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

Authors: Grégory Antoni
Journal: International Journal of Engineering Mathematics
 (2016)

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## 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.


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## Body

## 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 equationf:x∈Ω⊂R→f(x)∈R (with f∈C∞(Ω)) in the following form (see Figure 1):(1)fx=0,where C∞(Ω) denotes the class of infinitely differentiable functions in domain Ω, δ is the simple solution (so-called “simple true zero” or “simple root”) on interval I=[a,b]∈Ω⊂R, that is, f(δ)=0 with δ∈I.Figure 1
Schematic diagram associated with the problem under consideration: monotonically increasing (a) and decreasing (b) nonlinear functionfwith a simple rootδon interval[a,b].
(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 first-order term in Taylor series expansion associated with function f (i.e., the linearization of the considered function):(2)fx≈fp+f′px-p,∀x,p∈I,where f′(p) denotes the first-order derivative of functionfat pointp.The equation of tangent straight lineT passing at point p=xk∈I associated with function f is (∀x,y∈I; see Figure 2) as follows:(3)Tx=f′xkx-xk+fxk,∀k=0,…,K,where †k (resp.,†k+1) denotes kth (resp., (k+1)th) iteration associated with variable † (with K∈N).Figure 2
Schematic diagram with the specific entities used by NNT combined with CNM applied on monotonically increasing (a) and decreasing (b) nonlinear functionf.
(a) (b)Using (3), the linearization of (1) must check the following relation (∀x,y∈I and ∀xk∈I):(4)Tx=0⟺f′xkx-xk+fxk=0,∀k=0,…,K.Based on (4),(k+1)th iterative point xk+1∈I provided by CNM is solution x=xk+1 (∀x∈I) such as(5)xk+1=xk-fxkf′xk,∀k=0,…,K,∀f′xk≠0.


#### 2.2.2. Convergence Results

Considering Taylor series expansion of functionfand truncation error (ek) between kth iterative approximate solution (xk) and true zero (δ), that is, ek=xk-δ(with ∀xk∈I and δ∈I), we have(6)fxk=fδ+ek=fδ+c1ek+c2ek2+⋯+H.O.Twith(7)cn=fnδn!,∀k=0,…,K,where f[n](δ) denotes n-order derivative of function f at point δ(with n∈N), n! is n-factorial, H.O.T represent the Higher-Order Terms which check: H.O.T≡o(ekn) or H.O.T≡O(ekn+1), o(·) and O(·) are Landau notations associated with the asymptotic behavior of function f, and ⋆ is the square norm or Euclidean-norm associated with the quantity ⋆ (here, Euclidean distance reduces to absolute value ⋆, i.e., ⋆≡⋆).Since thatf(δ)=0, (6)-(7) can be written as follows (with∀xk∈I):(8)fxk=c1ek+c2ek2+⋯+H.O.T,∀k=0,…,K.The first-order derivative associated with functionf at point xk∈I (i.e., f′(xk)) checks:(9)f′xk=c1+2c2ek+3c3ek2+⋯+H.O.T,∀k=0,…,K.Combining (5), (8)-(9), and xk=δ+ek leads to the following:(10)ek+1=xk+1-δ=c2c1ek2+⋯+H.O.T,∀k=0,…,K.An iterative numerical scheme hasθth order of convergence with associated convergence rate τ to number δ (i.e., “true zero”) if there exists θ≥1 and τ≥0 [25] such as(11)limk→+∞⁡ek+1ekθ=τ,∀k=0,…,K,where |φ| is the absolute-value function of variableφ(such as|φ|=-φ when φ<0, φ=0 when φ=0, and φ=φ when φ>0) and “lim” is the limit operator. It is important to emphasize that (i) ifθ=1 and τ∈0,1 then the convergence is linear (or “first-order” type, e.g., bisection or false position method); (ii) ifθ=1andτ=1then the convergence is sublinear; (iii) ifθ=1andτ=0 (orθ>1,∀τ∈0,1) then the convergence is superlinear (e.g., secant method); (iv) ifθ=2(withτ∈0,1) then the convergence is quadratic (or “second-order” type, e.g., CNM); (v) ifθ=3 (withτ∈0,1) then the convergence is cubic (or “third-order” type, e.g., TMNM; see Section 3.1). θth order of convergence means that the number of significant digits isθ-fold at each iterationk, for example, in the case of CNM (resp., TMNM), where θ=2 (resp.,θ=3), the number of exact decimals doubles (resp., triples) at each iteration k.Using (10)-(11), we can see that CNM has quadratic convergence (θ=2) with a rate of convergence [24]:(12)τ=τCNM=c2c1>0,∀c1,c2≠0.


### 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 Section2.2, 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 lineN associated with the curve representing nonlinear equation f at point xk∈I (see [26, 27] and Figure 2):(13)Nx=-1f′xkx-xk+fxk,∀k=0,…,K,∀f′xk≠0.(ii)
We introduce straight lineH having direction vector w→ which depends on the sum of direction vectors u→+v→ associated with normal (u→) and tangent (v→) straight lines passing by point (xk,f(xk)) (with ∀xk∈I; see Figure 2), that is:(14)Hx=12f′xk-1f′xkx-xk+fxk,∀k=0,…,K,∀f′xk≠0,∀f′xk≠1with(15)w1=u1+v1=f′xk-1f′xk,w2=u2+v2=2,wherew1=u1+v1 and w2=u2+v2 are the components of direction vector w→ associated with straight line H in any orthonormal basis {e→1,e→2}.(iii)
Using CNM, we definekth iterative point zk∈I (with ∀x∈I; see (5) and also Figure 2):(16)zk=xk-fxkf′xk,∀k=0,…,K,∀f′xk≠0.(iv)
We introduce also straight lineD in the following form (with ∀x∈I; see Figure 2):(17)Dx=-1Mkx-zkwith(18)Mk=12f′xk-1f′xk,∀k=0,…,K,∀f′xk≠0,∀f′xk≠1.(v)
Combining (14)-(15) and (17)-(18), we adopt the solution x=qk∈I of the equation H(x)=D(x) (with ∀x∈I; see Figure 2), that is:(19)qk=1Mk+1/MkzkMk+Mkxk-fxk,∀k=0,…,K,∀Mk≠0.According to (16) and (19), kth iterative point qk∈I associated with NNT (see Figure 2) can be rewritten (with ∀xk∈I):(20)qk=xk-fxkMk+1/Mk1+1Mkf′xk,∀k=0,…,K,∀f′xk≠0,∀Mk≠0.The iterative numerical scheme associated with NNT is coupled with CNM and therefore(k+1)th iterative solution xk+1∈I can be written as follows (with, ∀qk,zk∈I; see Figure 2):(21a)xk+1=qkifA,(21b)xk+1=zkelse,∀k=0,…,Kwith the different conditions associated with the iterative solution [A] (with ∀qk,zk,xk∈I):(i)
First condition [A1]:sgn⁡qk=sgn⁡xk and sgn⁡f(qk)=sgn⁡f(xk) and f(qk)<f(zk).(ii)
Second condition [A2]:f(qk)<f(zk).


#### 2.3.2. Convergence Analysis

Similar to that in Section2.2.2, we analyze the convergence associated with NNT which is presented in Section 2.2.Using (9) leads to the following:(22)Mk=12f′xk-1f′xk=12c1-1c1+c2ek+32c3ek2+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.In line with (22), we have the following:(23)Mk+1Mk=12c1-1c1+2c1-1/c1+c2ek+32c3ek2+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.On the other hand, we have the following:(24)zkMk=2c1-1/c1δ+c2c1ek2+H.O.T,∀c1≠0,∀c1≠1,Mkxk=12c1-1c1δ+12c1-1c1+c2δek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.Combining (8) and (24) leads to the following:(25)zkMk+Mkxk-fxk=2c1-1/c1+12c1-1c1δ+12c1-1c1+c2δ-c1ek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.According to (19) and using (23) and (25), it holds that(26)ek+1=xk+1-δ=11/2c1-1/c1+2/c1-1/c112c1-1c1+c2δ-c1ek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.In line with (11) and (26), we can see that the rate of convergenceτassociated with NNT is(27)τ=τNNT=1/2c1-1/c1+c2δ-c11/2c1-1/c1+2/c1-1/c1,∀c1≠0,∀c1≠1and the order of convergence is of linear-type (i.e., θ=1) if 0<τNNT≤1 and quadratic-type (i.e., θ=2) if τNNT=0.By taking (12) and (27), the rate of convergenceτNNT-CNMof NNT combined with CNM is(28)τNNT-CNM=τNNTifAτCNMelse.It is important to emphasize that the associated convergence order is linear-type (θ=1) if 0<τNNT≤1 with condition [A] (i.e., [A1] or [A2]) and quadratic-type (θ=2) if τNNT=0 with condition [A] (i.e., [A1] or [A2]) and elsewhere with τCNM≠0.


## 2.1. Problem Statement

We consider scalar-valued nonlinear equationf:x∈Ω⊂R→f(x)∈R (with f∈C∞(Ω)) in the following form (see Figure 1):(1)fx=0,where C∞(Ω) denotes the class of infinitely differentiable functions in domain Ω, δ is the simple solution (so-called “simple true zero” or “simple root”) on interval I=[a,b]∈Ω⊂R, that is, f(δ)=0 with δ∈I.Figure 1
Schematic diagram associated with the problem under consideration: monotonically increasing (a) and decreasing (b) nonlinear functionfwith a simple rootδon interval[a,b].
(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 first-order term in Taylor series expansion associated with function f (i.e., the linearization of the considered function):(2)fx≈fp+f′px-p,∀x,p∈I,where f′(p) denotes the first-order derivative of functionfat pointp.The equation of tangent straight lineT passing at point p=xk∈I associated with function f is (∀x,y∈I; see Figure 2) as follows:(3)Tx=f′xkx-xk+fxk,∀k=0,…,K,where †k (resp.,†k+1) denotes kth (resp., (k+1)th) iteration associated with variable † (with K∈N).Figure 2
Schematic diagram with the specific entities used by NNT combined with CNM applied on monotonically increasing (a) and decreasing (b) nonlinear functionf.
(a) (b)Using (3), the linearization of (1) must check the following relation (∀x,y∈I and ∀xk∈I):(4)Tx=0⟺f′xkx-xk+fxk=0,∀k=0,…,K.Based on (4),(k+1)th iterative point xk+1∈I provided by CNM is solution x=xk+1 (∀x∈I) such as(5)xk+1=xk-fxkf′xk,∀k=0,…,K,∀f′xk≠0.


### 2.2.2. Convergence Results

Considering Taylor series expansion of functionfand truncation error (ek) between kth iterative approximate solution (xk) and true zero (δ), that is, ek=xk-δ(with ∀xk∈I and δ∈I), we have(6)fxk=fδ+ek=fδ+c1ek+c2ek2+⋯+H.O.Twith(7)cn=fnδn!,∀k=0,…,K,where f[n](δ) denotes n-order derivative of function f at point δ(with n∈N), n! is n-factorial, H.O.T represent the Higher-Order Terms which check: H.O.T≡o(ekn) or H.O.T≡O(ekn+1), o(·) and O(·) are Landau notations associated with the asymptotic behavior of function f, and ⋆ is the square norm or Euclidean-norm associated with the quantity ⋆ (here, Euclidean distance reduces to absolute value ⋆, i.e., ⋆≡⋆).Since thatf(δ)=0, (6)-(7) can be written as follows (with∀xk∈I):(8)fxk=c1ek+c2ek2+⋯+H.O.T,∀k=0,…,K.The first-order derivative associated with functionf at point xk∈I (i.e., f′(xk)) checks:(9)f′xk=c1+2c2ek+3c3ek2+⋯+H.O.T,∀k=0,…,K.Combining (5), (8)-(9), and xk=δ+ek leads to the following:(10)ek+1=xk+1-δ=c2c1ek2+⋯+H.O.T,∀k=0,…,K.An iterative numerical scheme hasθth order of convergence with associated convergence rate τ to number δ (i.e., “true zero”) if there exists θ≥1 and τ≥0 [25] such as(11)limk→+∞⁡ek+1ekθ=τ,∀k=0,…,K,where |φ| is the absolute-value function of variableφ(such as|φ|=-φ when φ<0, φ=0 when φ=0, and φ=φ when φ>0) and “lim” is the limit operator. It is important to emphasize that (i) ifθ=1 and τ∈0,1 then the convergence is linear (or “first-order” type, e.g., bisection or false position method); (ii) ifθ=1andτ=1then the convergence is sublinear; (iii) ifθ=1andτ=0 (orθ>1,∀τ∈0,1) then the convergence is superlinear (e.g., secant method); (iv) ifθ=2(withτ∈0,1) then the convergence is quadratic (or “second-order” type, e.g., CNM); (v) ifθ=3 (withτ∈0,1) then the convergence is cubic (or “third-order” type, e.g., TMNM; see Section 3.1). θth order of convergence means that the number of significant digits isθ-fold at each iterationk, for example, in the case of CNM (resp., TMNM), where θ=2 (resp.,θ=3), the number of exact decimals doubles (resp., triples) at each iteration k.Using (10)-(11), we can see that CNM has quadratic convergence (θ=2) with a rate of convergence [24]:(12)τ=τCNM=c2c1>0,∀c1,c2≠0.


## 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 f (i.e., the linearization of the considered function):(2)fx≈fp+f′px-p,∀x,p∈I,where f′(p) denotes the first-order derivative of functionfat pointp.The equation of tangent straight lineT passing at point p=xk∈I associated with function f is (∀x,y∈I; see Figure 2) as follows:(3)Tx=f′xkx-xk+fxk,∀k=0,…,K,where †k (resp.,†k+1) denotes kth (resp., (k+1)th) iteration associated with variable † (with K∈N).Figure 2
Schematic diagram with the specific entities used by NNT combined with CNM applied on monotonically increasing (a) and decreasing (b) nonlinear functionf.
(a) (b)Using (3), the linearization of (1) must check the following relation (∀x,y∈I and ∀xk∈I):(4)Tx=0⟺f′xkx-xk+fxk=0,∀k=0,…,K.Based on (4),(k+1)th iterative point xk+1∈I provided by CNM is solution x=xk+1 (∀x∈I) such as(5)xk+1=xk-fxkf′xk,∀k=0,…,K,∀f′xk≠0.


## 2.2.2. Convergence Results

Considering Taylor series expansion of functionfand truncation error (ek) between kth iterative approximate solution (xk) and true zero (δ), that is, ek=xk-δ(with ∀xk∈I and δ∈I), we have(6)fxk=fδ+ek=fδ+c1ek+c2ek2+⋯+H.O.Twith(7)cn=fnδn!,∀k=0,…,K,where f[n](δ) denotes n-order derivative of function f at point δ(with n∈N), n! is n-factorial, H.O.T represent the Higher-Order Terms which check: H.O.T≡o(ekn) or H.O.T≡O(ekn+1), o(·) and O(·) are Landau notations associated with the asymptotic behavior of function f, and ⋆ is the square norm or Euclidean-norm associated with the quantity ⋆ (here, Euclidean distance reduces to absolute value ⋆, i.e., ⋆≡⋆).Since thatf(δ)=0, (6)-(7) can be written as follows (with∀xk∈I):(8)fxk=c1ek+c2ek2+⋯+H.O.T,∀k=0,…,K.The first-order derivative associated with functionf at point xk∈I (i.e., f′(xk)) checks:(9)f′xk=c1+2c2ek+3c3ek2+⋯+H.O.T,∀k=0,…,K.Combining (5), (8)-(9), and xk=δ+ek leads to the following:(10)ek+1=xk+1-δ=c2c1ek2+⋯+H.O.T,∀k=0,…,K.An iterative numerical scheme hasθth order of convergence with associated convergence rate τ to number δ (i.e., “true zero”) if there exists θ≥1 and τ≥0 [25] such as(11)limk→+∞⁡ek+1ekθ=τ,∀k=0,…,K,where |φ| is the absolute-value function of variableφ(such as|φ|=-φ when φ<0, φ=0 when φ=0, and φ=φ when φ>0) and “lim” is the limit operator. It is important to emphasize that (i) ifθ=1 and τ∈0,1 then the convergence is linear (or “first-order” type, e.g., bisection or false position method); (ii) ifθ=1andτ=1then the convergence is sublinear; (iii) ifθ=1andτ=0 (orθ>1,∀τ∈0,1) then the convergence is superlinear (e.g., secant method); (iv) ifθ=2(withτ∈0,1) then the convergence is quadratic (or “second-order” type, e.g., CNM); (v) ifθ=3 (withτ∈0,1) then the convergence is cubic (or “third-order” type, e.g., TMNM; see Section 3.1). θth order of convergence means that the number of significant digits isθ-fold at each iterationk, for example, in the case of CNM (resp., TMNM), where θ=2 (resp.,θ=3), the number of exact decimals doubles (resp., triples) at each iteration k.Using (10)-(11), we can see that CNM has quadratic convergence (θ=2) with a rate of convergence [24]:(12)τ=τCNM=c2c1>0,∀c1,c2≠0.


## 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 Section2.2, 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 lineN associated with the curve representing nonlinear equation f at point xk∈I (see [26, 27] and Figure 2):(13)Nx=-1f′xkx-xk+fxk,∀k=0,…,K,∀f′xk≠0.(ii)
We introduce straight lineH having direction vector w→ which depends on the sum of direction vectors u→+v→ associated with normal (u→) and tangent (v→) straight lines passing by point (xk,f(xk)) (with ∀xk∈I; see Figure 2), that is:(14)Hx=12f′xk-1f′xkx-xk+fxk,∀k=0,…,K,∀f′xk≠0,∀f′xk≠1with(15)w1=u1+v1=f′xk-1f′xk,w2=u2+v2=2,wherew1=u1+v1 and w2=u2+v2 are the components of direction vector w→ associated with straight line H in any orthonormal basis {e→1,e→2}.(iii)
Using CNM, we definekth iterative point zk∈I (with ∀x∈I; see (5) and also Figure 2):(16)zk=xk-fxkf′xk,∀k=0,…,K,∀f′xk≠0.(iv)
We introduce also straight lineD in the following form (with ∀x∈I; see Figure 2):(17)Dx=-1Mkx-zkwith(18)Mk=12f′xk-1f′xk,∀k=0,…,K,∀f′xk≠0,∀f′xk≠1.(v)
Combining (14)-(15) and (17)-(18), we adopt the solution x=qk∈I of the equation H(x)=D(x) (with ∀x∈I; see Figure 2), that is:(19)qk=1Mk+1/MkzkMk+Mkxk-fxk,∀k=0,…,K,∀Mk≠0.According to (16) and (19), kth iterative point qk∈I associated with NNT (see Figure 2) can be rewritten (with ∀xk∈I):(20)qk=xk-fxkMk+1/Mk1+1Mkf′xk,∀k=0,…,K,∀f′xk≠0,∀Mk≠0.The iterative numerical scheme associated with NNT is coupled with CNM and therefore(k+1)th iterative solution xk+1∈I can be written as follows (with, ∀qk,zk∈I; see Figure 2):(21a)xk+1=qkifA,(21b)xk+1=zkelse,∀k=0,…,Kwith the different conditions associated with the iterative solution [A] (with ∀qk,zk,xk∈I):(i)
First condition [A1]:sgn⁡qk=sgn⁡xk and sgn⁡f(qk)=sgn⁡f(xk) and f(qk)<f(zk).(ii)
Second condition [A2]:f(qk)<f(zk).


### 2.3.2. Convergence Analysis

Similar to that in Section2.2.2, we analyze the convergence associated with NNT which is presented in Section 2.2.Using (9) leads to the following:(22)Mk=12f′xk-1f′xk=12c1-1c1+c2ek+32c3ek2+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.In line with (22), we have the following:(23)Mk+1Mk=12c1-1c1+2c1-1/c1+c2ek+32c3ek2+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.On the other hand, we have the following:(24)zkMk=2c1-1/c1δ+c2c1ek2+H.O.T,∀c1≠0,∀c1≠1,Mkxk=12c1-1c1δ+12c1-1c1+c2δek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.Combining (8) and (24) leads to the following:(25)zkMk+Mkxk-fxk=2c1-1/c1+12c1-1c1δ+12c1-1c1+c2δ-c1ek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.According to (19) and using (23) and (25), it holds that(26)ek+1=xk+1-δ=11/2c1-1/c1+2/c1-1/c112c1-1c1+c2δ-c1ek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.In line with (11) and (26), we can see that the rate of convergenceτassociated with NNT is(27)τ=τNNT=1/2c1-1/c1+c2δ-c11/2c1-1/c1+2/c1-1/c1,∀c1≠0,∀c1≠1and the order of convergence is of linear-type (i.e., θ=1) if 0<τNNT≤1 and quadratic-type (i.e., θ=2) if τNNT=0.By taking (12) and (27), the rate of convergenceτNNT-CNMof NNT combined with CNM is(28)τNNT-CNM=τNNTifAτCNMelse.It is important to emphasize that the associated convergence order is linear-type (θ=1) if 0<τNNT≤1 with condition [A] (i.e., [A1] or [A2]) and quadratic-type (θ=2) if τNNT=0 with condition [A] (i.e., [A1] or [A2]) and elsewhere with τCNM≠0.


## 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 Section2.2, 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 lineN associated with the curve representing nonlinear equation f at point xk∈I (see [26, 27] and Figure 2):(13)Nx=-1f′xkx-xk+fxk,∀k=0,…,K,∀f′xk≠0.(ii)
We introduce straight lineH having direction vector w→ which depends on the sum of direction vectors u→+v→ associated with normal (u→) and tangent (v→) straight lines passing by point (xk,f(xk)) (with ∀xk∈I; see Figure 2), that is:(14)Hx=12f′xk-1f′xkx-xk+fxk,∀k=0,…,K,∀f′xk≠0,∀f′xk≠1with(15)w1=u1+v1=f′xk-1f′xk,w2=u2+v2=2,wherew1=u1+v1 and w2=u2+v2 are the components of direction vector w→ associated with straight line H in any orthonormal basis {e→1,e→2}.(iii)
Using CNM, we definekth iterative point zk∈I (with ∀x∈I; see (5) and also Figure 2):(16)zk=xk-fxkf′xk,∀k=0,…,K,∀f′xk≠0.(iv)
We introduce also straight lineD in the following form (with ∀x∈I; see Figure 2):(17)Dx=-1Mkx-zkwith(18)Mk=12f′xk-1f′xk,∀k=0,…,K,∀f′xk≠0,∀f′xk≠1.(v)
Combining (14)-(15) and (17)-(18), we adopt the solution x=qk∈I of the equation H(x)=D(x) (with ∀x∈I; see Figure 2), that is:(19)qk=1Mk+1/MkzkMk+Mkxk-fxk,∀k=0,…,K,∀Mk≠0.According to (16) and (19), kth iterative point qk∈I associated with NNT (see Figure 2) can be rewritten (with ∀xk∈I):(20)qk=xk-fxkMk+1/Mk1+1Mkf′xk,∀k=0,…,K,∀f′xk≠0,∀Mk≠0.The iterative numerical scheme associated with NNT is coupled with CNM and therefore(k+1)th iterative solution xk+1∈I can be written as follows (with, ∀qk,zk∈I; see Figure 2):(21a)xk+1=qkifA,(21b)xk+1=zkelse,∀k=0,…,Kwith the different conditions associated with the iterative solution [A] (with ∀qk,zk,xk∈I):(i)
First condition [A1]:sgn⁡qk=sgn⁡xk and sgn⁡f(qk)=sgn⁡f(xk) and f(qk)<f(zk).(ii)
Second condition [A2]:f(qk)<f(zk).


## 2.3.2. Convergence Analysis

Similar to that in Section2.2.2, we analyze the convergence associated with NNT which is presented in Section 2.2.Using (9) leads to the following:(22)Mk=12f′xk-1f′xk=12c1-1c1+c2ek+32c3ek2+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.In line with (22), we have the following:(23)Mk+1Mk=12c1-1c1+2c1-1/c1+c2ek+32c3ek2+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.On the other hand, we have the following:(24)zkMk=2c1-1/c1δ+c2c1ek2+H.O.T,∀c1≠0,∀c1≠1,Mkxk=12c1-1c1δ+12c1-1c1+c2δek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.Combining (8) and (24) leads to the following:(25)zkMk+Mkxk-fxk=2c1-1/c1+12c1-1c1δ+12c1-1c1+c2δ-c1ek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.According to (19) and using (23) and (25), it holds that(26)ek+1=xk+1-δ=11/2c1-1/c1+2/c1-1/c112c1-1c1+c2δ-c1ek+⋯+H.O.T,∀k=0,…,K,∀c1≠0,∀c1≠1.In line with (11) and (26), we can see that the rate of convergenceτassociated with NNT is(27)τ=τNNT=1/2c1-1/c1+c2δ-c11/2c1-1/c1+2/c1-1/c1,∀c1≠0,∀c1≠1and the order of convergence is of linear-type (i.e., θ=1) if 0<τNNT≤1 and quadratic-type (i.e., θ=2) if τNNT=0.By taking (12) and (27), the rate of convergenceτNNT-CNMof NNT combined with CNM is(28)τNNT-CNM=τNNTifAτCNMelse.It is important to emphasize that the associated convergence order is linear-type (θ=1) if 0<τNNT≤1 with condition [A] (i.e., [A1] or [A2]) and quadratic-type (θ=2) if τNNT=0 with condition [A] (i.e., [A1] or [A2]) and elsewhere with τCNM≠0.


## 3. Numerical Examples

### 3.1. Some Preliminary Remarks

In this section, we propose to test, evaluate, and analyze the iterative numerical procedure presented in Section2.3 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):(29)(CC1):m≤Kmax,(CC2):fxk+1≤ϵre,(CC3):xk+1-xk≤ϵae,∀k=0,…,K,where Kmax denotes the maximum number of iterations and ϵre (resp.,ϵae) is the tolerance parameter associated with the residue (resp., approximation) error criterion. Here, the considered values for each CC are Kmax=10, ϵre=10-10, and ϵae=10-10.The iterative numerical scheme associated with TMNM (see [22, 26]) is(30)xk+1=xk-fxkf′xk-f′′xkfxk22f′xk3,∀k=0,…,K,∀f′xk≠0,where f′′(xk) denotes the second-order derivative of function f at point xk. It should be noted that order of convergence θ is cubic (i.e., θ=3) and rate of convergence τ is (see [22, 26])(31)τ=τTMNM=2c22-c3c1>0,∀c1≠0.


### 3.2. Examples

We consider the following scalar nonlinear equations:(32)f1x=5x4+3ln⁡x-1=0,f2x=3x3+5exp⁡x-10=0,f3x=5exp⁡x+x3cos⁡x-20=0,where exp⁡(·), ln⁡(·), and cos⁡(·) represent exponential, natural logarithm, and cosinus functions.The approximate numerical solution of roots associated with functionsf1, f2, and f3 are, respectively,(33)δ1≈0.7720250256⋯,δ2≈0.6192449540⋯,δ3≈1.3595986211⋯.


### 3.3. Results and Discussion

All the numerical results associated with scalar nonlinear functionsf1 to f3 are presented in Tables 1–6 and also in Figures 3–20. For each scalar nonlinear function fj (with j=1,2,3), we test two guest points x0 for starting iterative procedure: (i) in the case of function f1, we adopt the first-guest point x0=3 (resp., second-guest point x0=10-3) and we show different approximate solutions xk (with k=1,…,10) and also the evolution of both residue error fxk+1 and approximation error xk+1-xk 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 f2, we adopt first-guest point x0=10 (resp., second-guest point x0=-4) and we show different approximate solutions xk (with k=1,…,10) 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 functionf3, we adopt first-guest point x0=10 (resp., second-guest point x0=-1) and we show different approximate solutions xk (with k=1,…,10) 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 solutionxk of scalar nonlinear equation f1 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 3.0000000000 3.0000000000 3.0000000000 3.0000000000 1 2.2471426305 1.4942904056 (∗∗) 1.4942904056 (∗∗) 1.9644439737 2 1.6823826593 1.1286469200 (∗) 0.7631581663 (∗∗) 1.2884809024 3 1.2337388302 0.8906377962 (∗) 0.7721173336 (∗) 0.8906149334 4 0.9723200229 0.7878458749 (∗) 0.7720250356 (∗) 0.7746164630 5 0.8146707520 0.7723183427 (∗) 0.7720250256 (∗) 0.7720250503 6 0.7741430851 0.7720251266 (∗)  0.7720250256 7 0.7720302909 0.7720250256 (∗) 8 0.7720250256 9 0.7720250256 10Table 2
Approximate solutionxk of scalar nonlinear equation f2 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 0.0010000000 0.0010000000 0.0010000000 0.0010000000 1 0.0082410886 0.0154821756 (∗∗) 0.0154821756 (∗∗) 0.0344577770 2 0.0505339927 0.1548613133 (∗∗) 0.1548613133 (∗∗) 0.3980194000 3 0.2182202963 0.4938846484 (∗) 0.8311197179 (∗∗) 0.9031891893 4 0.6163033790 0.8261543800 (∗) 0.7760693275 (∗) 0.7755098327 5 0.7975366151 0.7754244611 (∗) 0.7720442199 (∗) 0.7720250858 6 0.7727863680 0.7720385875 (∗) 0.7720250260 (∗) 0.7720250256 7 0.7720257060 0.7720250258 (∗) 0.7720250256 (∗) 8 0.7720250256 0.7720250256 (∗) 9 10Table 3
Approximate solutionxk of scalar nonlinear equation f2 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 10.000000000 10.000000000 10.000000000 10.000000000 1 8.9811766535 7.9623533071 (∗∗) 7.9623533071 (∗∗) 8.4655416573 2 7.9456740785 5.8372149905 (∗∗) 5.8372149905 (∗∗) 6.8799380466 3 6.8825672128 3.5600762341 (∗∗) 3.5600762341 (∗∗) 5.2048513181 4 5.7803883041 1.4821734803 (∗∗) 1.4821734803 (∗∗) 3.4865001565 5 4.6404493806 0.9609028311 (∗) 0.4402289741 (∗∗) 2.0197617679 6 3.5056070182 0.6928062817 (∗) 0.6483081663 (∗) 1.0685736902 7 2.4744259573 0.6233620739 (∗) 0.6199084714 (∗) 0.6632122684 8 1.6461067325 0.6192585092 (∗) 0.6192453069 (∗) 0.6193191836 9 1.0635449602 0.6192449541 (∗) 0.6192449540 (∗) 0.6192449540 10 0.7354406002 0.6192449510 (∗)  0.6192449540Table 4
Approximate solutionxk of scalar nonlinear equation f2 (when guest point x0 = −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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 −4.0000000000 −4.0000000000 −4.0000000000 −4.0000000000 1 −2.5987493208 −1.1976336149 (∗∗) −1.1976336149 (∗∗) −2.1088093909 2 −1.5803235380 −0.2513579056 (∗) 0.6858578709 (∗∗) −0.7930763263 3 −0.6950023819 1.1303884977 (∗) 0.6226379112 (∗) 1.4022343921 4 0.5489290102 0.7665675348 (∗) 0.6192541648 (∗) 0.7698587570 5 0.6234126286 0.6349082108 (∗) 0.6192449540 (∗) 0.6217981372 6 0.6192588436 0.6194395336 (∗) 0.6192449540 (∗) 0.6192449694 7 0.6192449541 0.6192449843 (∗)  0.6192449540 8 0.6192449540 0.6192449540 (∗)  0.6192449540 9 10Table 5
Approximate solutionxk of scalar nonlinear equation f3 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 10.000000000 10.000000000 10.000000000 10.000000000 1 9.0104267607 8.0208535216 (∗∗) 8.0208535216 (∗∗) 8.5171506533 2 8.0146859130 5.9615544966 (∗∗) 5.9615544966 (∗∗) 6.9859301990 3 6.9847503720 3.9493415483 (∗∗) 3.9493415483 (∗∗) 5.3851997390 4 5.9189110435 2.4996523082 (∗∗) 2.4996523082 (∗∗) 4.0778432768 5 4.9180876708 1.7226271036 (∗) 0.9467658294 (∗∗) 3.0067861343 6 4.0738875311 1.3881795157 (∗) 1.4273487664 (∗) 1.8098899052 7 3.3404588095 1.3598140013 (∗) 1.3607723321 (∗) 1.3648044141 8 2.6091363141 1.3595986336 (∗) 1.3595989925 (∗) 1.3595986468 9 1.8194405679 1.3595986211 (∗) 1.3595986211 (∗) 1.3595986211 10 1.4045600984Table 6
Approximate solutionxk of scalar nonlinear equation f3 (when guest point x0 = −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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 −1.0000000000 −1.0000000000 −1.0000000000 −1.0000000000 1 6.1409303723 6.1409303723 (∗) 6.1409303723 (∗) 14.5343055115 2 5.1148208484 4.0887116616 (∗∗) 4.0887116616 (∗∗) 13.0337347682 3 4.2379703013 2.6194618995 (∗∗) 2.6194618995 (∗∗) 11.5318478151 4 3.4892766889 1.8291595323 (∗) 1.0397597354 (∗∗) 10.0370297219 5 2.7676801225 1.4064394475 (∗) 1.3966765848 (∗) 8.5556896133 6 1.9776897481 1.3601688047 (∗) 1.3599586541 (∗) 7.0203746257 7 1.4417009128 1.3595987088 (∗) 1.3595986561 (∗) 5.3974911835 8 1.3613036994 1.3595986211 (∗) 1.3595986211 (∗) 4.1321703085 9 1.3595994046   3.1258556869 10 1.3595986211   1.9485648236Figure 3
Evolution of approximate solutionxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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
Evolution of residue errorfxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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 5
Evolution of approximation errorxk+1-xk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=3) 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 solutionxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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
Evolution of residue errorfxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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 8
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=10-3) 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 solutionxk compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=10) 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|f(xk)| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=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 11
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=10) 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 solutionxk compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-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 13
Evolution of residue error|f(xk)| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-4) 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|xk+1-xk| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-4) 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 solutionxk compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=10) 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|f(xk)| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=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 17
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=10) 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 solutionsxk compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-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
Evolution of residue error|f(xk)| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=-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).Figure 20
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=-1) 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).


## 3.1. Some Preliminary Remarks

In this section, we propose to test, evaluate, and analyze the iterative numerical procedure presented in Section2.3 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):(29)(CC1):m≤Kmax,(CC2):fxk+1≤ϵre,(CC3):xk+1-xk≤ϵae,∀k=0,…,K,where Kmax denotes the maximum number of iterations and ϵre (resp.,ϵae) is the tolerance parameter associated with the residue (resp., approximation) error criterion. Here, the considered values for each CC are Kmax=10, ϵre=10-10, and ϵae=10-10.The iterative numerical scheme associated with TMNM (see [22, 26]) is(30)xk+1=xk-fxkf′xk-f′′xkfxk22f′xk3,∀k=0,…,K,∀f′xk≠0,where f′′(xk) denotes the second-order derivative of function f at point xk. It should be noted that order of convergence θ is cubic (i.e., θ=3) and rate of convergence τ is (see [22, 26])(31)τ=τTMNM=2c22-c3c1>0,∀c1≠0.


## 3.2. Examples

We consider the following scalar nonlinear equations:(32)f1x=5x4+3ln⁡x-1=0,f2x=3x3+5exp⁡x-10=0,f3x=5exp⁡x+x3cos⁡x-20=0,where exp⁡(·), ln⁡(·), and cos⁡(·) represent exponential, natural logarithm, and cosinus functions.The approximate numerical solution of roots associated with functionsf1, f2, and f3 are, respectively,(33)δ1≈0.7720250256⋯,δ2≈0.6192449540⋯,δ3≈1.3595986211⋯.


## 3.3. Results and Discussion

All the numerical results associated with scalar nonlinear functionsf1 to f3 are presented in Tables 1–6 and also in Figures 3–20. For each scalar nonlinear function fj (with j=1,2,3), we test two guest points x0 for starting iterative procedure: (i) in the case of function f1, we adopt the first-guest point x0=3 (resp., second-guest point x0=10-3) and we show different approximate solutions xk (with k=1,…,10) and also the evolution of both residue error fxk+1 and approximation error xk+1-xk 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 f2, we adopt first-guest point x0=10 (resp., second-guest point x0=-4) and we show different approximate solutions xk (with k=1,…,10) 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 functionf3, we adopt first-guest point x0=10 (resp., second-guest point x0=-1) and we show different approximate solutions xk (with k=1,…,10) 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 solutionxk of scalar nonlinear equation f1 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 3.0000000000 3.0000000000 3.0000000000 3.0000000000 1 2.2471426305 1.4942904056 (∗∗) 1.4942904056 (∗∗) 1.9644439737 2 1.6823826593 1.1286469200 (∗) 0.7631581663 (∗∗) 1.2884809024 3 1.2337388302 0.8906377962 (∗) 0.7721173336 (∗) 0.8906149334 4 0.9723200229 0.7878458749 (∗) 0.7720250356 (∗) 0.7746164630 5 0.8146707520 0.7723183427 (∗) 0.7720250256 (∗) 0.7720250503 6 0.7741430851 0.7720251266 (∗)  0.7720250256 7 0.7720302909 0.7720250256 (∗) 8 0.7720250256 9 0.7720250256 10Table 2
Approximate solutionxk of scalar nonlinear equation f2 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 0.0010000000 0.0010000000 0.0010000000 0.0010000000 1 0.0082410886 0.0154821756 (∗∗) 0.0154821756 (∗∗) 0.0344577770 2 0.0505339927 0.1548613133 (∗∗) 0.1548613133 (∗∗) 0.3980194000 3 0.2182202963 0.4938846484 (∗) 0.8311197179 (∗∗) 0.9031891893 4 0.6163033790 0.8261543800 (∗) 0.7760693275 (∗) 0.7755098327 5 0.7975366151 0.7754244611 (∗) 0.7720442199 (∗) 0.7720250858 6 0.7727863680 0.7720385875 (∗) 0.7720250260 (∗) 0.7720250256 7 0.7720257060 0.7720250258 (∗) 0.7720250256 (∗) 8 0.7720250256 0.7720250256 (∗) 9 10Table 3
Approximate solutionxk of scalar nonlinear equation f2 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 10.000000000 10.000000000 10.000000000 10.000000000 1 8.9811766535 7.9623533071 (∗∗) 7.9623533071 (∗∗) 8.4655416573 2 7.9456740785 5.8372149905 (∗∗) 5.8372149905 (∗∗) 6.8799380466 3 6.8825672128 3.5600762341 (∗∗) 3.5600762341 (∗∗) 5.2048513181 4 5.7803883041 1.4821734803 (∗∗) 1.4821734803 (∗∗) 3.4865001565 5 4.6404493806 0.9609028311 (∗) 0.4402289741 (∗∗) 2.0197617679 6 3.5056070182 0.6928062817 (∗) 0.6483081663 (∗) 1.0685736902 7 2.4744259573 0.6233620739 (∗) 0.6199084714 (∗) 0.6632122684 8 1.6461067325 0.6192585092 (∗) 0.6192453069 (∗) 0.6193191836 9 1.0635449602 0.6192449541 (∗) 0.6192449540 (∗) 0.6192449540 10 0.7354406002 0.6192449510 (∗)  0.6192449540Table 4
Approximate solutionxk of scalar nonlinear equation f2 (when guest point x0 = −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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 −4.0000000000 −4.0000000000 −4.0000000000 −4.0000000000 1 −2.5987493208 −1.1976336149 (∗∗) −1.1976336149 (∗∗) −2.1088093909 2 −1.5803235380 −0.2513579056 (∗) 0.6858578709 (∗∗) −0.7930763263 3 −0.6950023819 1.1303884977 (∗) 0.6226379112 (∗) 1.4022343921 4 0.5489290102 0.7665675348 (∗) 0.6192541648 (∗) 0.7698587570 5 0.6234126286 0.6349082108 (∗) 0.6192449540 (∗) 0.6217981372 6 0.6192588436 0.6194395336 (∗) 0.6192449540 (∗) 0.6192449694 7 0.6192449541 0.6192449843 (∗)  0.6192449540 8 0.6192449540 0.6192449540 (∗)  0.6192449540 9 10Table 5
Approximate solutionxk of scalar nonlinear equation f3 (when guest point x0 = 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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 10.000000000 10.000000000 10.000000000 10.000000000 1 9.0104267607 8.0208535216 (∗∗) 8.0208535216 (∗∗) 8.5171506533 2 8.0146859130 5.9615544966 (∗∗) 5.9615544966 (∗∗) 6.9859301990 3 6.9847503720 3.9493415483 (∗∗) 3.9493415483 (∗∗) 5.3851997390 4 5.9189110435 2.4996523082 (∗∗) 2.4996523082 (∗∗) 4.0778432768 5 4.9180876708 1.7226271036 (∗) 0.9467658294 (∗∗) 3.0067861343 6 4.0738875311 1.3881795157 (∗) 1.4273487664 (∗) 1.8098899052 7 3.3404588095 1.3598140013 (∗) 1.3607723321 (∗) 1.3648044141 8 2.6091363141 1.3595986336 (∗) 1.3595989925 (∗) 1.3595986468 9 1.8194405679 1.3595986211 (∗) 1.3595986211 (∗) 1.3595986211 10 1.4045600984Table 6
Approximate solutionxk of scalar nonlinear equation f3 (when guest point x0 = −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 xk is provided using (21a) (resp., (21b)) in Section 2.3.1.
Iteration (k) CNM CNM + NNT with condition [A1] CNM + NNT with condition [A2] TMNM 0 −1.0000000000 −1.0000000000 −1.0000000000 −1.0000000000 1 6.1409303723 6.1409303723 (∗) 6.1409303723 (∗) 14.5343055115 2 5.1148208484 4.0887116616 (∗∗) 4.0887116616 (∗∗) 13.0337347682 3 4.2379703013 2.6194618995 (∗∗) 2.6194618995 (∗∗) 11.5318478151 4 3.4892766889 1.8291595323 (∗) 1.0397597354 (∗∗) 10.0370297219 5 2.7676801225 1.4064394475 (∗) 1.3966765848 (∗) 8.5556896133 6 1.9776897481 1.3601688047 (∗) 1.3599586541 (∗) 7.0203746257 7 1.4417009128 1.3595987088 (∗) 1.3595986561 (∗) 5.3974911835 8 1.3613036994 1.3595986211 (∗) 1.3595986211 (∗) 4.1321703085 9 1.3595994046   3.1258556869 10 1.3595986211   1.9485648236Figure 3
Evolution of approximate solutionxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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
Evolution of residue errorfxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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 5
Evolution of approximation errorxk+1-xk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=3) 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 solutionxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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
Evolution of residue errorfxk compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=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 8
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f1 (when guest point x0=10-3) 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 solutionxk compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=10) 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|f(xk)| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=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 11
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=10) 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 solutionxk compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-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 13
Evolution of residue error|f(xk)| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-4) 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|xk+1-xk| compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-4) 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 solutionxk compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=10) 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|f(xk)| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=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 17
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=10) 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 solutionsxk compared to kth iteration for scalar nonlinear equation f2 (when guest point x0=-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
Evolution of residue error|f(xk)| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=-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).Figure 20
Evolution of approximation error|xk+1-xk| compared to kth iteration for scalar nonlinear equation f3 (when guest point x0=-1) 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.



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