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 first-order 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 first-order 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 first-order 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 scalar-valued nonlinear equation (with ) in the following form (see Figure 1):where denotes the class of infinitely differentiable functions in domain , is the simple solution (so-called “simple true zero” or “simple root”) on interval , that is, with .

(a)
(a)
(b)
(b)
Figure 1 
Schematic diagram associated with the problem under consideration: monotonically increasing (a) and decreasing (b) nonlinear functionwith a simple rooton interval.
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 first-order term in Taylor series expansion associated with function (i.e., the linearization of the considered function):where denotes the first-order 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)
(a)
(b)
(b)
Figure 2 
Schematic diagram with the specific entities used by NNT combined with CNM applied on monotonically increasing (a) and decreasing (b) nonlinear function.

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 Higher-Order Terms which check: or , and are Landau notations associated with the asymptotic behavior of function , and is the square norm or Euclidean-norm associated with the quantity (here, Euclidean distance reduces to absolute value , i.e., ).

Since that , (6)-(7) can be written as follows (with):

The first-order 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 absolute-value 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 “first-order” 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 “second-order” type, e.g., CNM); (v) if (with) then the convergence is cubic (or “third-order” type, e.g., TMNM; see Section 3.1). th order of convergence means that the number of significant digits is-fold 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 first-order 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 linear-type (i.e., ) if and quadratic-type (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 linear-type () if with condition A (i.e., [A1] or [A2]) and quadratic-type () 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 Third-order 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 second-order 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 first-guest point (resp., second-guest 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 Third-order 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 first-guest point (resp., second-guest 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 first-guest point (resp., second-guest 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.

Table 1 
Approximate solution of scalar nonlinear equation (when guest point = 3) obtained by Classical Newton’s Method (CNM) (cf. Section 2.2), Classical Newton’s Method coupled with New Numerical Technique (CNM + NNT) for conditions [A1] and [A2] (cf. Section 2.3), and Third-order Modified Newton’s Method (TMNM) (cf. Section 3.1). Notations: () (resp., ()) denotes that approximate solution is provided using (21a) (resp., (21b)) in Section 2.3.1.
Table 2 
Approximate solution of scalar nonlinear equation (when guest point = 10−3) obtained by Classical Newton’s Method (CNM) (cf. Section 2.2), Classical Newton’s Method coupled with New Numerical Technique (CNM + NNT) for conditions [A1] and [A2] (cf. Section 2.3), and Third-order Modified Newton’s Method (TMNM) (cf. Section 3.1). Notations: () (resp., ()) denotes that approximate solution is provided using (21a) (resp., (21b)) in Section 2.3.1.
Table 3 
Approximate solution of scalar nonlinear equation (when guest point = 10) obtained by Classical Newton’s Method (CNM) (cf. Section 2.2), Classical Newton’s Method coupled with New Numerical Technique (CNM + NNT) for conditions [A1] and [A2] (cf. Section 2.3), and Third-order Modified Newton’s Method (TMNM) (cf. Section 3.1). Notations: () (resp., ()) denotes that approximate solution is provided using (21a) (resp., (21b)) in Section 2.3.1.
Table 4 
Approximate solution of scalar nonlinear equation (when guest point = −4) obtained by Classical Newton’s Method (CNM) (cf. Section 2.2), Classical Newton’s Method coupled with New Numerical Technique (CNM + NNT) for conditions [A1] and [A2] (cf. Section 2.3), and Third-order Modified Newton’s Method (TMNM) (cf. Section 3.1). Notations: () (resp., ()) denotes that approximate solution is provided using (21a) (resp., (21b)) in Section 2.3.1.
Table 5 
Approximate solution of scalar nonlinear equation (when guest point = 10) obtained by Classical Newton’s Method (CNM) (cf. Section 2.2), Classical Newton’s Method coupled with New Numerical Technique (CNM + NNT) for conditions [A1] and [A2] (cf. Section 2.3), and Third-order Modified Newton’s Method (TMNM) (cf. Section 3.1). Notations: () (resp., ()) denotes that approximate solution is provided using (21a) (resp., (21b)) in Section 2.3.1.
Table 6 
Approximate solution of scalar nonlinear equation (when guest point = −1) obtained by Classical Newton’s Method (CNM) (cf. Section 2.2), Classical Newton’s Method coupled with New Numerical Technique (CNM + NNT) for conditions [A1] and [A2] (cf. Section 2.3), and Third-order Modified Newton’s Method (TMNM) (cf. Section 3.1). Notations: () (resp., ()) denotes that approximate solution is provided using (21a) (resp., (21b)) in Section 2.3.1.

Figure 3 
Evolution of approximate solution compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with circles), TMNM (green solid line with circles), and CNM + NNT with the condition [A1] (blue solid line with circles) and condition [A2] (red solid line with circles).

Figure 4 
Evolution of residue error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with diamonds), TMNM (green solid line with diamonds), and CNM + NNT with condition [A1] (blue solid line with diamonds) and condition [A2] (red solid line with diamonds).

Figure 5 
Evolution of approximation error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with squares), TMNM (green solid line with squares), and CNM + NNT with condition [A1] (blue solid line with squares) and condition [A2] (red solid line with squares).

Figure 6 
Evolution of approximate solution compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with circles), TMNM (green solid line with circles), and CNM + NNT with condition [A1] (blue solid line with circles) and condition [A2] (red solid line with circles).

Figure 7 
Evolution of residue error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with diamonds), TMNM (green solid line with diamonds), and CNM + NNT with condition [A1] (blue solid line with diamonds) and condition [A2] (red solid line with diamonds).

Figure 8 
Evolution of approximation error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with squares), TMNM (green solid line with squares), and CNM + NNT with condition [A1] (blue solid line with squares) and condition [A2] (red solid line with squares).

Figure 9 
Evolution of approximate solution compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with circles), TMNM (green solid line with circles), and CNM + NNT with condition [A1] (blue solid line with circles) and condition [A2] (red solid line with circles).

Figure 10 
Evolution of residue error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with diamonds), TMNM (green solid line with diamonds), and CNM + NNT with condition [A1] (blue solid line with diamonds) and condition [A2] (red solid line with diamonds).

Figure 11 
Evolution of approximation error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with squares), TMNM (green solid line with squares), and CNM + NNT with condition [A1] (blue solid line with squares) and condition [A2] (red solid line with squares).

Figure 12 
Evolution of approximate solution compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with circles), TMNM (green solid line with circles), and CNM + NNT with condition [A1] (blue solid line with circles) and condition [A2] (red solid line with circles).

Figure 13 
Evolution of residue error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with diamonds), TMNM (green solid line with diamonds), amd CNM + NNT with condition [A1] (blue solid line with diamonds) and condition [A2] (red solid line with diamonds).

Figure 14 
Evolution of approximation error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with squares), TMNM (green solid line with squares), and CNM + NNT with condition [A1] (blue solid line with squares) and condition [A2] (red solid line with squares).

Figure 15 
Evolution of approximate solution compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with circles), TMNM (green solid line with circles), and CNM + NNT with condition [A1] (blue solid line with circles) and condition [A2] (red solid line with circles).

Figure 16 
Evolution of residue error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with diamonds), TMNM (green solid line with diamonds), and CNM + NNT with condition [A1] (blue solid line with diamonds) and condition [A2] (red solid line with diamonds).

Figure 17 
Evolution of approximation error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with squares), TMNM (green solid line with squares), and CNM + NNT with condition [A1] (blue solid line with squares) and condition [A2] (red solid line with squares).

Figure 18 
Evolution of approximate solutions compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with circles), TMNM (green solid line with circles), and CNM + NNT with condition [A1] (blue solid line with circles) and condition [A2] (red solid line with circles).

Figure 19 
Evolution of residue error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with diamonds), TMNM (green solid line with diamonds), and CNM + NNT with condition [A1] (blue solid line with diamonds) and condition [A2] (red solid line with diamonds).

Figure 20 
Evolution of approximation error compared to th iteration for scalar nonlinear equation (when guest point ) obtained by CNM (black solid line with squares), TMNM (green solid line with squares), and CNM + NNT with condition [A1] (blue solid line with squares) and condition [A2] (red solid line with squares).

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 first-order 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.