Construction of Optimal Derivative-Free Techniques without Memory

Abstract

Construction of iterative processes without memory, which are both optimal according to the Kung-Traub hypothesis and derivative-free, is considered in this paper. For this reason, techniques with four and five function evaluations per iteration, which reach to the optimal orders eight and sixteen, respectively, are discussed theoretically. These schemes can be viewed as the generalizations of the recent optimal derivative-free family of Zheng et al. in (2011). This procedure also provides an n-step family using 𝑛+1 function evaluations per full cycle to possess the optimal order 2ⁿ. The analytical proofs of the main contributions are given and numerical examples are included to confirm the outstanding convergence speed of the presented iterative methods using only few function evaluations. The second aim of this work will be furnished when a hybrid algorithm for capturing all the zeros in an interval has been proposed. The novel algorithm could deal with nonlinear functions having finitely many zeros in an interval.

1. Introduction

The purpose of this study is to present some generalizations of both the celebrated second-order Steffensen’s method [1], which was advanced by the Danish mathematician Johan Frederik Steffensen (1873–1961) as follows 𝑥_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛+𝑓𝑥_𝑛−𝑓𝑥_𝑛⁻¹𝑓𝑥_𝑛²,𝑛=0,1,2,…, (1.1) with higher orders of convergence and optimal efficiency indices, and also the extension of the family of methods in [2]. In 1974, Kung and Traub [3] conjectured on the optimality of multipoint iterative schemes without memory as comes next. An iterative method for solving single variable nonlinear equation 𝑓(𝑥)=0, with 𝑛+1,𝑛≥1, evaluations per iteration reaches to the maximum order of convergence 2^𝑛 and the optimal efficiency index 2^{𝑛/(𝑛+1)}. Consequently, the efficiency of a 𝑝th-order method could be given by 𝑝^1/𝑑, where 𝑑 is the whole number of (functional) evaluations per iteration. Note that Steffensen’s method possesses 1.414 as its efficiency index. By considering this conjecture, per iteration the contributed techniques in this research should reach the order 8 with four and order 16 with five evaluations.

The paper unfolds the contents as follows. A collection of pointers to known literature on derivative-free techniques will be presented in Section 2. This is followed by Section 3, whereas the central results are given and also by Section 4, where an optimal sixteenth-order derivative-free technique is constructed. Results and discussion on the comparisons with other famous derivative-free methods are presented in Section 5, entitled by Computational Tests. The only difficulty of iterative methods of this type is in the choice of the initial guess. Using the programming package MATHEMATICA 8, we will provide an algorithm to capture all the real solutions of a nonlinear equation in a short piece of time by presenting a hybrid algorithm in Section 6. The conclusions have finally been drawn in Section 7.

2. Selections from the Literature

The literature related to the present paper is substantial, and we do not present a comprehensive survey. The references of the papers we cite should be consulted for further reading. Consider iterative techniques for finding a simple root of the nonlinear equation 𝑓(𝑥)=0, where 𝑓∶𝐷⊆ℝ→ℝ for an open interval 𝐷=(𝑎,𝑏) is a scalar function and it is sufficiently smooth in a neighborhood of 𝛼∈𝐷. The design of formulas for solving such equations is an absorbing task in numerical analysis [4]. The most frequent approach to solve the nonlinear equations consists of the implementation of rapidly convergent iterative methods starting from a reasonably good initial guess to the sought zero of 𝑓(𝑥).

Many iterative methods have been improved by using various techniques such as quadrature formulas and weight function; see, for example, [5, 6]. All these developments are aimed at increasing the local order of convergence with a special view of increasing their efficiency indices. The derivative-involved methods are well discussed in the literature; see, for example, [7–9] and the references therein. But another thing that should be mentioned is that for many particular choices of the function 𝑓, specially in hard problems, the calculations of the derivatives are not possible or it takes a deal of time.

Thus, in some situations the considered function 𝑓(𝑥) has an improper behavior or its derivative is close to 0, which causes that the applied iterative processes fail. That is why higher-order derivative-free methods are better root solvers and are in focus recently. The most important merit of Steffensen’s method is that it has quadratic convergence like Newton’s method. That is, both techniques estimate roots of the equation 𝑓(𝑥)=0 just as quickly. In this case, quickly means that the number of correct digits in the new obtained value doubles with each iteration for both. But the formula for Newton’s method requires a separate function for the derivative; Steffensen’s method does not. Now let us review some derivative-free techniques.

In 2010, an optimal fourth-order derivative-free method [10] was introduced by Liu et al. in the following form: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛²𝑓𝑥_𝑛+𝑓𝑥_𝑛−𝑓𝑥_𝑛,𝑥_𝑛+1=𝑦_𝑛−𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑦_𝑛,𝑧_𝑛+𝑓𝑥_𝑛,𝑧_𝑛𝑓[𝑥_𝑛,𝑦_𝑛]²𝑓𝑦_𝑛, (2.1) where 𝑧_𝑛=𝑥_𝑛+𝑓(𝑥_𝑛). This technique consists of three evaluations of the function per iteration to obtain the fourth-order convergence. We here remark that 𝑓[𝑥_𝑛,𝑦_𝑛],𝑓[𝑦_𝑛,𝑧_𝑛], and 𝑓[𝑥_𝑛,𝑧_𝑛] are divided differences. This scheme has 4^1/3≈1.587 as its efficiency index. The notation of divided differences will be used throughout this paper.

In 2011, Cordero et al. [11] proposed a sixth-order method, which is free from derivative 𝑦_𝑛=𝑥_𝑛−2𝑓𝑥_𝑛²𝑓𝑥_𝑛+𝑓𝑥_𝑛−𝑓𝑥_𝑛−𝑓𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−𝑦_𝑛−𝑥_𝑛2𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑦_𝑛−𝑥_𝑛2𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑧_𝑛, (2.2) and includes 5 evaluations of the function per iteration to reach the efficiency index 6^1/5≈1.495.

Zheng et al. in [2] presented the following eighth-order derivative-free family without memory: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑤_𝑛,𝑤_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑦_𝑛+𝑓𝑦_𝑛,𝑤_𝑛−𝑓𝑥_𝑛,𝑤_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑧_𝑛,𝑦_𝑛+𝑓𝑧_𝑛,𝑦_𝑛,𝑥_𝑛𝑧_𝑛−𝑦_𝑛+𝑓𝑧_𝑛,𝑦_𝑛,𝑥_𝑛,𝑤_𝑛𝑧_𝑛−𝑦_𝑛𝑧_𝑛−𝑥_𝑛, (2.3) which is optimal in the sense of Kung-Traub.

Soleymani [12] proposed the following optimal three-step iteration without memory, including four function evaluations, just like (2.3), by using the weight function approach: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑤_𝑛,𝑤_𝑛=𝑥_𝑛+𝑓𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑤_𝑛𝑓𝑦1+^𝑛𝑓𝑥_𝑛+𝑓𝑦_𝑛𝑓𝑤_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑦_𝑛𝑓𝑧1+^𝑛𝑓𝑥_𝑛+𝑓𝑧_𝑛𝑓𝑦_𝑛𝑓𝑧+2^𝑛𝑓𝑤_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛²+𝑓𝑦_𝑛𝑓𝑤_𝑛−1630+6𝑓𝑥_𝑛,𝑤_𝑛8+𝑓𝑥_𝑛,𝑤_𝑛5+𝑓𝑥_𝑛,𝑤_𝑛𝑓𝑦_𝑛𝑓𝑤_𝑛³. (2.4)

To see a recent paper including derivative-free methods with memory, refer to [13]. For further reading on this topic, refer to [14–19].

3. Construction of a New Eighth-Order Derivative-Free Class

In this paper, we derive a new way for constructing multistep methods of orders eight, sixteen, and so forth, requiring four, five, and so forth respectively, function evaluations per iteration. This means that the proposed techniques support the Kung-Traub hypothesis. Besides, the novel schemes do not use any derivative of a function 𝑓 whose zeros are sought, which is another advantage since it is preferable to avoid calculations of derivatives of 𝑓 in some situations. Let us consider a two-step cycle in which we have Steffensen’s method in the first step and Newton’s method in the second step as follows: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛²𝑓𝑥_𝑛+𝑓𝑥_𝑛−𝑓𝑥_𝑛,𝑥_𝑛+1=𝑦_𝑛−𝑓𝑦_𝑛𝑓^′𝑦_𝑛, (3.1) with four evaluations, that is, three evaluations of the function 𝑓(𝑥_𝑛),𝑓(𝑥_𝑛+𝑓(𝑥_𝑛)), and  𝑓(𝑦_𝑛) and one derivative evaluation 𝑓^′(𝑦_𝑛) per iteration. For simplicity, we assume that 𝑓(𝑥_𝑛+𝑓(𝑥_𝑛))=𝑓(𝑘_𝑛); that is, 𝑥_𝑛+𝑓(𝑥_𝑛)=𝑘_𝑛. At this time, the main challenge is to approximate 𝑓^′(𝑦_𝑛) as efficiently as possible such that the fourth-order convergence does not decrease and the efficiency index increases to 1.587 at the same time. To fulfill this aim, we must use all of the past three known data, that is, 𝑓(𝑥_𝑛),𝑓(𝑘_𝑛), and 𝑓(𝑦_𝑛). Now take into account the following approximation function for 𝑓(𝑡) in the domain 𝐷 [20]: 𝑓(𝑡)≈𝑤(𝑡)=𝑎₀+𝑎₁𝑡−𝑥_𝑛+𝑎₂𝑡−𝑥_𝑛²+𝑎₃𝑡−𝑥_𝑛³, (3.2) where its first derivative takes the form 𝑤^′(𝑡)=𝑎₁+2𝑎₂(𝑡−𝑥_𝑛)+3𝑎₃(𝑡−𝑥_𝑛)². Clearly, the unknown three parameters 𝑎₀,𝑎₁, and 𝑎₂ will be obtained by substituting of the known values in (3.2). Note that 𝑎₃ is a free real parameter. Hence, we have 𝑎₀=𝑓𝑥_𝑛,𝑎₂=𝑓𝑘_𝑛,𝑥_𝑛−𝑓𝑦_𝑛,𝑥_𝑛𝑘_𝑛−𝑦_𝑛−𝑎₃𝑥_𝑛+𝑦_𝑛+𝑧_𝑛,𝑎₁=𝑓𝑦_𝑛,𝑥_𝑛−𝑦_𝑛−𝑥_𝑛𝑎₂+𝑎₃𝑥_𝑛𝑦_𝑛+𝑥_𝑛𝑧_𝑛+𝑦_𝑛𝑧_𝑛. (3.3)

Accordingly, by considering 𝑎₁,𝑎₂ in the derivative form of (3.2), we attain the following two-step derivative-free technique: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛²𝑓𝑘_𝑛−𝑓𝑥_𝑛,𝑘_𝑛=𝑥_𝑛+𝑓𝑥_𝑛,𝑥_𝑛+1=𝑦_𝑛−𝑓𝑦_𝑛𝑎₁+2𝑎₂𝑦_𝑛−𝑥_𝑛+3𝑎₃𝑦_𝑛−𝑥_𝑛²,𝑎₃∈ℝ, (3.4) wherein there are three evaluations of the function per iteration only. Theorem 3.1 indicates that the order of (3.4) is four, and hence, it is an optimal derivative-free class with the efficiency index 1.587. Note that (3.4) is similar to the method given by Ren et al. in [20].

Theorem 3.1. Assume 𝑓(𝑥) to be a sufficiently continuous real function in the domain 𝐷. Then the sequence generated by (3.4) converges to the simple root 𝛼∈𝐷 with fourth-order convergence and it satisfies the follow-up error equation 𝑒_𝑛+1=𝑐₂𝑐²₂−𝑐₁𝑐₃+𝑎₃1+𝑐₁²𝑐³₁𝑒⁴_𝑛+𝑂𝑒⁵_𝑛, (3.5) wherein 𝑐_𝑗=(𝑓^(𝑗)(𝛼))/𝑗!,𝑗≥1, and 𝑒_𝑛=𝑥_𝑛−𝛼.

Proof. See [20].

The given approximation (3.3) of the first derivative of the function in the second step of our cycle can be applied on any optimal second-order derivative-free method to provide a new optimal fourth-order method. For example, if one chooses a two-step cycle in which the first derivative of the function in the first step estimated by backward finite difference, and after that applies the presented approximation in the second step, then another novel optimal fourth-order method could be obtained. This will be discussed more in the rest of the work.

Remark 3.2. The simplified form of the approximation for the derivative in the quotient of Newton’s iteration at the second step of our cycle is as follows: 𝑎₁+2𝑎₂𝑦_𝑛−𝑥_𝑛+3𝑎₃𝑦_𝑛−𝑥_𝑛²=𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛−𝑓𝑦_𝑛,𝑥_𝑛−𝑥_𝑛+𝑦_𝑛𝑘_𝑛−𝑦_𝑛+3𝑎₃𝑦_𝑛−𝑥_𝑛²=𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛. (3.6)
Inspired by the above approach, now we are about to construct new high-order derivative-free methods, which are optimal as well and can be considered as the generalizations of the methods in [2, 20, 21]. To construct such techniques, first we should consider an optimal fourth-order method in the first and second steps of a multistep cycle. Toward this end, we take into consideration the following three-step cycle in which Newton’s method is applied in the third step: 𝑦_𝑛=𝑥_𝑛−𝛽𝑓𝑥_𝑛²𝑓𝑘_𝑛−𝑓𝑥_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛,𝑎₃𝑥∈ℝ,_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓^′𝑧_𝑛. (3.7)
It is crystal clear that scheme (3.7) is an eighth-order method with five evaluations per iteration. The main question is that Is there any way to keep the order on eight but reduce the number of evaluation while the method be free from derivative? Fortunately, by using all four past known data, a very powerful approximation of 𝑓^′(𝑧_𝑛) will be obtained. Therefore, we approximate 𝑓(𝑡) in the domain 𝐷, by a new polynomial approximation of degree four with a free parameter 𝑏₄, as follows: 𝑓(𝑡)≈𝑙(𝑡)=𝑏₀+𝑏₁𝑡−𝑥_𝑛+𝑏₂𝑡−𝑥_𝑛²+𝑏₃𝑡−𝑥_𝑛³+𝑏₄𝑡−𝑥_𝑛⁴, (3.8) where its first derivative takes the following form: 𝑓^′(𝑡)≈𝑙^′(𝑡)=𝑏₁+2𝑏₂𝑡−𝑥_𝑛+3𝑏₃𝑡−𝑥_𝑛²+4𝑏₄𝑡−𝑥_𝑛³. (3.9)
Hopefully, we have four known values 𝑓(𝑥_𝑛),𝑓(𝑘_𝑛),𝑓(𝑦_𝑛), and 𝑓(𝑧_𝑛). Thus, by substituting these values in (3.8), we obtain the following linear system of four equations with four unknowns: 𝑓𝑥_𝑛=𝑏₀,𝑓𝑘_𝑛=𝑓𝑥_𝑛+𝑏₁𝑘_𝑛−𝑥_𝑛+𝑏₂𝑘_𝑛−𝑥_𝑛²+𝑏₃𝑘_𝑛−𝑥_𝑛³+𝑏₄𝑘_𝑛−𝑥_𝑛⁴,𝑓𝑦_𝑛=𝑓𝑥_𝑛+𝑏₁𝑦_𝑛−𝑥_𝑛+𝑏₂𝑦_𝑛−𝑥_𝑛²+𝑏₃𝑦_𝑛−𝑥_𝑛³+𝑏₄𝑦_𝑛−𝑥_𝑛⁴,𝑓𝑧_𝑛=𝑓𝑥_𝑛+𝑏₁𝑧_𝑛−𝑥_𝑛+𝑏₂𝑧_𝑛−𝑥_𝑛²+𝑏₃𝑧_𝑛−𝑥_𝑛³+𝑏₄𝑧_𝑛−𝑥_𝑛⁴, (3.10) and consequently we can find the unknown parameters by solving linear system (3.10). We attain 𝑏₀=𝑓𝑥_𝑛,𝑏₃=𝑓𝑘_𝑛,𝑥_𝑛−𝑓𝑧_𝑛,𝑥_𝑛/𝑘_𝑛−𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛−𝑓𝑧_𝑛,𝑥_𝑛/𝑦_𝑛−𝑧_𝑛𝑘_𝑛−𝑦_𝑛−𝑏₄𝑥_𝑛+𝑦_𝑛+𝑘_𝑛+𝑧_𝑛=𝑓𝑧_𝑛,𝑥_𝑛,𝑘_𝑛−𝑓𝑧_𝑛,𝑥_𝑛,𝑦_𝑛𝑘_𝑛−𝑦_𝑛−𝑏₄𝑥_𝑛+𝑦_𝑛+𝑘_𝑛+𝑧_𝑛,𝑏₂=𝑓𝑦_𝑛,𝑥_𝑛,𝑘_𝑛−𝑏₃𝑘_𝑛−2𝑥_𝑛+𝑦_𝑛+𝑏₄𝑥_𝑛𝑦_𝑛+𝑥_𝑛𝑘_𝑛+𝑥_𝑛𝑧_𝑛+𝑦_𝑛𝑘_𝑛+𝑦_𝑛𝑧_𝑛+𝑘_𝑛𝑧_𝑛,𝑏₁=𝑓𝑧_𝑛,𝑥_𝑛−𝑧_𝑛−𝑥_𝑛²𝑏₃−𝑧_𝑛−𝑥_𝑛𝑏₂. (3.11)
Now, an efficient, accurate, and optimal eighth-order family, which is free from any derivative, can be obtained in the following form: 𝑦_𝑛=𝑥_𝑛−𝛽𝑓𝑥_𝑛²𝑓𝑘_𝑛−𝑓𝑥_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛,𝑎₃𝑥∈ℝ,_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑏₁+2𝑏₂𝑧_𝑛−𝑥_𝑛+3𝑏₃𝑧_𝑛−𝑥_𝑛²+4𝑏₄𝑧_𝑛−𝑥_𝑛³,𝑏₄∈ℝ. (3.12)

Remark 3.3. The simplified form of the approximation for the derivative in the quotient of Newton’s iteration at the third step of our cycle is as comes next: 𝑏₁+2𝑏₂𝑧_𝑛−𝑥_𝑛+3𝑏₃𝑧_𝑛−𝑥_𝑛²+4𝑏₄𝑧_𝑛−𝑥_𝑛³=𝑓𝑥_𝑛,𝑧_𝑛+(𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛)𝑥_𝑛−𝑧_𝑛+𝑏₄𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛. (3.13)
Using Remark 3.3, we can propose the following optimal eighth-order derivative-free iteration in this paper with some free parameters: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑘_𝑛,𝑥_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛,𝑎₃𝑥∈ℝ,_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝑏₄𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛, (3.14) where 𝑏₄∈ℝ and it consists of four evaluations per iteration to reach the efficiency index 8^1/4≈1.682. Theorem 3.4 illustrates its error equation and order of convergence. The main contribution of this section lies in the following Theorem.

Theorem 3.4. Assume that the function 𝑓∶𝐷⊆ℝ→ℝ has a single root 𝛼∈𝐷, where 𝐷 is an open interval. Then the convergence order of the derivative-free iterative method defined by (3.14) is eight and it satisfies the following the error equation: 𝑒_𝑛+1=1𝑐⁷₁𝑐²₂𝑐²₂−𝑐₁𝑐₃+𝑎₃1+𝑐₁𝛽⁴𝑐³₂−𝑐₁𝑐₂𝑐₃+𝑎₃+𝑐²₁𝑐₄+𝑏₄𝑒⁸_𝑛+𝑂𝑒⁹_𝑛. (3.15)

Proof. To find the asymptotic error constant of (3.14), wherein 𝑐_𝑗=𝑓^(𝑗)(𝛼)/𝑗!,𝑗≥1, and 𝑒_𝑛=𝑥_𝑛−𝛼, we expand any terms of (3.14) around the simple root 𝛼 in the 𝑛th iterate. Thus, we write 𝑓(𝑥_𝑛)=𝑐₁𝑒_𝑛+𝑐₂𝑒²_𝑛+𝑐₃𝑒³_𝑛+𝑐₄𝑒⁴_𝑛+𝑐₅𝑒⁵_𝑛+𝑐₆𝑒⁶_𝑛+𝑐₇𝑒⁷_𝑛+𝑐₈𝑒⁸_𝑛+𝑂(𝑒⁹_𝑛). Accordingly, we attain 𝑦_𝑛=𝛼+𝑐₂1𝑐₁+𝛽𝑒²_𝑛+⋯+𝑂𝑒⁹_𝑛. (3.16) Now we should expand 𝑓(𝑦_𝑛) around the simple root by using (3.16). We have 𝑓𝑦_𝑛=𝑐₂+𝑐₁𝑐₂𝛽𝑒²_𝑛+⋯+𝑂𝑒⁹_𝑛. (3.17) Using (3.17) and the second step of (3.14), we attain 𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛=𝑐₂1𝑐₁+𝛽𝑒²_𝑛+⋯+𝑂𝑒⁹_𝑛. (3.18) Now, the Taylor expansion of the second step of (3.14), using (3.18), gives us 𝑧_𝑛𝑐−𝛼=²𝑐²₂−𝑐₁𝑐₃+𝑎₃1+𝑐₁𝛽²𝑐³₁𝑒⁴_𝑛+⋯+𝑂𝑒⁹_𝑛. (3.19) A similar Taylor expansion is required for continuing. Hence, it is needed to write the Taylor expansion of 𝑓(𝑧_𝑛) now. Thus, we have 𝑓𝑧_𝑛=𝑐₂𝑐²₂−𝑐₁𝑐₃+𝑎₃1+𝑐₁𝛽²𝑐²₁𝑒⁴_𝑛+…+𝑂𝑒⁹_𝑛. (3.20) Subsequently for the approximation of 𝑓^′(𝑧_𝑛), we obtain 𝑓[𝑥_𝑛,𝑧_𝑛] + (𝑓[𝑘_𝑛,𝑥_𝑛,𝑦_𝑛]−𝑓[𝑘_𝑛,𝑥_𝑛,𝑧_𝑛]−𝑓[𝑦_𝑛,𝑥_𝑛,𝑧_𝑛])(𝑥_𝑛−𝑧_𝑛)+𝑏₄(𝑧_𝑛−𝑥_𝑛)(𝑧_𝑛−𝑘_𝑛)(𝑧_𝑛−𝑦_𝑛)=𝑐₁+(𝑐₂(2𝑐³₂+𝑐²₁𝑐₄−2𝑐₁𝑐₂(𝑐₃+𝑎₃))(1+𝑐₁𝛽)²𝑒⁴_𝑛)/𝑐³₁+⋯+𝑂(𝑒⁹_𝑛). Furthermore, we attain 𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝑏₄𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛=𝑐₂𝑐²₂−𝑐₁𝑐₃+𝑎₃1+𝑐₁𝛽²𝑒⁴_𝑛𝑐³₁+⋯+𝑂𝑒⁹_𝑛. (3.21) Using (3.21) and the last step of (3.14), we have the error equation (3.15). This manifests that (3.14) is of optimal order eight with four function evaluations per iteration. Hence, the proof is complete and (3.14) reaches the efficiency index 8^1/4≈1.682.

Noting that (3.14) is an extension of the Steffensen and the Ren et al. methods, we should here pull the attention toward this fact that (3.14) is also a generalization of the family given by Zheng et al. in [2]. As a matter of fact, our scheme (3.14) in this paper includes two free parameters (more than that of Zheng et al.), which shows the generality of our technique. Clearly, choosing 𝑎₃=𝑏₄=0 will result in the Zheng et al. method (2.4). Thus, it is only an special element from the family (3.14).

Remark 3.5. The introduced approximation for the first derivative of the function in the third step of (3.7) can be implemented on any optimal derivative-free fourth-order method without memory for presenting new optimal eighth-order techniques free from derivative. Namely, using (2.1) and (3.13) results in the following optimal eighth-order derivative-free method: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑘_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑦_𝑛,𝑘_𝑛+𝑓𝑥_𝑛,𝑘_𝑛𝑓𝑥_𝑛,𝑦_𝑛²𝑓𝑦_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝛾𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛, (3.22) where 𝛾∈ℝ with the following error equation: 𝑒_𝑛+1=1𝑐⁷₁𝑐₂+𝑐₁𝑐₂𝛽²−𝑐₁𝑐₃1+𝑐₁𝛽+𝑐²₂2+𝑐₁𝛽×−𝑐₁𝑐₂𝑐₃1+𝑐₁𝛽+𝑐³₂2+𝑐₁𝛽+𝑐²₁1+𝑐₁𝛽𝑐₄+𝛾)𝑒⁸_𝑛+𝑂𝑒⁹_𝑛, (3.23) and if one chooses any of the derivative-free third-order methods without memory in the first two steps and applied the introduced estimation (3.13), then a sixth-order derivative-free technique will be attained. This also shows that the given approximation doubles the convergence rate.

To show the generality of the proposed class and by using Remark 3.5, in what follows, we give some other optimal three-step four-point root solvers without memory. For the first example and by using (11) in [13], we have 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑘_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑘_𝑛𝑓𝑦1+^𝑛𝑓𝑥_𝑛+𝑓𝑦_𝑛𝑓𝑘_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝛾𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛, (3.24) where 𝛾∈ℝ and 𝑒_𝑛+1=𝑥_𝑛+11−𝛼=𝑐⁷₁𝑐₂+𝑐₁𝑐₂𝛽²−𝑐₁𝑐₃1+𝑐₁𝛽+𝑐²₂5+𝑐₁𝛽5+𝑐₁𝛽×−𝑐₁𝑐₂𝑐₃1+𝑐₁𝛽+𝑐³₂5+𝑐₁𝛽5+𝑐₁𝛽+𝑐²₁1+𝑐₁𝛽𝑐₄+𝛾𝑒⁸_𝑛+𝑂𝑒⁹_𝑛. (3.25) Using a different fourth-order method according to [13], we get that 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑘_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑘_𝑛11−𝑓𝑦_𝑛/𝑓𝑥_𝑛−𝑓𝑦_𝑛/𝑓𝑘_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝛾𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛, (3.26) wherein 𝛾∈ℝ, with the following error relation: 𝑒_𝑛+1=𝑐²₂𝑐²₂−𝑐₁𝑐₃1+𝑐₁𝛽⁴𝑐³₂−𝑐₁𝑐₂𝑐₃+𝑐²₁𝑐₄+𝛾𝑐⁷₁𝑒⁸_𝑛+𝑂𝑒⁹_𝑛. (3.27)

We can also propose the following general family of three-step iterations using [22]: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑘_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑘_𝑛𝑥1+2+𝛽𝑓_𝑛,𝑘_𝑛1+𝛽𝑓𝑥_𝑛,𝑘_𝑛𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝛾𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛, (3.28) where (𝛾∈ℝ) 𝑒_𝑛+1=1𝑐⁷₁𝑐₂+𝑐₁𝑐₂𝛽²−𝑐₁𝑐₃1+𝑐₁𝛽+𝑐²₂5+𝑐₁𝛽5+𝑐₁×𝛽))))−𝑐₁𝑐₂𝑐₃1+𝑐₁𝛽+𝑐³₂5+𝑐₁𝛽5+𝑐₁𝛽)+𝑐²₁1+𝑐₁𝛽)𝑐₄+𝛾))𝑒⁸_𝑛+𝑂𝑒⁹_𝑛. (3.29)

Remark 3.6. Using backward finite difference approximation for the first derivative of the function in the first step of our cycles will end in other new methods as comes next: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑘_𝑛,𝑘_𝑛=𝑥_𝑛−𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑦_𝑛,𝑘_𝑛+𝑓𝑥_𝑛,𝑘_𝑛𝑓𝑥_𝑛,𝑦_𝑛²𝑓𝑦_𝑛,𝑥_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝛾𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛, (3.30) where (𝛾∈ℝ) 𝑒_𝑛+1=1𝑐⁷₁𝑐²₂−1+𝑐₁𝛽²𝑐₁𝑐₃1−𝑐₁𝛽+𝑐²₂−2+𝑐₁𝛽×𝑐₁𝑐₂𝑐₃1−𝑐₁𝛽+𝑐³₂−2+𝑐₁𝛽+𝑐²₁−1+𝑐₁𝛽𝑐₄+𝛾𝑒⁸_𝑛+𝑂𝑒⁹_𝑛. (3.31) We can also have 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑘_𝑛,𝑘_𝑛=𝑥_𝑛−𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛,𝑎₃𝑥∈ℝ,_𝑛+1=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝛾𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛, (3.32) with the following error equation (𝛾∈ℝ): 𝑒_𝑛+1=1𝑐⁷₁𝑐²₂𝑐²₂−𝑐₁𝑐₃+𝑎₃−1+𝑐₁𝛽⁴×𝑐³₂−𝑐₁𝑐₂𝑐₃+𝑎₃+𝑐²₁𝑐₄+𝛾𝑒⁸_𝑛+𝑂𝑒⁹_𝑛. (3.33)

4. Higher-Order Optimal Schemes

In this section, we take a special heed to generalize the novel scheme (3.14) by using the same idea. To increase the local order of convergence and the efficiency index more, we should consider cycles in which there are four, five, and so forth steps. Here, we consider a four-step cycle at which (3.14) is in the first three steps and Newton’s method is in the fourth step as follows (𝑎₃,𝑏₄∈ℝ): 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑘_𝑛,𝑥_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝑏₄𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝑓𝑤_𝑛𝑓^′𝑤_𝑛. (4.1)

It is easy to check that (4.1) is a sixteenth-order method with six evaluations per iteration and it has the efficiency index 16^1/6≈1.587. To make (4.1) optimal derivative-free multipoint technique with 16^1/5≈1.741 as the efficiency index, it is required to approximate 𝑓^′(𝑤_𝑛) effectively. To do this, we estimate 𝑓(𝑡) in the domain 𝐷, by a novel polynomial of degree four like the similar cases in (3.2) and (3.8) as follows: 𝑓(𝑡)≈𝑟(𝑡)=𝑟₀+𝑟₁𝑡−𝑥_𝑛+𝑟₂𝑡−𝑥_𝑛²+𝑟₃𝑡−𝑥_𝑛³+𝑟₄𝑡−𝑥_𝑛⁴+𝑟₅𝑡−𝑥_𝑛⁵, (4.2) where its first derivative takes the form as follows: 𝑓^′(𝑡)≈𝑟^′(𝑡) = 𝑟₁+2𝑟₂(𝑡−𝑥_𝑛)+3𝑟₃(𝑡−𝑥_𝑛)²+4𝑟₄(𝑡−𝑥_𝑛)³+5𝑟₅(𝑡−𝑥_𝑛)⁴. Substituting the known values in (4.2) gives us the unknown values by solving a system of linear equations in what follows: 𝑘_𝑛−𝑥_𝑛𝑘_𝑛−𝑥_𝑛²𝑘_𝑛−𝑥_𝑛³𝑘_𝑛−𝑥_𝑛⁴𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑥_𝑛²𝑦_𝑛−𝑥_𝑛³𝑦_𝑛−𝑥_𝑛⁴𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑥_𝑛²𝑧_𝑛−𝑥_𝑛³𝑧_𝑛−𝑥_𝑛⁴𝑤_𝑛−𝑥_𝑛𝑤_𝑛−𝑥_𝑛²𝑤_𝑛−𝑥_𝑛³𝑤_𝑛−𝑥_𝑛⁴𝑟₁𝑟₂𝑟₃𝑟₄=𝑓𝑘_𝑛−𝑓𝑥_𝑛−𝑟₅𝑘_𝑛−𝑥_𝑛⁴𝑓𝑦_𝑛−𝑓𝑥_𝑛−𝑟₅𝑦_𝑛−𝑥_𝑛⁴𝑓𝑧_𝑛−𝑓𝑥_𝑛−𝑟₅𝑧_𝑛−𝑥_𝑛⁴𝑓𝑤_𝑛−𝑓𝑥_𝑛−𝑟₅𝑤_𝑛−𝑥_𝑛⁴. (4.3)

Note that for the case of multipoint methods, it is more desirable to apply the symbolic calculations using one of the modern software in mathematics. Using Mathematica 8 gives us the unknown parameters by the command LinearSolve as in Algorithm 1.

LinearSolve[{{k-x,((k-x)∧2), ((k-x)∧3),((k-x)∧4) } , { y-x, ((y-x)∧2),((y-x)∧3), ((y-x)∧4) } , { z-x, ((z-x)∧2), ((z-x)∧3),((z-x)∧4) } , { w-x, ((w-x)∧2), ((w-x)∧3),((w-x)∧4)}}, { f[k]-f[x]-r5∗(k-x)∧4,f[y]-f[x]-r5∗(y-x)∧4, f[z]-f[x] -r5∗(z-x)∧4,f[w]-f[x]-r5∗(w-x)∧4}]//FullSimplify

Algorithm 1 

Here to cut the long story short, we just provide the following theorem.

Theorem 4.1. The four-step method 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑘_𝑛,𝑥_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑦_𝑛𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝑎₃𝑦_𝑛−𝑥_𝑛𝑦_𝑛−𝑘_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝑏₄𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝑓𝑤_𝑛𝑟₁+2𝑟₂𝑤_𝑛−𝑥_𝑛+3𝑟₃𝑤_𝑛−𝑥_𝑛²+4𝑟₄𝑤_𝑛−𝑥_𝑛³+5𝑟₅𝑤_𝑛−𝑥_𝑛⁴,𝑟₅∈ℝ, (4.4) converges to the simple root of 𝑓(𝑥)=0 in the domain 𝐷 with local sixteenth order of convergence, where the first three steps are the optimal eighth-order method (3.14) and 𝑟₁,𝑟₂,𝑟₃ and 𝑟₄ are given as follows: 𝑟₄=𝑓𝑘_𝑛,𝑥_𝑛−𝑓𝑧_𝑛,𝑥_𝑛𝑘_𝑛−𝑤_𝑛𝑘_𝑛−𝑦_𝑛𝑘_𝑛−𝑧_𝑛+𝑓𝑧_𝑛,𝑥_𝑛−𝑓𝑤_𝑛,𝑥_𝑛𝑘_𝑛−𝑤_𝑛𝑤_𝑛−𝑦_𝑛𝑤_𝑛−𝑧_𝑛+𝑓𝑦_𝑛,𝑥_𝑛−𝑓𝑧_𝑛,𝑥_𝑛𝑘_𝑛−𝑦_𝑛𝑤_𝑛−𝑦_𝑛𝑦_𝑛−𝑧_𝑛,𝑟₃=𝑓𝑘_𝑛,𝑥_𝑛−𝑓𝑧_𝑛,𝑥_𝑛/𝑘_𝑛−𝑧_𝑛+−𝑓𝑦_𝑛,𝑥_𝑛+𝑓𝑧_𝑛,𝑥_𝑛/𝑦_𝑛−𝑧_𝑛𝑘_𝑛−𝑦_𝑛,𝑟₂=−𝑓𝑘_𝑛,𝑥_𝑛+𝑓𝑦_𝑛,𝑥_𝑛+𝑘_𝑛−𝑥_𝑛²𝑟₃+𝑘_𝑛−𝑥_𝑛³𝑟₄−𝑟₃𝑥_𝑛−𝑦_𝑛²+𝑟₄𝑥_𝑛−𝑦_𝑛³𝑦_𝑛−𝑘_𝑛,𝑟₁=𝑓𝑤_𝑛,𝑥_𝑛−𝑟₂𝑤_𝑛−𝑥_𝑛−𝑟₃𝑤_𝑛−𝑥_𝑛²−𝑟₄𝑤_𝑛−𝑥_𝑛³. (4.5)

Proof. The proof of this theorem is similar to the proofs of Theorems 3.1 and 3.4. Hence, it is omitted.

Remark 4.2. The simplified form of the approximation for the derivative in the quotient of Newton’s iteration at the fourth step of our cycle (without the index 𝑛) is as comes next: 𝑟₁+2𝑟₂𝑤_𝑛−𝑥_𝑛+3𝑟₃𝑤_𝑛−𝑥_𝑛²+4𝑟₄𝑤_𝑛−𝑥_𝑛³+5𝑟₅𝑤_𝑛−𝑥_𝑛⁴=𝑓𝑧_𝑛,𝑤_𝑛+𝑓𝑤_𝑛,𝑧_𝑛,𝑦_𝑛𝑤_𝑛−𝑧_𝑛+𝑓𝑤_𝑛,𝑧_𝑛,𝑦_𝑛,𝑥_𝑛𝑤_𝑛−𝑧_𝑛𝑤_𝑛−𝑦_𝑛+𝑓𝑤_𝑛,𝑧_𝑛,𝑦_𝑛,𝑥_𝑛,𝑘_𝑛×𝑤_𝑛−𝑧_𝑛𝑤_𝑛−𝑦_𝑛𝑤_𝑛−𝑥_𝑛+𝑟₅𝑤_𝑛−𝑥_𝑛𝑤_𝑛−𝑘_𝑛𝑤_𝑛−𝑦_𝑛𝑤_𝑛−𝑧_𝑛=Υ(𝑛). (4.6)

Remark 4.3. Note that if we choose another optimal derivative-free eighth-order method in the first three steps of (4.4) and then implement the introduced approximation in the fourth-step, we will obtain a new technique of optimal sixteenth convergence rate to solve nonlinear scalar equations. In what follows, we give some of such optimal 16^th-order schemes: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑘_𝑛,𝑥_𝑛,𝑘_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑦_𝑛,𝑘_𝑛+𝑓𝑥_𝑛,𝑘_𝑛𝑓𝑥_𝑛,𝑦_𝑛²𝑓𝑦_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝑏₄𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝑓𝑤_𝑛Υ(𝑛), (4.7) where 𝑏₄,𝑟₅∈ℝ and 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓𝑘_𝑛,𝑥_𝑛,𝑘_𝑛=𝑥_𝑛−𝛽𝑓𝑥_𝑛{0},𝑧,𝛽∈ℝ⧵_𝑛=𝑦_𝑛−𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑦_𝑛,𝑘_𝑛+𝑓𝑥_𝑛,𝑘_𝑛𝑓𝑥_𝑛,𝑦_𝑛²𝑓𝑦_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛+𝑓𝑘_𝑛,𝑥_𝑛,𝑦_𝑛−𝑓𝑘_𝑛,𝑥_𝑛,𝑧_𝑛−𝑓𝑦_𝑛,𝑥_𝑛,𝑧_𝑛𝑥_𝑛−𝑧_𝑛+𝑏₄𝑧_𝑛−𝑥_𝑛𝑧_𝑛−𝑘_𝑛𝑧_𝑛−𝑦_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝑓𝑤_𝑛Υ(𝑛). (4.8)

Remark 4.4. If one considers an 𝑛-step derivative-free construction in which the first 𝑛−1 steps are any of the existing optimal derivative-free methods of order 2^𝑛−1, then by considering a polynomial approximation of degree 𝑛+1 (with one free parameter, as in the idea above), as discussed above, an optimal derivative-free method of order 2^𝑛 with the optimal efficiency index 2 will be attained.

Now, let us list the efficiency indices of some of the well-known methods with the contributed schemes in this study. The results of comparisons are made in Table 1. As it shows, our schemes are totally better than the existing methods in the literature in terms of efficiency index.

Table 1 

Comparisons of efficiency indices for different types of derivative-free schemes.

5. Computational Tests

The practical utilities of the main contributions in this paper are given by solving a couple of numerical examples and comparison with other well-known methods of different orders in this section. The main purpose of demonstrating the new derivative-free methods for nonlinear equations is 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 techniques.

We have computed the root of each test function for an initial approximation in the neighborhood 𝐷. We remark that we have chosen for comparison only the methods which do not require the computation of derivatives of the function to carry out iterations. We have used the fourth-order scheme of Ren et al. (RM4), the eighth-order derivative-free family of Zheng et al. (2.3) with 𝛽=1, the eighth-order derivative-free method of Soleymani (2.4), and our novel contributed techniques (3.14) with 𝑎₃=1,𝑏₄=−3, (3.22) with 𝛾=0, and (3.30) with 𝛾=0. We here recall the family of derivative-free methods, which was given by Kung and Traub (KT8) [3] as comes next: 𝑦_𝑛=𝑥_𝑛+𝛽𝑓𝑥_𝑛,𝑧_𝑛=𝑦_𝑛𝑓𝑥−𝛽^𝑛𝑓𝑦_𝑛𝑓𝑦_𝑛−𝑓𝑥_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛1𝑓𝑦_𝑛,𝑥_𝑛−1𝑓𝑧_𝑛,𝑦_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛𝑓𝑧_𝑛𝑓𝑤_𝑛−𝑓𝑥_𝑛×1𝑓𝑤_𝑛−𝑓𝑦_𝑛1𝑓𝑤_𝑛,𝑧_𝑛−1𝑓𝑧_𝑛,𝑦_𝑛−1𝑓𝑧_𝑛−𝑓𝑥_𝑛1𝑓𝑧_𝑛,𝑦_𝑛−1𝑓𝑦_𝑛,𝑥_𝑛, (5.1) where 𝛽∈ℝ⧵{0}; we have used this scheme in numerical comparison by choosing 𝛽=1. It is completely clear that (4.4) performs better than any of the other methods; hence we do not include it in numerical comparisons. The test functions and their roots are given in Table 2.

Table 2 

The test functions considered in this study.

The results are summarized in Tables 3 and 4, in terms of the required number of iterations to obtain an approximation of the root. In our numerical comparisons, we have used 2000-digit floating point in our calculations.

Table 3 

Results of comparisons for different methods after two full iterations.

Table 4 

Results of comparisons for different methods after three full iterations.

As Tables 3 and 4 manifest, our contributed methods are accurate and efficient in contrast by the other high-order schemes in terms of required number of iterations. It is well known that convergence of iteration formula is guaranteed only when the initial approximation is sufficiently near to the root. This issue will be the main body of the next section. However, our class of derivative-free method is much better and has better convergence rate in contrast to the existing high-order derivative-free methods in the literature.

6. A Hybrid Algorithm to Capture All the Solutions

An important aspect in implementing the iterative methods for the solution of nonlinear equations and systems relies on the choice of the initial approximation. There are a few known ways in the literature [23] to extract a starting point for the solutions of nonlinear functions. In practice, users need to find out robust approximations for all the zeros in an interval. Thus, to remedy this and to respond on this need, we provide a way to extract all the real zeros of nonlinear function in the interval 𝐷=[𝑎,𝑏]. We use the command Reduce[] in MATHEMATICA 8 [24, 25]. Hence, we give a hybrid algorithm including two main steps, a predictor and a corrector. In the predictor step, we extract initial approximations for all the zeros in an interval up to 8 decimal places. Then in the corrector step, an eighth-order method, for example, (3.28), will be used to boost up the accuracy of the starting points up to any tolerance. We also give some significant cautions for applying on different test functions.

In what follows, we keep going by choosing an oscillatory function 𝑓(𝑥)=−cos(2−𝑥²)+log(𝑥/7)+(1/10) in the domain 𝐷=[0,15].

Let us define the function and the domain for imposing the Reduce[] command as in Algorithm 2.

ClearAll[f, x, rts, initialValues, tol, a, b, j, NumberOfGuesses, FD, nMax, beta, gamma, y, z, k, fx, fk, fy, fz] f[x_]:= Log[x/7] - Cos[x∧2 - 2] + 1/10 a = 0; b = 15; Plot[f[x], { x, a, b } , Background -> LightBlue,   PlotStyle -> {Brown, Thick}, PlotRange -> All,   PerformanceGoal -> "Quality"] rts = Reduce[f[x] == 0 && a < x < b, x];

Algorithm 2 

One may note that Reduce[] works with function of exact arithmetic. Hence, if a nonlinear function is the floating point arithmetic, that is, has inexact coefficients, thus we should write it in the exact arithmetic when we enter it into the above piece of code.

Now we store the list of initial approximations in initialValues, by the following piece of code, which also sort the initial points. The tol will specify that the accuracy of each member of the provided sequence to be correct up to utmost tol, decimal places (Algorithm 3).

tol = 8; initialValues = Sort[N[x /. { ToRules[rts]}, tol]] Length[initialValues] Accuracy[initialValues]

Algorithm 3 

It is obvious that 𝑓 is so oscillatory, and by the above predictor piece of Mathematica code, we attain that it has 69 real solutions. Note that the graph of the function 𝑓 has been drawn in Figure 1. Now it is time to use one of the derivative-free methods given in Section 2 or 3 for correcting the elements of the obtained list. We use the method (3.28) in Algorithm 4.

nMax = 2; beta = SetAccuracy[1, 2000]; gamma = SetAccuracy[0, 2000]; For[j = 1, j <= NumberOfGuesses, j++,  {x = SetAccuracy[initialValues[[j]], 2000],    Do[     fx = SetAccuracy[f[x], 2000];     k = SetAccuracy[x + beta fx, 2000];     fk = SetAccuracy[f[k], 2000];     FD = SetAccuracy[(fk - fx)/(beta fx), 2000];     y = SetAccuracy[x - fx/FD, 2000];     fy = SetAccuracy[f[y], 2000];     z = SetAccuracy[y - (fy/FD)*(1 + ((2 + beta FD)/(1         + beta FD)) (fy/fx)), 2000];     fz = SetAccuracy[f[z], 2000];     FDxz = SetAccuracy[(fz - fx)/(z - x), 2000];     FDxy = SetAccuracy[(fx - fy)/(x - y), 2000];     FDxk = SetAccuracy[(fk - fx)/(k - x), 2000];     x = SetAccuracy[z - fz/(FDxz + ((FDxk - FDxy)/(k - y)        - (FDxk - FDxz)/(k - z)        - (FDxy - FDxz)/(y - z))*(x - z)    + gamma (z - x)*(z - k)*(z - y)), 2000];     , {n, nMax}];  Print[Column[{    N[x, 128],    N[f[x]]}    , Background -> {{LightGreen, LightGray}},    Frame -> True]];  }   ];

Algorithm 4 

Figure 1 

The plot of the function 𝑓(𝑥) including the zeros.

In Algorithm 4, we have the ability to choose the nonzero free parameter 𝛽, as well as the parameter gamma. Note that for optimal eighth-order methods and with such an accurate list of initial points, we believe that only two full iterations are required. Hence, we have selected nMax=2. Note that if a user needs much more accuracy, thus higher number of steps should be taken. It should be remarked that in order to work with such a high accuracy, we must then choose more than 2000 decimal places arithmetic in our calculations.

Due to page limitations, we are not able to list the results for all 69 zeros. However, running the above algorithm could capture all the real zeros of the nonlinear functions. We also remark three points. One is that for many oscillatory function or for nonsmooth functions, the best way is to first divide the whole interval into some subintervals and then find all the zeros of the function on the subintervals. And second, in case of having a root cluster, that is, when the zeros are concentrated on a very small area, then it would be better to increase the first tolerance of our algorithm in the predictor step, to find reliable starting points and then start the process. And last, if the nonlinear function has an exact solution, that is to say, an integer be the solution of a nonlinear function, then the first step of our algorithm finds this exact solution, and an error-like message would be generated by applying our second step.

For instance, the function 𝑔(𝑥)=(𝑥²−4)sin(100𝑥) on the interval 𝐷=[0,10] has 319 real solutions in which one of them (its plot is given in Figure 2), that is, 2, is an exact one. Thus, the first step of the previously mentioned algorithm finds the following very efficient list of starting points in which 2, is the exact solution: {}.0.031415927,0.062831853,0.094247780,0.12566371,0.15707963,0.18849556,0.21991149,0.25132741,0.28274334,0.31415927,0.34557519,0.37699112,0.40840704,0.43982297,0.47123890,0.50265482,0.53407075,0.56548668,0.59690260,0.62831853,0.65973446,0.69115038,0.72256631,0.75398224,0.78539816,0.81681409,0.84823002,0.87964594,0.91106187,0.94247780,0.97389372,1.0053096,1.0367256,1.0681415,1.0995574,1.1309734,1.1623893,1.1938052,1.2252211,1.2566371,1.2880530,1.3194689,1.3508848,1.3823008,1.4137167,1.4451326,1.4765485,1.5079645,1.5393804,1.5707963,1.6022123,1.6336282,1.6650441,1.6964600,1.7278760,1.7592919,1.7907078,1.8221237,1.8535397,1.8849556,1.9163715,1.9477874,1.9792034,2.0000000,2.0106193,2.0420352,2.0734512,2.1048671,2.1362830,2.1676989,2.1991149,2.2305308,2.2619467,2.2933626,2.3247786,2.3561945,2.3876104,2.4190263,2.4504423,2.4818582,2.5132741,2.5446900,2.5761060,2.6075219,2.6389378,2.6703538,2.7017697,2.7331856,2.7646015,2.7960175,2.8274334,2.8588493,2.8902652,2.9216812,2.9530971,2.9845130,3.0159289,3.0473449,3.0787608,3.1101767,3.1415927,3.1730086,3.2044245,3.2358404,3.2672564,3.2986723,3.3300882,3.3615041,3.3929201,3.4243360,3.4557519,3.4871678,3.5185838,3.5499997,3.5814156,3.6128316,3.6442475,3.6756634,3.7070793,3.7384953,3.7699112,3.8013271,3.8327430,3.8641590,3.8955749,3.9269908,3.9584067,3.9898227,4.0212386,4.0526545,4.0840704,4.1154864,4.1469023,4.1783182,4.2097342,4.2411501,4.2725660,4.3039819,4.3353979,4.3668138,4.3982297,4.4296456,4.4610616,4.4924775,4.5238934,4.5553093,4.5867253,4.6181412,4.6495571,4.6809731,4.7123890,4.7438049,4.7752208,4.8066368,4.8380527,4.8694686,4.9008845,4.9323005,4.9637164,4.9951323,5.0265482,5.0579642,5.0893801,5.1207960,5.1522120,5.1836279,5.2150438,5.2464597,5.2778757,5.3092916,5.3407075,5.3721234,5.4035394,5.4349553,5.4663712,5.4977871,5.5292031,5.5606190,5.5920349,5.6234508,5.6548668,5.6862827,5.7176986,5.7491146,5.7805305,5.8119464,5.8433623,5.8747783,5.9061942,5.9376101,5.9690260,6.0004420,6.0318579,6.0632738,6.0946897,6.1261057,6.1575216,6.1889375,6.2203535,6.2517694,6.2831853,6.3146012,6.3460172,6.3774331,6.4088490,6.4402649,6.4716809,6.5030968,6.5345127,6.5659286,6.5973446,6.6287605,6.6601764,6.6915924,6.7230083,6.7544242,6.7858401,6.8172561,6.8486720,6.8800879,6.9115038,6.9429198,6.9743357,7.0057516,7.0371675,7.0685835,7.0999994,7.1314153,7.1628313,7.1942472,7.2256631,7.2570790,7.2884950,7.3199109,7.3513268,7.3827427,7.4141587,7.4455746,7.4769905,7.5084064,7.5398224,7.5712383,7.6026542,7.6340701,7.6654861,7.6969020,7.7283179,7.7597339,7.7911498,7.8225657,7.8539816,7.8853976,7.9168135,7.9482294,7.9796453,8.0110613,8.0424772,8.0738931,8.1053090,8.1367250,8.1681409,8.1995568,8.2309728,8.2623887,8.2938046,8.3252205,8.3566365,8.3880524,8.4194683,8.4508842,8.4823002,8.5137161,8.5451320,8.5765479,8.6079639,8.6393798,8.6707957,8.7022117,8.7336276,8.7650435,8.7964594,8.8278754,8.8592913,8.8907072,8.9221231,8.9535391,8.9849550,9.0163709,9.0477868,9.0792028,9.1106187,9.1420346,9.1734505,9.2048665,9.2362824,9.2676983,9.2991143,9.3305302,9.3619461,9.3933620,9.4247780,9.4561939,9.4876098,9.5190257,9.5504417,9.5818576,9.6132735,9.6446894,9.6761054,9.7075213,9.7389372,9.7703532,9.8017691,9.8331850,9.8646009,9.8960169,9.9274328,9.9588487,9.9902646 (6.1)

Figure 2 

The plot of the function 𝑔(𝑥) including the zeros.

Now we are able to solve nonlinear equations with finitely many roots in an interval and find all the real zeros in a short piece of time. Finding robust ways, to capture the complex solutions along working with complex nonlinear functions, can be taken into account as future works.

7. Concluding Comments

In a word, a general way for producing optimal methods based upon the still unproved hypothesis of Kung-Traub [3] has been discussed fully. In these methods, it is necessary to begin with one initial approximation 𝑥₀. The convergence results of the iterative sequences generated by new algorithms were provided. Our contributed techniques without memory reach the optimal efficiency indices 8^1/4≈1.682 and 16^1/5≈1.741.

In the limitation case, based on a polynomial of degree 𝑛+1 for a method with 𝑛 steps, we can obtain an iterative scheme with efficiency index 2, which is the highest possible efficiency in this field of study for methods without memory. By numerical experiments above, we verify that new methods are effective and accurate. We conclude that the new methods presented in this paper are competitive with other recognized efficient equation solvers.

Note that construction of some ways, which launch 𝑛-step Steffensen-type methods with memory with the order up to 2^𝑛+2^𝑛−1 (50% of an improvement) requiring the same computational cost and applying elegant and quite natural accelerating procedure, could be considered as future works.

Note that the second aim of this paper was given by presenting a robust algorithm for solving nonlinear equations with finitely many roots in an interval. In fact, a way for extracting very robust initial approximations as the seeds to start the iterative Steffensen-type methods was presented. Now we are able to capture all the solutions with any desire number of decimal places.

Acknowledgment

The second author is an IEEE member.