An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations

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.


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][2][3][4][5][6][7][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][10][11][12][13][14][15][16][17][18][19][20][21][22][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.
Figure 2: Schematic diagram with the specific entities used by NNT combined with CNM applied on monotonically increasing (a) and decreasing (b) nonlinear function .

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 function  at point .
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 ⃗  + ⃗ V associated with normal ( ⃗ ) and tangent ( ⃗ V) straight lines passing by point (  , (  )) (with ∀  ∈ ; see Figure 2), that is: with where (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 ( + 1)th iterative solution  +1 ∈  can be written as follows (with, ∀  ,   ∈ ; see Figure 2): with the different conditions associated with the iterative solution [A] (with ∀  ,   ,   ∈ ):

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 Approximative solution (x k In line with (11) and ( 26), we can see that the rate of convergence  associated with NNT is , and the order of convergence is of linear-type (i.e.,  = 1) if 0 <  NNT ≤ 1 and quadratic-type (i.e.,  = 2) if  NNT = 0.By taking ( 12) and ( 27), the rate of convergence  NNT-CNM of NNT combined with CNM is It is important to emphasize that the associated convergence order is linear-type ( = 1) if 0 <  NNT ≤ 1 with condition [A] (i.e., [A1] or [A2]) and quadratic-type ( = 2) if  NNT = 0 with condition [A] (i.e., [A1] or [A2]) and elsewhere with  CNM ̸ = 0.  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][30][31][32]).where  max denotes the maximum number of iterations and  re (resp.,  ae ) is the tolerance parameter associated with the residue (resp., approximation) error criterion.Here, the considered values for each CC are  max = 10,  re = 10 −10 , and  ae = 10 −10 .

International Journal of Engineering Mathematics
The iterative numerical scheme associated with TMNM (see [22,26]) is International Journal of Engineering Mathematics where   (  ) denotes the second-order derivative of function  at point   .It should be noted that order of convergence  is cubic (i.e.,  = 3) and rate of convergence  is (see [22,26])

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.

Figure 1 :
Figure 1: Schematic diagram associated with the problem under consideration: monotonically increasing (a) and decreasing (b) nonlinear function  with a simple root  on interval [, ].

Figure 3 :
Figure 3: Evolution of approximate solution   compared to th iteration for scalar nonlinear equation  1 (when guest point  0 = 3) 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 :Figure 5 :
Figure 4: Evolution of residue error |(  )| compared to th iteration for scalar nonlinear equation  1 (when guest point  0 = 3) 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 6 :
Figure 6: Evolution of approximate solution   compared to th iteration for scalar nonlinear equation  1 (when guest point  0 = 10 −3 ) 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 :
Figure 7: Evolution of residue error |(  )| compared to th iteration for scalar nonlinear equation  1 (when guest point  0 = 10 −3 ) 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 10 :Figure 11 :
Figure 10: Evolution of residue error |(  )| compared to th iteration for scalar nonlinear equation  2 (when guest point  0 = 10) 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 12 :Figure 13 :
Figure 12: Evolution of approximate solution   compared to th iteration for scalar nonlinear equation  2 (when guest point  0 = −4) 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 18 :
Figure 18: Evolution of approximate solutions   compared to th iteration for scalar nonlinear equation  2 (when guest point  0 = −1) 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 :
Figure 19: Evolution of residue error |(  )| compared to th iteration for scalar nonlinear equation  3 (when guest point  0 = −1) 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).