdorder Modified Newton’s Method (TMNM; see Section 3.1) in Table 1 (resp., Table 2) and Figures 3–5 (resp., Figures 6–8); (ii) in the case of function , we adopt firstguest point (resp., secondguest point ) and we show different approximate solutions (with ) provided by CNM, CNM + NNT with both conditions [A1] and [A2], and TMNM in Table 3 (resp., Table 4) and Figures 9–11 (resp., Figures 12–14); (iii) in the case of function, we adopt firstguest point (resp., secondguest point ) and we show different approximate solutions (with ) provided by CNM, CNM + NNT with both conditions [A1] and [A2], and TMNM in Table 5 (resp., Table 6) and Figures 15–17 (resp., Figures 18–20). The obtained numerical results emphasize clearly that NNT combined with CNM is able to provide in the vast majority of cases a better approximate solution than that supplied only by CNM. In addition, on all results presented, condition [A2] seems to be more convenient than condition [A1]. For summary, NNT presents the main advantages of being a procedure: (i) “accurate” in terms of achieved approximate solution; (ii) “straightforward” to implement in computation software.






4. Concluding Comments
This study is devoted to a New Numerical Technique (NNT) to improve the accuracy of approximate solution provided by Classical Newton’s Method (CNM) and afford to have better numerical evaluation of the roots associated with the scalar nonlinear equations. As in CNM, this NNT requires only the determination of the firstorder derivative of the nonlinear function under consideration. The predictive capabilities associated with NNT are shown on some examples.
Competing Interests
The author declares that they have no competing interests.
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Copyright
Copyright © 2016 Grégory Antoni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.