# New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Authors: Rajinder Thukral
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based onn evaluations could achieve optimal convergence order of 2n−1. Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for n=4. Numerical comparisons are made to demonstrate the performance of the methods presented.


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## Body

## 1. Introduction

In this paper, we present a new family of the eighth-order methods to find a simple rootα of the nonlinear equation:(1.1)f(x)=0,
where f:D⊂ℝ→ℝ is a scalar function on an open interval D and it is sufficiently smooth in a neighbourhood of α.  It is well known that the techniques to solve nonlinear equations have many applications in science and engineering.  We will compare our new methods with well-known methods, namely, the classical Steffensen method for its simplicity [1, 2] and recently introduced eighth-order methods [3–5].The eighth-order methods presented in this paper are derivative-free and only use four evaluations of the function per iteration.  In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based onn evaluations, could achieve optimal convergence order 2n-1.  Thus, we present  new derivative-free methods which agree with the Kung and Traub conjecture for n=4. In addition, these new eighth-order derivative-free methods have an equivalent  efficiency index to the established eighth-order derivative based methods presented in [3–5]. Furthermore, the new eighth-order derivative-free methods have a better efficiency index than the sixth-order  derivative-free methods presented recently in [6, 7] and  in view of this fact, the new methods are significantly better when compared with the established methods.  Consequently, we have found that the new eighth-order derivative-free methods are consistent, stable, and convergent.This paper is organised as follows. In Section2, we describe the eighth-order methods that are free from derivatives and prove the important fact that the methods obtained preserve their convergence order. In Section 3, we will briefly state the established methods in order to compare the effectiveness of the new methods. Finally, in Section 4 we demonstrate the performance of each of the methods described.


## 2. Development of the Eighth-Order Derivative-Free Methods and Analysis of Convergence

In this section, we will define a new family of eighth-order derivative-free methods. In order to establish the order of convergence of these new methods, we state three essential definitions.Definition 2.1.
Letf(x) be a real function with a simple root α and let {xn} be a sequence of real numbers that converge towards α.  The order of convergence m is given by
(2.1)limn→∞xn+1-α(xn-α)m=ζ≠0,
where ζ is the asymptotic error constant and m∈ℝ+.Definition 2.2.
Suppose thatxn-1,xn and xn+1  are three successive iterations closer to the root α  of  (1.1).  Then, the computational order of convergence [8] may be approximated by
(2.2)COC≈ln|(xn+1-α)(xn-α)-1|ln|(xn-α)(xn-1-α)-1|,
where n∈ℕ.Definition 2.3.
Letβ be the number of function evaluations of the new method.  The efficiency of the new method is measured by the concept of efficiency index [9, 10] and defined as
(2.3)μ1/β,
where μ is the order of the method.


### 2.1. The Eighth-Order Derivative-Free Methods

In this subsection, we will define the new eighth-order derivative-free iterative method. In fact, we define different types of eighth-order method by varying the parametersβi,ϕj,ωk, and ξl. Therefore, the general formula of the new eighth-order method for determining the simple root of (1.1) is given as:
(2.4)wn=xn+βi-1f(xn),(2.5)yn=xn-(f(xn)2f(wn)-f(xn)),(2.6)zn=yn-ϕj(xn-ynf(xn)-f(yn))f(yn),(2.7)xn+1=zn-ωkξl(f(zn)-f(yn)zn-yn-f(yn)-f(xn)yn-xn+f(zn)-f(xn)zn-xn)-1f(zn),
where n∈ℕ, β∈ℝ+, provided that the denominators (2.5)–(2.7) are not equal to zero.The parameters used in the above eighth-order method are given as:(2.8)βi=i-1,i∈ℝ+,ϕ1=(1-f(yn)f(wn))-1,ϕ2=(1+f(yn)f(wn)),ω1=(1-f(zn)f(wn))-1,ω2=(1+f(zn)f(wn)+(f(zn)f(wn))2),ξ1=(1-2f(yn)3f(wn)2f(xn)),ξ2=(1+2f(yn)3f(wn)2f(xn))-1.In order to obtain a solution of the formula (2.7), we take one parameter from each set given above. Simply varying these parameters, we have many variants of eighth-order derivative-free methods.  Furthermore, we will demonstrate the performance of the eighth-order methods with the parameters given in (2.8).  To obtain the solution of (1.1) by the new derivative-free methods, we must set a particular initial approximation x0, ideally close to the simple root.  In numerical mathematics, it is very useful and essential to know the behaviour of an approximate method. Therefore, we will prove the order of convergence of the new eighth-order method.Theorem 2.4.
Assume that the functionf:D⊂ℝ→ℝ  for an open interval D  has a simple root α∈D.  Let f(x) be sufficiently smooth in the interval D,  the initial approximation x0 is sufficiently close to α then the order of convergence of the new derivative-free method defined by (2.7) is eight.Proof.
Letα  be a simple root of f(x), that is, f(α)=0 and f'(α)≠0, and the error is expressed as
(2.9)e=x-α.
Using the Taylor expansion, we have(2.10)f(xn)=f(α)+f'(α)en+2-1f′′(α)en2+6-1f′′′(α)en3+24-1fiv(α)en4+···.
Takingf(α)=0 and simplifying, expression (2.10) becomes
(2.11)f(xn)=c1en+c2en2+c3en3+c4en4+···,
where  n∈ℕ and
(2.12)ck=f(k)(α)(k!)fork=1,2,3,4,….
Expanding the Taylor series off(wn) and substituting f(xn) given by (2.10), we have
(2.13)f(wn)=c1(1+c1β)en+(3βc1c2+β2c12c2+c2)en2+···.
Substituting (2.11) and (2.13) in the expression (2.5) gives us
(2.14)yn-α=xn-α-(xn-wnf(xn)-f(wn))f(xn)=(c2c1)(βc1+1)en2+···.
The expansion of f(yn) about α is given as
(2.15)f(yn)=[c1(yn-α)+c2(yn-α)2+c3(yn-α)3+···].
Simplifying (2.15), we have
(2.16)f(yn)=c2(c1β+1)en2+(βc13c3-2c22+3βc12c3+2c1c3-β2c12c22-2βc1c22c1)en3+···.
The expansion of the particular term used in (2.6) is given as
(2.17)ϕ1=(1-f(yn)f(wn))-1=1+(c2c1)en+(βc12c3-2βc1c22+βc1c3-2c22λc12)en2+···.
Substituting appropriate expressions in (2.6), we obtain
(2.18)zn-α=yn-α-(1-f(yn)f(wn))(xn-ynf(xn)-f(yn))f(yn).
The Taylor series expansion of f(zn) about α is given as
(2.19)f(zn)=[c1(zn-α)+c2(zn-α)2+c3(zn-α)3+···].
Simplifying (2.19), we have
(2.20)f(zn)=(2c23-c1c2c3+4βc1c23+2β2c12c23-2βc12c2c3-c13c2c3c13)en4+···.
In order to evaluate the essential terms of (2.7), we expand term by term
(2.21)(f(yn)-f(xn)yn-xn)=c1+c2en+(c1c3+βc1c22+c22c1)en2+···,(f(zn)-f(yn)zn-yn)=c1+(βc1c22+c22c1)en2+···,(f(zn)-f(xn)zn-xn)=c1+c2en+c3en2+···.
Collecting the above terms
(2.22)ψ=[(f(yn)-f(xn)yn-xn)-(f(yn)-f(xn)yn-xn)+(f(zn)-f(xn)zn-xn)]-1=1c1+(c2c3+βc1c2c3c13)en3+···,ω1=(1-f(zn)f(wn))-1=1-(βc12c2c3-2βc1c23+c1c2c3-2c23c13)en3+···,ξ1=(1-2f(yn)3f(wn)2f(xn))=1-(2βc1c23+2c23c13)en3+···,ω1ξ1=1-(βc1c2c3+c2c3c12)en3+···,(2.23)ψω1ξ1=1c1+(β2c12c24+6βc12c22c3+3β2c13c22c3+βc1c24-2βc13c2c4+3c24-c12c2c4+3c1c22c3-β2c14c2c4c15)×en4+···.
Substituting appropriate expressions in (2.7), we obtain
(2.24)en+1=zn-α-ψω1ξ1f(zn).
Simplifying (2.24), we obtain the error equation
(2.25)en+1=c1-7[8β3c15c23c32-10c27-32β2c12c27-βc13c25c3+12β2c14c23c32+8βc13c23c32-β4c17c22c3c4+8βc13c24c4+12β2c14c24c4-9β3c14c25c3-8β2c13c25c3-3β4c15c25c3+2β4c16c24c4-6β2c15c22c3c4-4βc14c22c3c4-4β3c16c22c3c4+2β4c16c23c32-c13c22c3c4-2β4c14c27-30βc1c27-14β3c13c27+c1c25c3+2c12c23c32+2c12c24c4+8β3c15c24c4]en8.
The expression (2.25) establishes the asymptotic error constant for the eighth order of convergence for the new eighth-order derivative-free method defined by (2.7).


## 2.1. The Eighth-Order Derivative-Free Methods

In this subsection, we will define the new eighth-order derivative-free iterative method. In fact, we define different types of eighth-order method by varying the parametersβi,ϕj,ωk, and ξl. Therefore, the general formula of the new eighth-order method for determining the simple root of (1.1) is given as:
(2.4)wn=xn+βi-1f(xn),(2.5)yn=xn-(f(xn)2f(wn)-f(xn)),(2.6)zn=yn-ϕj(xn-ynf(xn)-f(yn))f(yn),(2.7)xn+1=zn-ωkξl(f(zn)-f(yn)zn-yn-f(yn)-f(xn)yn-xn+f(zn)-f(xn)zn-xn)-1f(zn),
where n∈ℕ, β∈ℝ+, provided that the denominators (2.5)–(2.7) are not equal to zero.The parameters used in the above eighth-order method are given as:(2.8)βi=i-1,i∈ℝ+,ϕ1=(1-f(yn)f(wn))-1,ϕ2=(1+f(yn)f(wn)),ω1=(1-f(zn)f(wn))-1,ω2=(1+f(zn)f(wn)+(f(zn)f(wn))2),ξ1=(1-2f(yn)3f(wn)2f(xn)),ξ2=(1+2f(yn)3f(wn)2f(xn))-1.In order to obtain a solution of the formula (2.7), we take one parameter from each set given above. Simply varying these parameters, we have many variants of eighth-order derivative-free methods.  Furthermore, we will demonstrate the performance of the eighth-order methods with the parameters given in (2.8).  To obtain the solution of (1.1) by the new derivative-free methods, we must set a particular initial approximation x0, ideally close to the simple root.  In numerical mathematics, it is very useful and essential to know the behaviour of an approximate method. Therefore, we will prove the order of convergence of the new eighth-order method.Theorem 2.4.
Assume that the functionf:D⊂ℝ→ℝ  for an open interval D  has a simple root α∈D.  Let f(x) be sufficiently smooth in the interval D,  the initial approximation x0 is sufficiently close to α then the order of convergence of the new derivative-free method defined by (2.7) is eight.Proof.
Letα  be a simple root of f(x), that is, f(α)=0 and f'(α)≠0, and the error is expressed as
(2.9)e=x-α.
Using the Taylor expansion, we have(2.10)f(xn)=f(α)+f'(α)en+2-1f′′(α)en2+6-1f′′′(α)en3+24-1fiv(α)en4+···.
Takingf(α)=0 and simplifying, expression (2.10) becomes
(2.11)f(xn)=c1en+c2en2+c3en3+c4en4+···,
where  n∈ℕ and
(2.12)ck=f(k)(α)(k!)fork=1,2,3,4,….
Expanding the Taylor series off(wn) and substituting f(xn) given by (2.10), we have
(2.13)f(wn)=c1(1+c1β)en+(3βc1c2+β2c12c2+c2)en2+···.
Substituting (2.11) and (2.13) in the expression (2.5) gives us
(2.14)yn-α=xn-α-(xn-wnf(xn)-f(wn))f(xn)=(c2c1)(βc1+1)en2+···.
The expansion of f(yn) about α is given as
(2.15)f(yn)=[c1(yn-α)+c2(yn-α)2+c3(yn-α)3+···].
Simplifying (2.15), we have
(2.16)f(yn)=c2(c1β+1)en2+(βc13c3-2c22+3βc12c3+2c1c3-β2c12c22-2βc1c22c1)en3+···.
The expansion of the particular term used in (2.6) is given as
(2.17)ϕ1=(1-f(yn)f(wn))-1=1+(c2c1)en+(βc12c3-2βc1c22+βc1c3-2c22λc12)en2+···.
Substituting appropriate expressions in (2.6), we obtain
(2.18)zn-α=yn-α-(1-f(yn)f(wn))(xn-ynf(xn)-f(yn))f(yn).
The Taylor series expansion of f(zn) about α is given as
(2.19)f(zn)=[c1(zn-α)+c2(zn-α)2+c3(zn-α)3+···].
Simplifying (2.19), we have
(2.20)f(zn)=(2c23-c1c2c3+4βc1c23+2β2c12c23-2βc12c2c3-c13c2c3c13)en4+···.
In order to evaluate the essential terms of (2.7), we expand term by term
(2.21)(f(yn)-f(xn)yn-xn)=c1+c2en+(c1c3+βc1c22+c22c1)en2+···,(f(zn)-f(yn)zn-yn)=c1+(βc1c22+c22c1)en2+···,(f(zn)-f(xn)zn-xn)=c1+c2en+c3en2+···.
Collecting the above terms
(2.22)ψ=[(f(yn)-f(xn)yn-xn)-(f(yn)-f(xn)yn-xn)+(f(zn)-f(xn)zn-xn)]-1=1c1+(c2c3+βc1c2c3c13)en3+···,ω1=(1-f(zn)f(wn))-1=1-(βc12c2c3-2βc1c23+c1c2c3-2c23c13)en3+···,ξ1=(1-2f(yn)3f(wn)2f(xn))=1-(2βc1c23+2c23c13)en3+···,ω1ξ1=1-(βc1c2c3+c2c3c12)en3+···,(2.23)ψω1ξ1=1c1+(β2c12c24+6βc12c22c3+3β2c13c22c3+βc1c24-2βc13c2c4+3c24-c12c2c4+3c1c22c3-β2c14c2c4c15)×en4+···.
Substituting appropriate expressions in (2.7), we obtain
(2.24)en+1=zn-α-ψω1ξ1f(zn).
Simplifying (2.24), we obtain the error equation
(2.25)en+1=c1-7[8β3c15c23c32-10c27-32β2c12c27-βc13c25c3+12β2c14c23c32+8βc13c23c32-β4c17c22c3c4+8βc13c24c4+12β2c14c24c4-9β3c14c25c3-8β2c13c25c3-3β4c15c25c3+2β4c16c24c4-6β2c15c22c3c4-4βc14c22c3c4-4β3c16c22c3c4+2β4c16c23c32-c13c22c3c4-2β4c14c27-30βc1c27-14β3c13c27+c1c25c3+2c12c23c32+2c12c24c4+8β3c15c24c4]en8.
The expression (2.25) establishes the asymptotic error constant for the eighth order of convergence for the new eighth-order derivative-free method defined by (2.7).


## 3. The Established Eighth-Order Methods

The eight particular eighth-order derivative-based methods considered are given in [3–5].  Since these methods are well established, we will state the essential expressions used in order to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new eighth-order derivative-free method.


### 3.1. The Bi, Wu, and Ren Methods

The first of the established eighth-order methods was presented by Bi et al. [3].Method 1.
(3.1)
y
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(3.2)
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(3.3)
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z
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x
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z
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where [zn,yn]=(f(zn)-f(yn))/(zn-yn),  γ∈ℝ, f(yn) is given by (3.1), x0 is the initial approximation and provided that the denominators of (3.1)–(3.3) are not equal to zero.Method 2.
(3.4)
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(3.5)
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z
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where γ,μ∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.4) and (3.5) are not equal to zero.Method 3.
(3.6)
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(3.7)
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where γ,μ∈ℝ, f(yn) is given by (3.1), x0 is the initial approximation and provided that the denominators of (3.6) and (3.7) are not equal to zero.Method 4.
(3.8)
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(3.9)
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where γ∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.8) and (3.9) are not equal to zero.


### 3.2. The Sharma Methods

The three particular eighth-order methods considered are given in [4].  Since these methods are well established, we will state the essential expressions used in order to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new iterative eighth-order method.Method 5.
(3.10)
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(3.11)
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where γ∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.10) and (3.11) are not equal to zero.Method 6.
(3.12)
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(3.13)
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where γ,β∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.12) and (3.13) are not equal to zero.Method 7.
(3.14)
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(
y
n
)
f
'
(
x
n
)
)
,
(3.15)
x
n
+
1
=
z
n
-
[
1
+
γ
f
(
z
n
)
f
(
x
n
)
]
(
1
/
γ
)
(
f
[
x
n
,
y
n
]
f
(
z
n
)
f
[
y
n
,
z
n
]
f
[
x
n
,
z
n
]
)
,
where γ∈ℝ, f(yn) is given by (3.1), x0 is the initial approximation and provided that the denominator of (3.14) and (3.15) are not equal to zero.


### 3.3. The Thukral Eighth-Order Method

The following eighth-order method is actually presented in [5] and since it is well established, we will state the essential expressions used in order to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new iterative eighth-order method. The Newton-type eighth-order iterative method is expressed as
(3.16)zn=xn-f(xn)2+f(yn)2f'(xn)(f(xn)-f(yn)),(3.17)xn+1=zn-[(1+μn21-μn)2-2(μn)2-6(μn)3+f(zn)f(yn)+4f(zn)f(xn)](f(zn)f'(xn)),
where μn=(f(yn)/f(xn))n∈ℕ, f(yn) is given by (3.1)  and provided that the denominators of (3.16) and (3.17) are not equal to zero.


## 3.1. The Bi, Wu, and Ren Methods

The first of the established eighth-order methods was presented by Bi et al. [3].Method 1.
(3.1)
y
n
=
x
n
-
f
(
x
n
)
f
'
(
x
n
)
,
(3.2)
z
n
=
y
n
-
[
2
f
(
x
n
)
-
f
(
y
n
)
2
f
(
x
n
)
-
5
f
(
y
n
)
]
(
f
(
y
n
)
f
'
(
x
n
)
)
,
(3.3)
x
n
+
1
=
z
n
-
[
f
(
x
n
)
+
(
γ
+
2
)
f
(
z
n
)
f
(
x
n
)
+
γ
f
(
z
n
)
]
(
f
(
z
n
)
f
[
z
n
,
y
n
]
+
f
[
z
n
,
x
n
,
x
n
]
(
z
n
-
y
n
)
)
,
where [zn,yn]=(f(zn)-f(yn))/(zn-yn),  γ∈ℝ, f(yn) is given by (3.1), x0 is the initial approximation and provided that the denominators of (3.1)–(3.3) are not equal to zero.Method 2.
(3.4)
z
n
=
y
n
-
[
1
+
2
f
(
y
n
)
f
(
x
n
)
+
5
(
f
(
y
n
)
f
(
x
n
)
)
2
+
(
f
(
y
n
)
f
(
x
n
)
)
3
]
(
f
(
y
n
)
f
'
(
x
n
)
)
,
(3.5)
x
n
+
1
=
z
n
-
[
f
(
x
n
)
+
(
γ
+
2
)
f
(
z
n
)
f
(
x
n
)
+
γ
f
(
z
n
)
]
(
f
(
z
n
)
f
[
z
n
,
y
n
]
+
f
[
z
n
,
x
n
,
x
n
]
(
z
n
-
y
n
)
)
,
where γ,μ∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.4) and (3.5) are not equal to zero.Method 3.
(3.6)
z
n
=
y
n
-
[
1
-
2
f
(
y
n
)
f
(
x
n
)
-
(
f
(
y
n
)
f
(
x
n
)
)
2
+
(
f
(
y
n
)
f
(
x
n
)
)
3
]
-
1
(
f
(
y
n
)
f
'
(
x
n
)
)
,
(3.7)
x
n
+
1
=
z
n
-
[
f
(
x
n
)
+
(
γ
+
2
)
f
(
z
n
)
f
(
x
n
)
+
γ
f
(
z
n
)
]
(
f
(
z
n
)
f
[
z
n
,
y
n
]
+
f
[
z
n
,
x
n
,
x
n
]
(
z
n
-
y
n
)
)
,
where γ,μ∈ℝ, f(yn) is given by (3.1), x0 is the initial approximation and provided that the denominators of (3.6) and (3.7) are not equal to zero.Method 4.
(3.8)
z
n
=
y
n
-
[
f
(
x
n
)
-
3
f
(
y
n
)
f
(
x
n
)
]
(
-
2
/
3
)
(
f
(
y
n
)
f
'
(
x
n
)
)
,
(3.9)
x
n
+
1
=
z
n
-
[
f
(
x
n
)
+
(
γ
+
2
)
f
(
z
n
)
f
(
x
n
)
+
γ
f
(
z
n
)
]
(
f
(
z
n
)
f
[
z
n
,
y
n
]
+
f
[
z
n
,
x
n
,
x
n
]
(
z
n
-
y
n
)
)
,
where γ∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.8) and (3.9) are not equal to zero.


## 3.2. The Sharma Methods

The three particular eighth-order methods considered are given in [4].  Since these methods are well established, we will state the essential expressions used in order to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new iterative eighth-order method.Method 5.
(3.10)
z
n
=
y
n
-
[
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
]
(
f
(
y
n
)
f
'
(
x
n
)
)
,
(3.11)
x
n
+
1
=
z
n
-
[
1
+
f
(
z
n
)
f
(
x
n
)
+
γ
(
f
(
z
n
)
f
(
x
n
)
)
2
]
(
f
[
x
n
,
y
n
]
f
(
z
n
)
f
[
y
n
,
z
n
]
f
[
x
n
,
z
n
]
)
,
where γ∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.10) and (3.11) are not equal to zero.Method 6.
(3.12)
z
n
=
y
n
-
[
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
]
(
f
(
y
n
)
f
'
(
x
n
)
)
,
(3.13)
x
n
+
1
=
z
n
-
[
f
(
x
n
)
+
(
γ
+
1
)
f
(
z
n
)
f
(
x
n
)
+
γ
f
(
z
n
)
]
(
f
[
x
n
,
y
n
]
f
(
z
n
)
f
[
y
n
,
z
n
]
f
[
x
n
,
z
n
]
)
,
where γ,β∈ℝ, f(yn) is given by (3.1) and provided that the denominators of (3.12) and (3.13) are not equal to zero.Method 7.
(3.14)
z
n
=
y
n
-
[
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
]
(
f
(
y
n
)
f
'
(
x
n
)
)
,
(3.15)
x
n
+
1
=
z
n
-
[
1
+
γ
f
(
z
n
)
f
(
x
n
)
]
(
1
/
γ
)
(
f
[
x
n
,
y
n
]
f
(
z
n
)
f
[
y
n
,
z
n
]
f
[
x
n
,
z
n
]
)
,
where γ∈ℝ, f(yn) is given by (3.1), x0 is the initial approximation and provided that the denominator of (3.14) and (3.15) are not equal to zero.


## 3.3. The Thukral Eighth-Order Method

The following eighth-order method is actually presented in [5] and since it is well established, we will state the essential expressions used in order to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new iterative eighth-order method. The Newton-type eighth-order iterative method is expressed as
(3.16)zn=xn-f(xn)2+f(yn)2f'(xn)(f(xn)-f(yn)),(3.17)xn+1=zn-[(1+μn21-μn)2-2(μn)2-6(μn)3+f(zn)f(yn)+4f(zn)f(xn)](f(zn)f'(xn)),
where μn=(f(yn)/f(xn))n∈ℕ, f(yn) is given by (3.1)  and provided that the denominators of (3.16) and (3.17) are not equal to zero.


## 4. Application of the New Derivative-Free Iterative Methods

To demonstrate the performance of the new eighth-order methods, we take six particular nonlinear equations.  We will determine the consistency and stability of results by examining the convergence of the new derivative-free iterative methods.  The findings are generalised by illustrating the effectiveness of the eighth-order methods for determining the simple root of a nonlinear equation.  Consequently, we will give estimates of the approximate solution produced by the eighth-order methods and list the errors obtained by each of the methods.  The numerical computations listed in the tables were performed on an algebraic system called Maple.  In addition, we need to set a particular value of the parameters used in all the eighth-order formula given in this paper.  Therefore, we takei=γ=β=1 as an arbitrary value. In fact, the errors displayed are of absolute value.Remark 4.1.
The family of three-point methods requires four function evaluations and has the order of convergence eight.  Therefore, this family is of optimal order and supports the Kung-Traub conjecture [11].  To determine the efficiency index of these new derivative-free methods, we will use the definition (2.2).  Hence, the efficiency index of the eighth-order derivative-free methods given is 84≈1.68.Remark 4.2.
In Tables1–6, it is observed that the new eighth-order derivative-free methods are competitive with the existing eighth-order derivative-based methods. Furthermore, in these tables we have omitted the insignificant approximations by the various methods and the absolute errors |xn-α|≤10-1500 in the first three iterations are given in Tables 1–6.Table 1
Errors occurring in the estimates of the root of (4.1) by the methods described.
Methods
|
x
1
-
α
|
|
x
2
-
α
|
|
x
3
-
α
|
COC
(2.7)
0.690
e
-
3
0.933
e
-
34
0.255
e
-
373
11.0000
(3.3)
0.228
e
-
1
0.281
e
-
13
0.213
e
-
108
7.9863
(3.5)
–-
–-
–-
–-
(3.7)
0.795
e
-
1
0.229
e
-
9
0.295
e
-
77
8.0425
(3.9)
0.281
e
-
2
0.104
e
-
20
0.386
e
-
168
8.0266
(3.11)
0.414
e
-
2
0.164
e
-
18
0.997
e
-
150
7.9992
(3.13)
0.654
e
-
2
0.625
e
-
17
0.452
e
-
137
7.9986
(3.15)
0.523
e
-
2
0.105
e
-
17
0.295
e
-
143
7.9990
(3.17)
–-
–-
–-
–-Table 2
Errors occurring in the estimates of the root of (4.2) by the methods described.
Methods
|
x
1
-
α
|
|
x
2
-
α
|
|
x
3
-
α
|
COC
(2.7)
0.193
e
-
14
0.982
e
-
127
0.443
e
-
1025
8.0000
(3.3)
0.244
e
-
12
0.248
e
-
108
0.279
e
-
876
8.0000
(3.5)
0.104
e
-
11
0.169
e
-
102
0.806
e
-
829
8.0675
(3.7)
0.289
e
-
12
0.119
e
-
107
0.102
e
-
870
8.0693
(3.9)
0.321
e
-
12
0.320
e
-
107
0.308
e
-
867
8.0689
(3.11)
0.109
e
-
11
0.220
e
-
102
0.591
e
-
828
7.9999
(3.13)
0.109
e
-
11
0.220
e
-
102
0.599
e
-
828
8.0003
(3.15)
0.109
e
-
11
0.220
e
-
102
0.595
e
-
828
7.9999
(3.17)
0.743
e
-
11
0.760
e
-
95
0.905
e
-
767
8.0000Table 3
Errors occurring in the estimates of the root of (4.3) by the methods described.
Methods
|
x
1
-
α
|
|
x
2
-
α
|
|
x
3
-
α
|
COC
(2.7)
0.241
e
-
4
0.872
e
-
37
0.258
e
-
296
7.9859
(3.3)
0.813
e
-
2
0.152
e
-
17
0.264
e
-
143
7.9961
(3.5)
0.416
0.370
0.661
e
-
4
0.8817
(3.7)
0.750
e
-
2
0.599
e
-
18
0.110
e
-
146
8.0254
(3.9)
0.128
e
-
2
0.355
e
-
24
0.123
e
-
196
8.0317
(3.11)
0.153
e
-
2
0.101
e
-
22
0.367
e
-
184
7.9999
(3.13)
0.205
e
-
2
0.104
e
-
21
0.447
e
-
176
7.9998
(3.15)
0.178
e
-
2
0.332
e
-
22
0.502
e
-
180
7.9998
(3.17)
–-
–-
–-
–-Table 4
Errors occurring in the estimates of the root of (4.4) by the methods described.
Methods
|
x
1
-
α
|
|
x
2
-
α
|
|
x
3
-
α
|
COC
(2.7)
0.228
e
-
5
0.453
e
-
44
0.110
e
-
353
7.9901
(3.3)
0.216
e
-
4
0.143
e
-
39
0.526
e
-
321
7.9940
(3.5)
0.180
e
-
5
0.500
e
-
48
0.176
e
-
388
7.9940
(3.7)
0.217
e
-
4
0.149
e
-
39
0.736
e
-
321
7.9940
(3.9)
0.218
e
-
4
0.153
e
-
39
0.891
e
-
321
8.0000
(3.11)
0.295
e
-
5
0.235
e
-
46
0.376
e
-
375
8.0428
(3.13)
0.295
e
-
5
0.235
e
-
46
0.372
e
-
375
8.0551
(3.15)
0.481
e
-
3
0.217
e
-
24
0.391
e
-
195
8.0553
(3.17)
0.127
e
-
4
0.959
e
-
42
0.101
e
-
338
7.9892Table 5
Errors occurring in the estimates of the root of (4.5) by the methods described.
Methods
|
x
1
-
α
|
|
x
2
-
α
|
|
x
3
-
α
|
COC
(2.7)
0.933
e
-
2
0.595
e
-
9
0.211
e
-
66
7.7073
(3.3)
0.225
e
-
1
0.363
e
-
8
0.179
e
-
62
7.9964
(3.5)
0.924
e
-
2
0.358
e
-
10
0.102
e
-
77
7.4660
(3.7)
0.247
e
-
2
0.132
e
-
15
0.930
e
-
122
7.6851
(3.9)
0.408
0.310
e
-
1
0.256
e
-
7
2.7679
(3.11)
0.474
e
-
3
0.192
e
-
24
0.141
e
-
195
7.9668
(3.13)
0.489
e
-
3
0.246
e
-
24
0.103
e
-
194
7.9666
(3.15)
0.481
e
-
3
0.217
e
-
24
0.391
e
-
195
7.9666
(3.17)
0.609
e
-
2
0.428
e
-
12
0.259
e
-
93
7.9992Table 6
Errors occurring in the estimates of the root of (4.6) by the methods described.
Methods
|
x
1
-
α
|
|
x
2
-
α
|
|
x
3
-
α
|
COC
(2.7)
0.320
e
-
2
0.116
e
-
10
0.114
e
-
75
7.5078
(3.3)
0.778
e
-
2
0.270
e
-
12
0.725
e
-
96
7.9904
(3.5)
1.43
–-
–-
–-
(3.7)
0.282
–-
–-
–-
(3.9)
0.119
e
-
2
0.568
e
-
19
0.153
e
-
149
7.7503
(3.11)
0.790
e
-
3
0.193
e
-
20
0.250
e
-
161
7.9997
(3.13)
0.115
e
-
2
0.390
e
-
19
0.693
e
-
151
7.9996
(3.15)
0.957
e
-
3
0.896
e
-
20
0.535
e
-
156
7.9997
(3.17)
–-
–-
–-
–-


### 4.1. Numerical Example 1

In our first example, we will demonstrate the convergence of the new eighth-order derivative-free methods for the following nonlinear equation:(4.1)f(x)=e-x-cos(x),
and the exact value of the simple root of (4.1) is α=-0.666273126….  In Table 1 are the errors obtained by each of the methods described, based on the initial approximation x0=3-1.


### 4.2. Numerical Example 2

In our second example, we will demonstrate the convergence of new eighth-order derivative-free methods for a different type of nonlinear equation:(4.2)f(x)=ln(x2+x+2)-x+1,
and the exact value of the simple root of (4.2) is α=4.15259074….  In Table 2 are the errors obtained by each of the methods described, based on the initial approximation  x0=4.4.


### 4.3. Numerical Example 3

In this subsection, we take another nonlinear equation.  We will demonstrate the convergence of the new eighth-order derivative-free methods for the following nonlinear equation:(4.3)f(x)=sin(x)2-x2+1,
and the exact value of the simple root of (4.3) is α=1.40449165….    In Table 3 are the errors obtained by each of the methods described, based on the initial approximation x0=1.


### 4.4. Numerical Example 4

In the next examples, we take another different type of nonlinear equation.  We will demonstrate the convergence of new eighth-order derivative-free methods for the following nonlinear equation:(4.4)f(x)=exp(-x2+x+2)-cos(x+1)+x3+1,
and the exact value of the simple root of (4.4) is α=-1.  In Table 4 are the errors obtained by each of the methods described, based on the initial approximation  x0=-2-1.


### 4.5. Numerical Example 5

In this subsection, we take another nonlinear equation.  We will demonstrate the convergence of the new eighth-order derivative-free methods for the following nonlinear equation:(4.5)f(x)=x11+x+1,
and the exact value of the simple root of (4.5) is α=-0.8443975….  In Table 5 are the errors obtained by each of the methods described, based on the initial approximation x0=-1.


### 4.6. Numerical Example 6

In the last but not least of the examples, we take another different type of nonlinear equation.  We will demonstrate the convergence of new eighth-order derivative-free methods for the following nonlinear equation:(4.6)f(x)=(x-2)(x10+x+1)e-x-1,
and the exact value of the simple root of (4.6) is α=2.  In Table 6 are the errors obtained by each of the methods described, based on the initial approximation x0=1.9.


## 4.1. Numerical Example 1

In our first example, we will demonstrate the convergence of the new eighth-order derivative-free methods for the following nonlinear equation:(4.1)f(x)=e-x-cos(x),
and the exact value of the simple root of (4.1) is α=-0.666273126….  In Table 1 are the errors obtained by each of the methods described, based on the initial approximation x0=3-1.


## 4.2. Numerical Example 2

In our second example, we will demonstrate the convergence of new eighth-order derivative-free methods for a different type of nonlinear equation:(4.2)f(x)=ln(x2+x+2)-x+1,
and the exact value of the simple root of (4.2) is α=4.15259074….  In Table 2 are the errors obtained by each of the methods described, based on the initial approximation  x0=4.4.


## 4.3. Numerical Example 3

In this subsection, we take another nonlinear equation.  We will demonstrate the convergence of the new eighth-order derivative-free methods for the following nonlinear equation:(4.3)f(x)=sin(x)2-x2+1,
and the exact value of the simple root of (4.3) is α=1.40449165….    In Table 3 are the errors obtained by each of the methods described, based on the initial approximation x0=1.


## 4.4. Numerical Example 4

In the next examples, we take another different type of nonlinear equation.  We will demonstrate the convergence of new eighth-order derivative-free methods for the following nonlinear equation:(4.4)f(x)=exp(-x2+x+2)-cos(x+1)+x3+1,
and the exact value of the simple root of (4.4) is α=-1.  In Table 4 are the errors obtained by each of the methods described, based on the initial approximation  x0=-2-1.


## 4.5. Numerical Example 5

In this subsection, we take another nonlinear equation.  We will demonstrate the convergence of the new eighth-order derivative-free methods for the following nonlinear equation:(4.5)f(x)=x11+x+1,
and the exact value of the simple root of (4.5) is α=-0.8443975….  In Table 5 are the errors obtained by each of the methods described, based on the initial approximation x0=-1.


## 4.6. Numerical Example 6

In the last but not least of the examples, we take another different type of nonlinear equation.  We will demonstrate the convergence of new eighth-order derivative-free methods for the following nonlinear equation:(4.6)f(x)=(x-2)(x10+x+1)e-x-1,
and the exact value of the simple root of (4.6) is α=2.  In Table 6 are the errors obtained by each of the methods described, based on the initial approximation x0=1.9.


## 5. Remarks and Conclusion

We have demonstrated the performance of a new family of eighth-order derivative-free methods. Convergence analysis proves that the new methods preserve their order of convergence. There are two major advantages of the eighth-order derivative-free methods.  Firstly, we do not have to evaluate the derivative of the functions; therefore, they are especially efficient where the computational cost of the derivative is expensive, and secondly we have established a higher order of convergence method than the existing derivative-free methods [6, 7].  We have examined the effectiveness of the new derivative-free methods by showing the accuracy of the simple root of a nonlinear equation.  The main purpose of demonstrating the new eighth-order methods for six types of nonlinear equations was purely to illustrate the accuracy of the approximate solution and the computational order of convergence.



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*Source: 493456-2012-10-03.xml*