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

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 of 2^𝑛−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.

1. Introduction

In this paper, we present a new family of the eighth-order methods to find a simple root 𝛼 of the nonlinear equation:𝑓(𝑥)=0, (1.1) 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 on n evaluations, could achieve optimal convergence order 2^𝑛−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 Section 2, 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. Let 𝑓(𝑥) be a real function with a simple root 𝛼 and let {𝑥_𝑛} be a sequence of real numbers that converge towards 𝛼. The order of convergence m is given by lim_𝑛→∞𝑥_𝑛+1−𝛼𝑥_𝑛−𝛼^𝑚=𝜁≠0, (2.1) where 𝜁 is the asymptotic error constant and 𝑚∈ℝ⁺.

Definition 2.2. Suppose that 𝑥_𝑛−1,𝑥_𝑛 and 𝑥_𝑛+1 are three successive iterations closer to the root 𝛼 of (1.1). Then, the computational order of convergence [8] may be approximated by 𝑥COC≈ln_𝑛+1−𝛼𝑥_𝑛−𝛼⁻¹ln𝑥_𝑛−𝛼𝑥_𝑛−1−𝛼⁻¹, (2.2) where 𝑛∈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 [9, 10] and defined as 𝜇^1/𝛽, (2.3) 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 parameters 𝛽_𝑖,𝜙_𝑗,𝜔_𝑘, and 𝜉_𝑙. Therefore, the general formula of the new eighth-order method for determining the simple root of (1.1) is given as: 𝑤_𝑛=𝑥_𝑛+𝛽_𝑖⁻¹𝑓𝑥_𝑛,𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛²𝑓𝑤_𝑛−𝑓𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−𝜙_𝑗𝑥_𝑛−𝑦_𝑛𝑓𝑥_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝜔_𝑘𝜉_𝑙𝑓𝑧_𝑛−𝑓𝑦_𝑛𝑧_𝑛−𝑦_𝑛−𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑦_𝑛−𝑥_𝑛+𝑓𝑧_𝑛−𝑓𝑥_𝑛𝑧_𝑛−𝑥_𝑛⁻¹𝑓𝑧_𝑛, (2.4)(2.5)(2.6)(2.7) 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: 𝛽_𝑖=𝑖⁻¹,𝑖∈ℝ⁺,𝜙₁=𝑓𝑦1−^𝑛𝑓𝑤_𝑛⁻¹,𝜙₂=𝑓𝑦1+^𝑛𝑓𝑤_𝑛,𝜔₁=𝑓𝑧1−^𝑛𝑓𝑤_𝑛⁻¹,𝜔₂=𝑓𝑧1+^𝑛𝑓𝑤_𝑛+𝑓𝑧_𝑛𝑓𝑤_𝑛²,𝜉₁=𝑦1−2𝑓_𝑛³𝑓𝑤_𝑛²𝑓𝑥_𝑛,𝜉₂=𝑦1+2𝑓_𝑛³𝑓𝑤_𝑛²𝑓𝑥_𝑛⁻¹. (2.8)

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. 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=𝑥−𝛼. (2.9)
Using the Taylor expansion, we have 𝑓𝑥_𝑛=𝑓(𝛼)+𝑓^{′(𝛼)𝑒𝑛}+2⁻¹𝑓^{′′(𝛼)𝑒2𝑛}+6⁻¹𝑓^{′′′(𝛼)𝑒3𝑛}+24⁻¹𝑓^𝑖𝑣(𝛼)𝑒⁴_𝑛+⋅⋅⋅. (2.10)
Taking 𝑓(𝛼)=0 and simplifying, expression (2.10) becomes 𝑓𝑥_𝑛=𝑐₁𝑒_𝑛+𝑐₂𝑒²_𝑛+𝑐₃𝑒³_𝑛+𝑐₄𝑒⁴_𝑛+⋅⋅⋅, (2.11) where 𝑛∈ℕ and 𝑐_𝑘=𝑓^(𝑘)(𝛼)(𝑘!)for𝑘=1,2,3,4,…. (2.12)
Expanding the Taylor series of 𝑓(𝑤_𝑛) and substituting 𝑓(𝑥_𝑛) given by (2.10), we have 𝑓𝑤_𝑛=𝑐₁1+𝑐₁𝛽𝑒_𝑛+3𝛽𝑐₁𝑐₂+𝛽²𝑐²₁𝑐₂+𝑐₂𝑒²_𝑛+⋅⋅⋅. (2.13)
Substituting (2.11) and (2.13) in the expression (2.5) gives us 𝑦_𝑛−𝛼=𝑥_𝑛𝑥−𝛼−^𝑛−𝑤_𝑛𝑓𝑥_𝑛−𝑓𝑤_𝑛𝑓𝑥_𝑛=𝑐₂𝑐₁𝛽𝑐₁+1𝑒²_𝑛+⋅⋅⋅. (2.14) The expansion of 𝑓(𝑦_𝑛) about 𝛼 is given as 𝑓𝑦_𝑛=𝑐₁𝑦_𝑛−𝛼+𝑐₂𝑦_𝑛−𝛼²+𝑐₃𝑦_𝑛−𝛼³+⋅⋅⋅. (2.15)
Simplifying (2.15), we have 𝑓𝑦_𝑛=𝑐₂𝑐₁𝛽+1𝑒²_𝑛+𝛽𝑐³₁𝑐₃−2𝑐²₂+3𝛽𝑐²₁𝑐₃+2𝑐₁𝑐₃−𝛽²𝑐²₁𝑐²₂−2𝛽𝑐₁𝑐²₂𝑐₁𝑒³_𝑛+⋅⋅⋅. (2.16)
The expansion of the particular term used in (2.6) is given as 𝜙₁=𝑓𝑦1−^𝑛𝑓𝑤_𝑛⁻¹𝑐=1+²𝑐₁𝑒_𝑛+𝛽𝑐²₁𝑐₃−2𝛽𝑐₁𝑐²₂+𝛽𝑐₁𝑐₃−2𝑐²₂𝜆𝑐²₁𝑒²_𝑛+⋅⋅⋅. (2.17) Substituting appropriate expressions in (2.6), we obtain 𝑧_𝑛−𝛼=𝑦_𝑛𝑓𝑦−𝛼−1−^𝑛𝑓𝑤_𝑛𝑥_𝑛−𝑦_𝑛𝑓𝑥_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛. (2.18) The Taylor series expansion of 𝑓(𝑧_𝑛) about 𝛼 is given as 𝑓𝑧_𝑛=𝑐₁𝑧_𝑛−𝛼+𝑐₂𝑧_𝑛−𝛼²+𝑐₃𝑧_𝑛−𝛼³+⋅⋅⋅. (2.19)
Simplifying (2.19), we have 𝑓𝑧_𝑛=2𝑐³₂−𝑐₁𝑐₂𝑐₃+4𝛽𝑐₁𝑐³₂+2𝛽²𝑐²₁𝑐³₂−2𝛽𝑐²₁𝑐₂𝑐₃−𝑐³₁𝑐₂𝑐₃𝑐³₁𝑒⁴_𝑛+⋅⋅⋅. (2.20)
In order to evaluate the essential terms of (2.7), we expand term by term 𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑦_𝑛−𝑥_𝑛=𝑐₁+𝑐₂𝑒_𝑛+𝑐₁𝑐₃+𝛽𝑐₁𝑐²₂+𝑐²₂𝑐₁𝑒²_𝑛𝑓𝑧+⋅⋅⋅,_𝑛−𝑓𝑦_𝑛𝑧_𝑛−𝑦_𝑛=𝑐₁+𝛽𝑐₁𝑐²₂+𝑐²₂𝑐₁𝑒²_𝑛𝑓𝑧+⋅⋅⋅,_𝑛−𝑓𝑥_𝑛𝑧_𝑛−𝑥_𝑛=𝑐₁+𝑐₂𝑒_𝑛+𝑐₃𝑒²_𝑛+⋅⋅⋅. (2.21) Collecting the above terms 𝑓𝑦𝜓=^𝑛−𝑓𝑥_𝑛𝑦_𝑛−𝑥_𝑛−𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑦_𝑛−𝑥_𝑛+𝑓𝑧_𝑛−𝑓𝑥_𝑛𝑧_𝑛−𝑥_𝑛⁻¹=1𝑐₁+𝑐₂𝑐₃+𝛽𝑐₁𝑐₂𝑐₃𝑐³₁𝑒³_𝑛𝜔+⋅⋅⋅,₁=𝑓𝑧1−^𝑛𝑓𝑤_𝑛⁻¹=1−𝛽𝑐²₁𝑐₂𝑐₃−2𝛽𝑐₁𝑐³₂+𝑐₁𝑐₂𝑐₃−2𝑐³₂𝑐³₁𝑒³_𝑛𝜉+⋅⋅⋅,₁=𝑦1−2𝑓_𝑛³𝑓𝑤_𝑛²𝑓𝑥_𝑛=1−2𝛽𝑐₁𝑐³₂+2𝑐³₂𝑐³₁𝑒³_𝑛𝜔+⋅⋅⋅,₁𝜉₁=1−𝛽𝑐₁𝑐₂𝑐₃+𝑐₂𝑐₃𝑐²₁𝑒³_𝑛+⋅⋅⋅,𝜓𝜔₁𝜉₁=1𝑐₁+𝛽²𝑐²₁𝑐⁴₂+6𝛽𝑐²₁𝑐²₂𝑐₃+3𝛽²𝑐³₁𝑐²₂𝑐₃+𝛽𝑐₁𝑐⁴₂−2𝛽𝑐³₁𝑐₂𝑐₄+3𝑐⁴₂−𝑐²₁𝑐₂𝑐₄+3𝑐₁𝑐²₂𝑐₃−𝛽²𝑐⁴₁𝑐₂𝑐₄𝑐⁵₁×𝑒⁴_𝑛+⋅⋅⋅. (2.22)(2.23) Substituting appropriate expressions in (2.7), we obtain 𝑒_𝑛+1=𝑧_𝑛−𝛼−𝜓𝜔₁𝜉₁𝑓𝑧_𝑛. (2.24) Simplifying (2.24), we obtain the error equation 𝑒_𝑛+1=𝑐₁⁻⁷8𝛽³𝑐⁵₁𝑐³₂𝑐²₃−10𝑐⁷₂−32𝛽²𝑐²₁𝑐⁷₂−𝛽𝑐³₁𝑐⁵₂𝑐₃+12𝛽²𝑐⁴₁𝑐³₂𝑐²₃+8𝛽𝑐³₁𝑐³₂𝑐²₃−𝛽⁴𝑐⁷₁𝑐²₂𝑐₃𝑐₄+8𝛽𝑐³₁𝑐⁴₂𝑐₄+12𝛽²𝑐⁴₁𝑐⁴₂𝑐₄−9𝛽³𝑐⁴₁𝑐⁵₂𝑐₃−8𝛽²𝑐³₁𝑐⁵₂𝑐₃−3𝛽⁴𝑐⁵₁𝑐⁵₂𝑐₃+2𝛽⁴𝑐⁶₁𝑐⁴₂𝑐₄−6𝛽²𝑐⁵₁𝑐²₂𝑐₃𝑐₄−4𝛽𝑐⁴₁𝑐²₂𝑐₃𝑐₄−4𝛽³𝑐⁶₁𝑐²₂𝑐₃𝑐₄+2𝛽⁴𝑐⁶₁𝑐³₂𝑐²₃−𝑐³₁𝑐²₂𝑐₃𝑐₄−2𝛽⁴𝑐⁴₁𝑐⁷₂−30𝛽𝑐₁𝑐⁷₂−14𝛽³𝑐³₁𝑐⁷₂+𝑐₁𝑐⁵₂𝑐₃+2𝑐²₁𝑐³₂𝑐²₃+2𝑐²₁𝑐⁴₂𝑐₄+8𝛽³𝑐⁵₁𝑐⁴₂𝑐₄𝑒⁸_𝑛. (2.25) 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. 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−2𝑓𝑥_𝑛−𝑓𝑦_𝑛2𝑓𝑥_𝑛−5𝑓𝑦_𝑛𝑓𝑦_𝑛𝑓^{′(𝑥𝑛)},𝑥_𝑛+1=𝑧_𝑛−𝑓𝑥_𝑛+𝛾+2𝑓𝑧_𝑛𝑓𝑥_𝑛+𝛾𝑓𝑧_𝑛𝑓𝑧_𝑛𝑓𝑧_𝑛,𝑦_𝑛+𝑓𝑧_𝑛,𝑥_𝑛,𝑥_𝑛𝑧_𝑛−𝑦_𝑛, (3.1)(3.2)(3.3) where [𝑧_𝑛,𝑦_𝑛]=(𝑓(𝑧_𝑛)−𝑓(𝑦_𝑛))/(𝑧_𝑛−𝑦_𝑛),  𝛾∈ℝ, 𝑓(𝑦_𝑛) 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. 𝑧_𝑛=𝑦_𝑛−𝑓𝑦1+2^𝑛𝑓𝑥_𝑛𝑓𝑦+5^𝑛𝑓𝑥_𝑛²+𝑓𝑦_𝑛𝑓𝑥_𝑛³𝑓𝑦_𝑛𝑓^{′(𝑥𝑛)},𝑥_𝑛+1=𝑧_𝑛−𝑓𝑥_𝑛+𝛾+2𝑓𝑧_𝑛𝑓𝑥_𝑛+𝛾𝑓𝑧_𝑛𝑓𝑧_𝑛𝑓𝑧_𝑛,𝑦_𝑛+𝑓𝑧_𝑛,𝑥_𝑛,𝑥_𝑛𝑧_𝑛−𝑦_𝑛, (3.4)(3.5) where 𝛾,𝜇∈ℝ, 𝑓(𝑦_𝑛) is given by (3.1) and provided that the denominators of (3.4) and (3.5) are not equal to zero.

Method 3. 𝑧_𝑛=𝑦_𝑛−𝑓𝑦1−2^𝑛𝑓𝑥_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛²+𝑓𝑦_𝑛𝑓𝑥_𝑛³⁻¹𝑓𝑦_𝑛𝑓^{′(𝑥𝑛)},𝑥_𝑛+1=𝑧_𝑛−𝑓𝑥_𝑛+𝛾+2𝑓𝑧_𝑛𝑓𝑥_𝑛+𝛾𝑓𝑧_𝑛𝑓𝑧_𝑛𝑓𝑧_𝑛,𝑦_𝑛+𝑓𝑧_𝑛,𝑥_𝑛,𝑥_𝑛𝑧_𝑛−𝑦_𝑛, (3.6)(3.7) where 𝛾,𝜇∈ℝ, 𝑓(𝑦_𝑛) 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. 𝑧_𝑛=𝑦_𝑛−𝑓𝑥_𝑛−3𝑓𝑦_𝑛𝑓𝑥_𝑛^(−2/3)𝑓𝑦_𝑛𝑓^{′(𝑥𝑛)}𝑥_𝑛+1,=𝑧_𝑛−𝑓𝑥_𝑛+𝛾+2𝑓𝑧_𝑛𝑓𝑥_𝑛+𝛾𝑓𝑧_𝑛𝑓𝑧_𝑛𝑓𝑧_𝑛,𝑦_𝑛+𝑓𝑧_𝑛,𝑥_𝑛,𝑥_𝑛𝑧_𝑛−𝑦_𝑛, (3.8)(3.9) where 𝛾∈ℝ, 𝑓(𝑦_𝑛) 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. 𝑧_𝑛=𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛−2𝑓𝑦_𝑛𝑓𝑦_𝑛𝑓^{′(𝑥𝑛)},𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧1+^𝑛𝑓𝑥_𝑛𝑓𝑧+𝛾^𝑛𝑓𝑥_𝑛²𝑓𝑥_𝑛,𝑦_𝑛𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛, (3.10)(3.11) where 𝛾∈ℝ, 𝑓(𝑦_𝑛) is given by (3.1) and provided that the denominators of (3.10) and (3.11) are not equal to zero.

Method 6. 𝑧_𝑛=𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛−2𝑓𝑦_𝑛𝑓𝑦_𝑛𝑓^{′(𝑥𝑛)},𝑥_𝑛+1=𝑧_𝑛−𝑓𝑥_𝑛+𝛾+1𝑓𝑧_𝑛𝑓𝑥_𝑛+𝛾𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑦_𝑛𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛, (3.12)(3.13) where 𝛾,𝛽∈ℝ, 𝑓(𝑦_𝑛) is given by (3.1) and provided that the denominators of (3.12) and (3.13) are not equal to zero.

Method 7. 𝑧_𝑛=𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛−2𝑓𝑦_𝑛𝑓𝑦_𝑛𝑓^{′(𝑥𝑛)},𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧1+𝛾^𝑛𝑓𝑥_𝑛^(1/𝛾)𝑓𝑥_𝑛,𝑦_𝑛𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛, (3.14)(3.15) where 𝛾∈ℝ, 𝑓(𝑦_𝑛) 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 𝑧_𝑛=𝑥_𝑛−𝑓𝑥_𝑛²+𝑓𝑦_𝑛²𝑓^{′(𝑥𝑛)(𝑓(𝑥𝑛)−𝑓(𝑦𝑛))},𝑥_𝑛+1=𝑧_𝑛−1+𝜇²_𝑛1−𝜇_𝑛²−2𝜇_𝑛²−6𝜇_𝑛³+𝑓𝑧_𝑛𝑓𝑦_𝑛𝑓𝑧+4^𝑛𝑓𝑥_𝑛𝑓𝑧_𝑛𝑓^{′(𝑥𝑛)}, (3.16)(3.17) where 𝜇_𝑛=(𝑓(𝑦_𝑛)/𝑓(𝑥_𝑛))𝑛∈ℕ, 𝑓(𝑦_𝑛) 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 take 𝑖=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 √⁴8≈1.68.

Remark 4.2. In Tables 1–6, 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 |𝑥_𝑛−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.

Table 2 

Errors occurring in the estimates of the root of (4.2) by the methods described.

Table 3 

Errors occurring in the estimates of the root of (4.3) by the methods described.

Table 4 

Errors occurring in the estimates of the root of (4.4) by the methods described.

Table 5 

Errors occurring in the estimates of the root of (4.5) by the methods described.

Table 6 

Errors occurring in the estimates of the root of (4.6) by the methods described.

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: 𝑓(𝑥)=𝑒^−𝑥−cos(𝑥), (4.1) 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: 𝑓(𝑥)=ln𝑥²+𝑥+2−𝑥+1, (4.2) 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: 𝑓(𝑥)=sin(𝑥)²−𝑥²+1, (4.3) 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: 𝑓(𝑥)=exp−𝑥²+𝑥+2−cos(𝑥+1)+𝑥³+1, (4.4) 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: 𝑓(𝑥)=𝑥¹¹+𝑥+1, (4.5) 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: 𝑓(𝑥)=(𝑥−2)𝑥¹⁰+𝑥+1𝑒^−𝑥−1, (4.6) 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.