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 on 𝑛 evaluations, could achieve optimal convergence order 2𝑛1. 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.

1. Introduction

Consider iterative methods for finding a simple root 𝛼 of the nonlinear equation𝑓(𝑥)=0,(1.1) where𝑓𝐷 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. In this paper, a new family of three-point derivative-free methods of the optimal order eight is constructed by combining optimal two-step fourth-order methods and a modified third step. In order to obtain these new derivative-free methods, we replace derivatives with suitable approximations based on divided difference. In fact, it is well known that the various methods have been used in order to approximate the derivatives by the Newton interpolation, the Hermite interpolation, the Lagrange interpolation, and ration function [1, 2].

The prime motive of this study is to develop a class of very efficient three-step derivative-free methods for solving nonlinear equations. 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 on 𝑛 evaluations, could achieve optimal convergence order 2𝑛1.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 Kung and Traub eighth-order derivative-free method presented in [3]. Furthermore, the new eighth-order derivative-free methods have a better efficiency index than the three-step sixth-order derivative-free methods presented recently in [4, 5], 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 Section 2 we construct 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 Kung and Traub method 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. Methods and Convergence Analysis

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 the three essential definitions.

Definition 2.1. Let 𝑓(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 lim𝑛𝑥𝑛+1𝛼𝑥𝑛𝛼𝑚=𝜁0,(2.1) where 𝜁 is the asymptotic error constant and 𝑚+.

Definition 2.2. Suppose that 𝑥𝑛1,𝑥𝑛, and 𝑥𝑛+1 are three successive iterations closer to the root 𝛼 of (1.1). Then, the computational order of convergence [6] may be approximated by |||𝑥COCln𝑛+1𝑥𝛼𝑛𝛼1||||||𝑥ln𝑛𝑥𝛼𝑛1𝛼1|||,(2.2) where 𝑛.

Definition 2.3. Let 𝛽 be the number of function evaluations of the new method. The efficiency of the new method is measured by the concept of efficiency index [7, 8] and defined as 𝜇1/𝛽,(2.3) where 𝜇 is the order of the method.

2.1. The Eighth-Order Derivative-Free Method (RT)

We consider the iteration scheme of the form𝑦𝑛=𝑥𝑛𝑓𝑥𝑛𝑥𝑓𝑛,𝑧𝑛=𝑦𝑛𝑓𝑦𝑛𝑦𝑓𝑛,𝑥𝑛+1=𝑧𝑛𝑓𝑧𝑛𝑧𝑓𝑛.(2.4)

This scheme consists of three steps in which the Newton method is repeated. It is clear that formula (2.4) requires six evaluations per iteration and has an efficiency index of 81/6=1.414, which is the same as the classical Newton method. In fact, scheme (2.4) does not increase the computational efficiency. The purpose of this paper is to establish new derivative-free methods with optimal order; hence, we reduce the number of evaluations to four by using some suitable approximation of the derivatives. To derive higher efficiency index, we consider approximating the derivatives by divided difference method. Therefore, the derivatives in (2.4) are replaced by𝑥𝑓𝑛𝑤𝑓𝑛,𝑥𝑛=𝑓𝑤𝑛𝑥𝑓𝑛𝑤𝑛𝑥𝑛=𝑓𝑤𝑛𝑥𝑓𝑛𝑓𝑥𝑛,𝑦𝑓𝑛𝑓𝑥𝑛,𝑦𝑛𝑓𝑤𝑛,𝑦𝑛𝑓𝑤𝑛,𝑥𝑛,𝑧𝑓𝑛𝑓𝑦𝑛,𝑧𝑛𝑥𝑓𝑛,𝑦𝑛𝑥+𝑓𝑛,𝑧𝑛.(2.5) Substituting (2.5) into (2.4), we get𝑤𝑛=𝑥𝑛𝑥+𝛽𝑓𝑛,𝑦𝑛=𝑥𝑛𝑓𝑥𝑛2𝑓𝑤𝑛𝑥𝑓𝑛,𝑧𝑛=𝑦𝑛𝑓𝑤𝑛,𝑥𝑛𝑓𝑦𝑛𝑓𝑥𝑛,𝑦𝑛𝑓𝑤𝑛,𝑦𝑛,𝑥𝑛+1=𝑧𝑛𝑓𝑧𝑛𝑓𝑦𝑛,𝑧𝑛𝑥𝑓𝑛,𝑦𝑛𝑥+𝑓𝑛,𝑧𝑛.(2.6)

The first step of formula (2.6) is the classical Steffensen second-order method [9], and the second step is the new fourth-order method. Furthermore, we have found that the third step does not produce an optimal order of convergence. Therefore, we have introduced two weight functions in the third step in order to achieve the desired eighth-order derivative-free method. The two weight functions are expressed as𝑓𝑧1𝑛𝑓𝑤𝑛1,𝑦1+2𝑓𝑛3𝑓𝑤𝑛2𝑓𝑥𝑛1.(2.7) Then the iteration scheme (2.4) in its final form is given as𝑤𝑛=𝑥𝑛𝑥+𝛽𝑓𝑛,𝑦𝑛=𝑥𝑛𝑓𝑥𝑛2𝑓𝑤𝑛𝑥𝑓𝑛,𝑧𝑛=𝑦𝑛𝑓𝑤𝑛,𝑥𝑛𝑓𝑦𝑛𝑓𝑥𝑛,𝑦𝑛𝑓𝑤𝑛,𝑦𝑛,𝑥𝑛+1=𝑧𝑛𝑓𝑧1𝑛𝑓𝑤𝑛1𝑦1+2𝑓𝑛3𝑓𝑤𝑛2𝑓𝑥𝑛1𝑓𝑧𝑛𝑓𝑦𝑛,𝑧𝑛𝑥𝑓𝑛,𝑦𝑛𝑥+𝑓𝑛,𝑧𝑛,(2.8) where 𝑛, 𝛽+, provided that the denominators in (2.8) are not equal to zero.

Thus the scheme (2.8) defines a new family of multipoint methods with two weight functions. To obtain the solution of (1.1) by the new derivative-free methods, we must set a particular initial approximation𝑥0, 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 function𝑓𝐷 for an open interval D has a simple root 𝛼𝐷. Letting 𝑓(𝑥) be sufficiently smooth in the interval D and the initial approximation 𝑥0 is sufficiently close to 𝛼, then the order of convergence of the new derivative-free method defined by (2.8) is eight.

Proof. Let 𝛼 be a simple root of 𝑓(𝑥), that is, 𝑓(𝛼)=0 and 𝑓(𝛼)0, and the error is expressed as 𝑒=𝑥𝛼.(2.9)
Using the Taylor expansion, we have 𝑓𝑥𝑛=𝑓(𝛼)+𝑓(𝛼)𝑒𝑛+21𝑓(𝛼)𝑒2𝑛+61𝑓(𝛼)𝑒3𝑛+241𝑓𝑖𝑣(𝛼)𝑒4𝑛+.(2.10)
Taking 𝑓(𝛼)=0 and simplifying, expression (2.10) becomes 𝑓𝑥𝑛=𝑐1𝑒𝑛+𝑐2𝑒2𝑛+𝑐3𝑒3𝑛+𝑐4𝑒4𝑛+,(2.11) where 𝑛 and 𝑐𝑘=𝑓(𝑘)(𝛼)(𝑘!)for𝑘=1,2,3,4,.(2.12)
Expanding the Taylor series of 𝑓(𝑤𝑛) and substituting 𝑓(𝑥𝑛) given by (2.11), we have 𝑓𝑤𝑛=𝑐11+𝑐1𝛽𝑒𝑛+3𝛽𝑐1𝑐2+𝛽2𝑐21𝑐2+𝑐2𝑒2𝑛+.(2.13)
Substituting (2.11) and (2.13) in expression (2.8) gives us 𝑦𝑛𝛼=𝑥𝑛𝑥𝛼𝛽𝑓𝑛2𝑓𝑤𝑛𝑥𝑓𝑛=𝑐2𝑐1𝛽𝑐1𝑒+12𝑛+.(2.14)
The expansion of 𝑓(𝑦𝑛) about 𝛼 is given as 𝑓𝑦𝑛=𝑐1𝑦𝑛𝛼+𝑐2𝑦𝑛𝛼2+𝑐3𝑦𝑛𝛼3+.(2.15)
Simplifying (2.15), we have 𝑓𝑦𝑛=𝑐2𝑐1𝑒𝛽+12𝑛+𝛽𝑐31𝑐32𝑐22+3𝛽𝑐21𝑐3+2𝑐1𝑐3𝛽2𝑐21𝑐222𝛽𝑐1𝑐22𝑐1𝑒3𝑛+.(2.16)
The expansion of the particular term used in (2.8) is given as 𝑓𝑤𝑛,𝑥𝑛=𝑓𝑤𝑛𝑥𝑓𝑛𝑤𝑛𝑥𝑛=𝑐1+2𝑐2+𝛽𝑐1𝑐2𝑒𝑛+3𝑐3+3𝛽𝑐1𝑐3+𝛽2𝑐21𝑐3+𝛽𝑐22𝑒2𝑛𝑓𝑤+,𝑛,𝑦𝑛=𝑓𝑤𝑛𝑦𝑓𝑛𝑤𝑛𝑦𝑛=𝑐1+𝑐2+𝛽𝑐1𝑐2𝑒𝑛+𝛽𝑐21𝑐3+𝑐22+2𝛽𝑐1𝑐22+2𝛽𝑐21𝑐3+𝑐31𝑐3𝑐1𝑒2𝑛𝑓𝑥+,𝑛,𝑦𝑛=𝑓𝑥𝑛𝑦𝑓𝑛𝑥𝑛𝑦𝑛=𝑐1+𝑐2𝑒𝑛+𝑐1𝑐3+𝑐22+𝛽𝑐1𝑐22𝑐1𝑒2𝑛𝑓𝑤+,𝑛,𝑥𝑛𝑓𝑥𝑛,𝑦𝑛𝑓𝑤𝑛,𝑦𝑛=1𝑐1+𝛽𝑐21𝑐33𝑐223𝛽𝑐1𝑐22+𝑐1𝑐3𝑐31𝑒2𝑛+.(2.17)
Substituting appropriate expressions in (2.8), we obtain 𝑧𝑛𝛼=𝑦𝑛𝑓𝑤𝛼𝑛,𝑥𝑛𝑓𝑦𝑛𝑓𝑥𝑛,𝑦𝑛𝑓𝑤𝑛,𝑦𝑛.(2.18)
The Taylor series expansion of 𝑓(𝑧𝑛) about 𝛼 is given as 𝑓𝑧𝑛=𝑐1𝑧𝑛𝛼+𝑐2𝑧𝑛𝛼2+𝑐3𝑧𝑛𝛼3+.(2.19)
Simplifying (2.18), we have 𝑓𝑧𝑛=2𝑐32𝑐1𝑐2𝑐3+4𝛽𝑐1𝑐32+2𝛽2𝑐21𝑐322𝛽𝑐21𝑐2𝑐3𝑐31𝑐2𝑐3𝑐21𝑒4𝑛+.(2.20)
In order to evaluate the essential terms of (2.8), we expand term by term 𝑓𝑦𝑛,𝑧𝑛=𝑓𝑦𝑛𝑧𝑓𝑛𝑦𝑛𝑧𝑛=𝑐1+𝛽𝑐1𝑐22+𝑐22𝑐1𝑒2𝑛𝑓𝑥+,𝑛,𝑧𝑛=𝑓𝑥𝑛𝑧𝑓𝑛𝑥𝑛𝑧𝑛=𝑐1+𝑐2𝑒𝑛+𝑐3𝑒2𝑛+.(2.21)
Collecting the above terms, 𝑓𝑦𝜓=𝑛,𝑧𝑛𝑥𝑓𝑛,𝑦𝑛𝑥+𝑓𝑛,𝑧𝑛=1=1𝑐1+𝑐2𝑐3+𝛽𝑐1𝑐2𝑐3𝑐31𝑒3𝑛𝑓𝑧+,𝜔=1𝑛𝑓𝑤𝑛1=1𝛽𝑐21𝑐2𝑐32𝛽𝑐1𝑐32+𝑐1𝑐2𝑐32𝑐32𝑐31𝑒3𝑛𝑦+,𝜉=1+2𝑓𝑛3𝑓𝑤𝑛2𝑓𝑥𝑛1=12𝛽𝑐1𝑐32+2𝑐32𝑐31𝑒3𝑛+,𝜔𝜉=1𝛽𝑐1𝑐2𝑐3+𝑐2𝑐3𝑐21𝑒3𝑛=1+,𝜓𝜔𝜉𝑐1+𝛽2𝑐21𝑐42+6𝛽𝑐21𝑐22𝑐3+3𝛽2𝑐31𝑐22𝑐3+𝛽𝑐1𝑐422𝛽𝑐31𝑐2𝑐4+2𝑐42𝑐21𝑐2𝑐4+3𝑐1𝑐22𝑐3𝛽2𝑐41𝑐2𝑐4𝑐51×𝑒4𝑛+.(2.22)
Substituting appropriate expressions in (2.8), we obtain 𝑒𝑛+1=𝑧𝑛𝑧𝛼𝜓𝜔𝜉𝑓𝑛.(2.23)
Simplifying (2.23), we obtain the error equation 𝑒𝑛+1=𝑐1730𝛽2𝑐21𝑐222𝑐21𝑐42𝑐4+26𝛽𝑐1𝑐72+14𝛽3𝑐31𝑐722𝑐21𝑐32𝑐23+9𝛽2𝑐31𝑐52𝑐312𝛽2𝑐41𝑐42𝑐48𝛽3𝑐51𝑐42𝑐4+3𝛽𝑐21𝑐42𝑐38𝛽𝑐31𝑐42𝑐4+9𝛽3𝑐41𝑐52𝑐312𝛽2𝑐41𝑐32𝑐238𝛽𝑐31𝑐32𝑐238𝛽3𝑐31𝑐32𝑐23+8𝑐72+2𝛽4𝑐41𝑐72+𝑐31𝑐22𝑐3𝑐4+𝛽4𝑐71𝑐22𝑐3𝑐42𝛽4𝑐61𝑐42𝑐4+4𝛽𝑐41𝑐22𝑐3𝑐4+6𝛽2𝑐51𝑐22𝑐3𝑐4+4𝛽3𝑐51𝑐22𝑐3𝑐42𝛽4𝑐61𝑐32𝑐23+3𝛽4𝑐51𝑐52𝑐3𝑒8𝑛.(2.24)
Expression (2.24) establishes the asymptotic error constant for the eighth order of convergence for the new eighth-order derivative-free method defined by (2.8).

2.2. Method 2: Liu 1

The second of three-step eighth-order derivative-free method is constructed by combining the two-step fourth-order method presented by Liu et al. [2], and the third step is developed to achieve the eighth order. As before, we have introduced two weight functions in the third step in order to achieve the desired eighth-order method. In this particular case the two weight functions are expressed as𝑓𝑧1𝑛𝑓𝑤𝑛1,𝑦1+2𝑓𝑛3𝑓𝑤𝑛2𝑓𝑥𝑛1.(2.25) Then the iteration scheme based on Liu et al. method is given as𝑧𝑛=𝑦𝑛𝑓𝑦𝑛𝑓𝑥𝑛,𝑦𝑛𝑤+𝑓𝑛,𝑦𝑛𝑤𝑓𝑛,𝑥𝑛𝑥,(2.26)𝑛+1=𝑧𝑛𝑓𝑧1𝑛𝑓𝑤𝑛1𝑓𝑦1+𝑛3𝑓𝑥𝑛𝑓𝑧𝑛21𝑓𝑧𝑛𝑓𝑦𝑛,𝑧𝑛𝑥𝑓𝑛,𝑦𝑛𝑥+𝑓𝑛,𝑧𝑛,(2.27) where 𝑤𝑛,𝑦𝑛 are given in (2.8) and 𝑥0 is the initial approximation provided that the denominators of (2.26)-(2.27) are not equal to zero.

Theorem 2.5. Assume that the function f is sufficiently differentiable and f has a simple root 𝛼𝐷. If the initial approximation 𝑥0 is sufficiently close to 𝛼, then the method defined by (2.27) converges to 𝛼with eighth order.

Proof. Using appropriate expressions in the proof of Theorem 2.4 and substituting them into (2.27), we obtain the asymptotic error constant 𝑒𝑛+1=𝑐1712𝛽2𝑐21𝑐72𝑐21𝑐42𝑐4+10𝛽𝑐1𝑐72𝑐1𝑐52𝑐3+6𝛽3𝑐31𝑐722𝑐21𝑐32𝑐236𝛽2𝑐41𝑐42𝑐44𝛽3𝑐51𝑐42𝑐42𝛽𝑐21𝑐52𝑐34𝛽𝑐31𝑐42𝑐4+2𝛽3𝑐41𝑐52𝑐312𝛽2𝑐41𝑐32𝑐238𝛽𝑐31𝑐32𝑐23+3𝑐72+𝛽4𝑐41𝑐72+𝑐31𝑐22𝑐3𝑐4+𝛽4𝑐71𝑐22𝑐3𝑐4𝛽4𝑐61𝑐42𝑐4+4𝛽𝑐41𝑐22𝑐3𝑐4+6𝛽2𝑐51𝑐22𝑐3𝑐4+4𝛽3𝑐61𝑐22𝑐3𝑐42𝛽4𝑐61𝑐32𝑐23+𝛽4𝑐51𝑐52𝑐38𝛽3𝑐51𝑐32𝑐23𝑒8𝑛.(2.28)
Expression (2.28) establishes the asymptotic error constant for the eighth order of convergence for the new eighth-order derivative-free method defined by (2.27).

2.3. Method 3: Liu 2

The third of three-step eighth-order derivative-free method is constructed by combining the two-step fourth-order method presented by Liu et al. [2], and the third step is developed to achieve the eighth-order. As before, we have introduced two weight functions in the third step in order to achieve the desired eighth-order method. In this particular case the two weight functions are expressed as𝑓𝑧1𝑛𝑓𝑤𝑛1,𝑦(2.29)1+2𝑓𝑛3𝑓𝑤𝑛2𝑓𝑥𝑛1.(2.30) Then the iteration scheme based on Liu et al. method is given as𝑧𝑛=𝑦𝑛𝑓𝑦𝑛𝑓𝑥𝑛,𝑦𝑛𝑤𝑓𝑛,𝑦𝑛𝑤+𝑓𝑛,𝑥𝑛𝑓𝑥𝑛,𝑦𝑛2,𝑥(2.31)𝑛+1=𝑧𝑛𝑓𝑧1𝑛𝑓𝑤𝑛1𝑓𝑦1𝑛𝑓𝑧𝑛3𝑓𝑦1+𝑛3𝑓𝑥𝑛𝑓𝑧𝑛21×𝑓𝑧𝑛𝑓𝑦𝑛,𝑧𝑛𝑥𝑓𝑛,𝑦𝑛𝑥+𝑓𝑛,𝑧𝑛,(2.32) where 𝑤𝑛,𝑦𝑛 are given in (2.8) and 𝑥0 is the initial approximation provided that the denominators of (2.31)-(2.32) are not equal to zero.

Theorem 2.6. Assume that the function f is sufficiently differentiable and f has a simple root 𝛼𝐷. If the initial approximation 𝑥0 is sufficiently close to 𝛼, then the method defined by (2.32) converges to 𝛼with eighth order.

Proof. Using appropriate expressions in the proof of Theorem 2.4 and substituting them into (2.32), we obtain the asymptotic error constant 𝑒𝑛+1=𝑐17𝛽4𝑐71𝑐22𝑐3𝑐4𝛽4𝑐61𝑐42𝑐42𝛽4𝑐61𝑐32𝑐23+𝛽4𝑐51𝑐52𝑐3+𝛽4𝑐41𝑐72+4𝛽3𝑐61𝑐22𝑐3𝑐45𝛽3𝑐51𝑐42𝑐4+3𝛽3𝑐41𝑐52𝑐3+8𝛽3𝑐31𝑐72+6𝛽2𝑐51𝑐22𝑐3𝑐412𝛽2𝑐41𝑐32𝑐239𝛽2𝑐41𝑐42𝑐4+3𝛽2𝑐31𝑐52𝑐3+21𝛽2𝑐21𝑐72+4𝛽𝑐41𝑐22𝑐3𝑐47𝛽𝑐31𝑐42𝑐48𝛽𝑐31𝑐32𝑐23+𝛽𝑐21𝑐52𝑐3+22𝛽𝑐1𝑐722𝑐21𝑐32𝑐232𝑐21𝑐42𝑐4+8𝑐72+𝑐31𝑐22𝑐3𝑐48𝛽3𝑐51𝑐32𝑐23𝑒8𝑛.(2.33)
Expression (2.33) establishes the asymptotic error constant for the eighth order of convergence for the new eighth-order derivative-free method defined by (2.32).

2.4. Method 4: SKK

The fourth of three-step eighth-order derivative-free method is constructed by combining the two-point fourth-order method presented by Khattri and Agarwal [10], and the third point is developed to achieve the eighth order. Here also, we have introduced two weight functions in the third step in order to achieve the desired eighth-order method. In this particular case the two weight functions are expressed as𝑓𝑧1𝑛𝑓𝑤𝑛1,𝑦1+3𝑓𝑛3𝑓𝑤𝑛2𝑓𝑥𝑛1.(2.34) Then the iteration scheme based on the Khattri and Agarwal method is given as𝑤𝑛=𝑥𝑛𝑥𝛽𝑓𝑛𝑦,(2.35)𝑛=𝑥𝑛𝑓𝑥𝑛2𝑓𝑥𝑛𝑤𝑓𝑛𝑧,(2.36)𝑛=𝑦𝑛𝑓𝑥𝑛𝑓𝑦𝑛𝑓𝑥𝑛𝑤𝑓𝑛𝑓𝑦1+𝑛𝑓𝑥𝑛+𝑓𝑦𝑛𝑓𝑥𝑛2+𝑓𝑦𝑛𝑓𝑤𝑛+𝑓𝑦𝑛𝑓𝑤𝑛2𝑥,(2.37)𝑛+1=𝑧𝑛𝑓𝑤𝑛𝑓𝑤𝑛𝑧𝑓𝑛𝑦1+3𝑓𝑛3𝑓𝑥𝑛𝑓𝑧𝑛21𝑓𝑧𝑛𝑓𝑦𝑛,𝑧𝑛𝑥𝑓𝑛,𝑦𝑛𝑥+𝑓𝑛,𝑧𝑛,(2.38) where 𝑤𝑛,𝑦𝑛 are given in (2.8) and 𝑥0 is the initial approximation provided that the denominators of (2.35)–(2.37) are not equal to zero.

Theorem 2.7. Assume that the function f is sufficiently differentiable and f has a simple root 𝛼𝐷. If the initial approximation 𝑥0 is sufficiently close to 𝛼, then the method defined by (2.38) converges to 𝛼with eighth order.

Proof. Using appropriate expressions in the proof of Theorem 2.4 and substituting them into (2.38), we obtain the asymptotic error constant 𝑒𝑛+1=𝑐178𝛽𝑐31𝑐32𝑐23+𝛽4𝑐51𝑐52𝑐32𝛽4𝑐61𝑐32𝑐23+15𝑐72+60𝛽2𝑐21𝑐724𝛽𝑐41𝑐22𝑐3𝑐4+6𝛽2𝑐51𝑐22𝑐3𝑐4+𝑐31𝑐22𝑐3𝑐44𝛽3𝑐61𝑐22𝑐3𝑐4+15𝛽4𝑐41𝑐72+𝛽4𝑐71𝑐22𝑐3𝑐43𝛽4𝑐61𝑐42𝑐48𝛽𝑐21𝑐52𝑐33𝑐21𝑐42𝑐4+12𝛽𝑐31𝑐42𝑐418𝛽2𝑐41𝑐42𝑐4+12𝛽3𝑐51𝑐42𝑐42𝑐21𝑐32𝑐23+𝑐1𝑐52𝑐348𝛽𝑐1𝑐7239𝛽3𝑐31𝑐72+𝛽5𝑐61𝑐52𝑐33𝛽5𝑐51𝑐7211𝛽3𝑐41𝑐52𝑐3+8𝛽3𝑐51𝑐32𝑐23+16𝛽𝑐31𝑐52𝑐312𝛽2𝑐41𝑐32𝑐23𝑒8𝑛.(2.39)
Expression (2.38) establishes the asymptotic error constant for the eighth order of convergence for the new eighth-order derivative-free method defined by (2.38).

3. The Kung-Traub Eighth-Order Derivative-Free Method

The classical Kung-Traub eighth-order derivative-free method considered is given in [3]. Since this method 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 eighth-order derivative-free methods. The Kung-Traub method is given as𝑧𝑛=𝑦𝑛𝑓𝑥𝑛𝑓𝑤𝑛𝑓𝑦𝑛𝑥𝑓𝑛1𝑓[]1𝑤,𝑥𝑓[],𝑥𝑤,𝑦𝑛+1=𝑧𝑛𝑓𝑤𝑛𝑓𝑥𝑛𝑓𝑦𝑛𝑓𝑧𝑛𝑥𝑓𝑛×1𝑓𝑧𝑛𝑤𝑓𝑛1𝑓[]1𝑦,𝑧𝑓[]1𝑤,𝑦𝑓𝑦𝑛𝑥𝑓𝑛1𝑓[]1𝑤,𝑦𝑓[],𝑤,𝑥(3.1) where 𝑤𝑛,𝑦𝑛 are given in (2.8) and 𝑥0 is the initial approximation provided that the denominators of (3.1) 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 ten 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 fact, the errors displayed are of absolute value, and insignificant approximations by the various methods have been omitted in Tables 1, 2, and 3.

Remark 4.1. The family of three-step 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 [3]. To determine the efficiency index of these new derivative-free methods, we will use Definition 2.3. Hence, the efficiency index of the eighth-order derivative-free methods given is 481.682.

Remark 4.2. The test functions and their exact root 𝛼 are displayed in Table 1. The differences between the root 𝛼 and the approximation 𝑥𝑛 for test functions with initial approximation 𝑥0 are displayed in Table 2. In fact, 𝑥𝑛 is calculated by using the same total number of function evaluations (TNFEs) for all methods. Here, the TNFE for all the methods is 12. Furthermore, the computational order of convergence (COC) is displayed in Table 3.

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 [4, 5]. 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 derivative-free methods for many different types of nonlinear equations was purely to illustrate the accuracy of the approximate solution, the stability of the convergence, and the consistency of the results and to determine the efficiency of the new iterative method. In addition, it should be noted that like all other iterative methods, the new methods have their own domain of validity and in certain circumstances should not be used.