Computing Simple Roots by an Optimal Sixteenth-Order Class

Abstract

The problem considered in this paper is to approximate the simple zeros of the function 𝑓(𝑥) by iterative processes. An optimal 16th order class is constructed. The class is built by considering any of the optimal three-step derivative-involved methods in the first three steps of a four-step cycle in which the first derivative of the function at the fourth step is estimated by a combination of already known values. Per iteration, each method of the class reaches the efficiency index √⁵16≈1.741, by carrying out four evaluations of the function and one evaluation of the first derivative. The error equation for one technique of the class is furnished analytically. Some methods of the class are tested by challenging the existing high-order methods. The interval Newton's method is given as a tool for extracting enough accurate initial approximations to start such high-order methods. The obtained numerical results show that the derived methods are accurate and efficient.

1. Introduction

Consider that the function 𝑓∶𝐼⊆ℝ→ℝ is a sufficiently differentiable scalar function. We assume that 𝑟∈𝐼 be a simple zero of the nonlinear function, that is, 𝑓(𝑟)=0 and 𝑓^′(𝑟)≠0. There is a vast literature on simple zeros of such nonlinear functions by iterative processes, see, for example, [1–3]. In 1974, Kung and Traub [4] conjectured that a multipoint iterative method without memory for solving single variable nonlinear equations consisting of 𝑛+1 evaluations per iteration has the maximal convergence rate 2^𝑛 and, subsequently, the maximal efficiency index will be ^𝑛+√¹2^𝑛. By taking into consideration this concept, many authors, see, for example, [5–7], have tried to produce optimal multistep methods.

Kung and Traub in [4] presented the following 𝑛-point optimal iterative process consisting of (𝑛+1)-evaluation per full cycle: 𝑞₂𝑓𝑥_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,⋮𝑞_𝑗+1𝑓𝑥_𝑛=𝑆_𝑗(0),(1.1) for 𝑗=2,…,𝑛−1, where 𝑆_𝑗(𝑦) is the inverse Hermite interpolatory polynomial of degree at most 𝑗 such that 𝑆_𝑗𝑓𝑥_𝑛=𝑥_𝑛,𝑆^′_𝑗𝑓𝑥_𝑛=1𝑓^′𝑥_𝑛,𝑆_𝑗𝑓𝑞_𝜆𝑓𝑥_𝑛=𝑞_𝜆𝑥_𝑛,𝜆=2,…,𝑗.(1.2)

This technique is defined by 𝑥_𝑛+1=𝑞_𝑛(𝑓)(𝑥_𝑛) starting with an initial point 𝑥₀. Recently, Neta and Petković applied the concept of inverse interpolation again in [8] to approximate the first derivative of the function in the third and fourth steps of a three- and four-step cycle. They obtained the following optimal 8th order technique: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−𝑓𝑥_𝑛+𝑡𝑓𝑦_𝑛𝑓𝑥_𝑛+(𝑡−2)𝑓𝑦_𝑛𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑦_𝑛+𝑐𝑓𝑥_𝑛²−𝑑𝑓𝑥_𝑛³,(1.3) with 𝑡∈ℝ, and 𝑐,𝑑 as comes next 1𝑑=𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛−𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑥_𝑛−1𝑓𝑧_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛−𝑓𝑧_𝑛𝑓𝑧_𝑛,𝑥_𝑛+1𝑓^′𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛−𝑓𝑧_𝑛−1𝑓^′𝑥_𝑛𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛−𝑓𝑧_𝑛,1𝑐=𝑓𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑦_𝑛,𝑥_𝑛−1𝑓^′𝑥_𝑛𝑓𝑦_𝑛−𝑓𝑥_𝑛−𝑑𝑓𝑦_𝑛−𝑓𝑥_𝑛.(1.4)

They also presented an optimal 16th order technique in the following structure consisting of four evaluations of the function and one evaluation of the first derivative: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−𝑓𝑥_𝑛+𝑡𝑓𝑦_𝑛𝑓𝑥_𝑛+(𝑡−2)𝑓𝑦_𝑛𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑤_𝑛=𝑦_𝑛+𝑐𝑓𝑥_𝑛²−𝑑𝑓𝑥_𝑛³,𝑥_𝑛+1=𝑦_𝑛+𝑐^′𝑓𝑥_𝑛²−𝑑^′𝑓𝑥_𝑛³+𝑔^′𝑓𝑥_𝑛⁴,(1.5) wherein (without the index 𝑛) 𝑔^′=𝜑_𝑤−𝜑_𝑧/𝑓(𝑤)−𝑓(𝑧)−𝜑_𝑦−𝜑_𝑧/𝑓𝑦−𝑓(𝑧)𝑓(𝑤)𝑦,𝑑−𝑓_′=𝜑_𝑤−𝜑_𝑧𝑓(𝑤)−𝑓(𝑧)−𝑔^′𝑓(𝑤)(𝑥)(𝑧,𝑐−2𝑓+𝑓)_′=𝜑_𝑤−𝑑^′𝑓(𝑤)−𝑓(𝑥)−𝑔^′𝑓(𝑤)−𝑓(𝑥)²,(1.6) and also 𝜑_𝑤=1/(𝑓[𝑤,𝑥](𝑓(𝑤)−𝑓(𝑥)))−1/(𝑓^′(𝑥_𝑛)(𝑓(𝑤)−𝑓(𝑥))),  𝜑_𝑧=1/(𝑓[𝑧,𝑥](𝑓(𝑧)−𝑓(𝑥)))−1/(𝑓^′(𝑥_𝑛)(𝑓(𝑧)−𝑓(𝑥))), and 𝜑_𝑦=1/(𝑓[𝑦,𝑥](𝑓(𝑦)−𝑓(𝑥)))−1/(𝑓^′(𝑥_𝑛)(𝑓(𝑦)−𝑓(𝑥))).

For further reading, one may refer to [9, 10]. Contents of the paper are summarized in what follows. In the next section, our novel contribution is constructed by considering an optimal eighth-order method in the first three steps of a four-step cycle, in which the derivative in the quotient of the new Newton’s iteration is estimated such that the order remains at 16, namely, optimal according to the Kung-Traub hypothesis. The new class of methods is supported with detailed proof in this section to verify the construction theoretically.

Then, it will be observed that the computed results listed in Table 2 completely support the theory of convergence and efficiency analyses discussed in the paper. Section 4 reminds the well-known Interval Newton method in the efficient programming package Mathematica 8 to extract enough accurate initial guesses to start the process. Finally, a short conclusion is given in the last section.

2. Main Results

In this section, we derive our main results by providing a class of four-step iterative methods, which agrees with Kung-Traub hypothesis. In order to build the class, we consider the following four-step four-point iteration, in which the first three steps are any of the optimal three-step three-point without memory derivative-involved methods: 𝑓𝑥optimal8th-ordermethod=^𝑛and𝑓^′𝑥_𝑛𝑓𝑦areavailable,_𝑛𝑓𝑧isavailable,_𝑛𝑥isavailable,_𝑛+1=𝑤_𝑛−𝑓𝑤_𝑛𝑓^′𝑤_𝑛.(2.1)

It can be seen that per full cycle of the structure (2.1), it includes four evaluations of the function and two evaluations of the first derivative to reach the convergence rate 16. At this time, we should approximate the first derivative of the function at the fourth step such that the convergence order does not decrease. In order to do this effectively, we have to use all of the known data from the past steps, that is, 𝑓(𝑥_𝑛), 𝑓^′(𝑥_𝑛), 𝑓(𝑦_𝑛), 𝑓(𝑧_𝑛), and 𝑓(𝑤_𝑛). Herein, we take into account the nonlinear fraction given by (without the index 𝑛) 𝑝(𝑡)=𝑏₁+𝑏₂(𝑡−𝑥)+𝑏₃(𝑡−𝑥)²+𝑏₄(𝑡−𝑥)³1+𝑏₅().𝑡−𝑥(2.2)

This nonlinear fraction is inspired by Pade approximant in essence. The approximation function should meet the interpolation conditions 𝑓(𝑥_𝑛)=𝑝(𝑥_𝑛), 𝑓^′(𝑥_𝑛)=𝑝^′(𝑥_𝑛), 𝑓(𝑦_𝑛)=𝑝(𝑦_𝑛), 𝑓(𝑧_𝑛)=𝑝(𝑧_𝑛), and 𝑓(𝑤_𝑛)=𝑝(𝑤_𝑛). Note that the first derivative of (2.2) takes the following form: 𝑝^′(𝑡)=𝑏₂−𝑏₁𝑏₅+2𝑏₃𝑡−𝑥_𝑛+3𝑏₄+𝑏₃𝑏₅𝑡−𝑥_𝑛²+2𝑏₄𝑏₅𝑡−𝑥_𝑛³1+𝑏₅𝑡−𝑥_𝑛².(2.3)

Substituting the known data in (2.2) and (2.3) will result in obtaining the five unknown parameters. It is obvious that 𝑏₁=𝑓(𝑥_𝑛), hence we obtain the following system of four linear equations with four unknowns: 𝑏₂−𝑓𝑥_𝑛𝑏₅=𝑓^′𝑥_𝑛,𝑦_𝑛−𝑥_𝑛𝑏₂+𝑦_𝑛−𝑥_𝑛²𝑏₃+𝑦_𝑛−𝑥_𝑛³𝑏₄−𝑦_𝑛−𝑥_𝑛𝑓𝑦_𝑛𝑏₅=𝑓𝑦_𝑛−𝑓𝑥_𝑛,𝑧_𝑛−𝑥_𝑛𝑏₂+𝑧_𝑛−𝑥_𝑛²𝑏₃+𝑧_𝑛−𝑥_𝑛³𝑏₄−𝑧_𝑛−𝑥_𝑛𝑓𝑧_𝑛𝑏₅=𝑓𝑧_𝑛−𝑓𝑥_𝑛,𝑤_𝑛−𝑥_𝑛𝑏₂+𝑤_𝑛−𝑥_𝑛²𝑏₃+𝑤_𝑛−𝑥_𝑛³𝑏₄−𝑤_𝑛−𝑥_𝑛𝑓𝑤_𝑛𝑏₅=𝑓𝑤_𝑛−𝑓𝑥_𝑛.(2.4) Solving (2.4) and simplifying (without the index 𝑛 and using divided differences) yields in 𝑏₅=𝑓[𝑧,𝑥]𝑊𝑌𝑤−𝑦+𝑍𝑓[𝑤,𝑥]𝑦−𝑧𝑌+𝑓𝑦,𝑥(𝑧−𝑤)𝑊−𝑤−𝑦(𝑤−𝑧)𝑦−𝑧𝑓^′(𝑥)𝑓(𝑤)+(𝑓(𝑥)𝑦𝑓(𝑧,𝑏𝑌𝑍𝑧−𝑦𝑤−𝑧)𝑤−𝑦𝑦−𝑧+𝑊𝑍𝑓+𝑊𝑌𝑦−𝑤)₂=𝑓^′(𝑥)+𝑓(𝑥)𝑏₅,𝑏₄=(𝑧−𝑥)𝑓𝑦,𝑥+𝑥−𝑦𝑓[𝑧,𝑥]+𝑦−𝑧𝑏₂+(𝑧−𝑥)𝑓𝑦+𝑥−𝑦𝑓(𝑧)𝑏₅(),𝑏𝑥−𝑦𝑥−𝑧𝑦−𝑧₃=𝑓[𝑤,𝑥,𝑥]+𝑓[𝑤,𝑥]𝑏₅−(𝑤−𝑥)𝑏₄,(2.5) wherein 𝑦−𝑥=𝑌, 𝑧−𝑥=𝑍, 𝑤−𝑥=𝑊. Now we have a powerful approximation of the first derivative of the function in the fourth step of (2.1), which doubles the convergence rate of the optimal 8th order methods. Therefore, we attain the following class, in which we have four evaluations of the function and one evaluation of the first-order derivative: 𝑓𝑥Anyoptimal8th-ordermethod=^𝑛,and𝑓^′𝑥_𝑛𝑓𝑦areavailable,_𝑛𝑓𝑧isavailable,_𝑛𝑥isavailable,_𝑛+1=𝑤_𝑛−1+𝑏₅(𝑤_𝑛−𝑥_𝑛)²𝑓𝑤_𝑛𝑓^′𝑥_𝑛+2𝑏₃𝑤_𝑛−𝑥_𝑛+3𝑏₄+𝑏₃𝑏₅𝑤_𝑛−𝑥_𝑛²+2𝑏₄𝑏₅𝑤_𝑛−𝑥_𝑛³.(2.6)

Now using any of the optimal three-point three-step methods, we could obtain a novel 16th order technique which satisfies the Kung-Traub hypothesis as well. Using the optimal 8th order method given by Wang and Liu in [11] ends in the following four-step method: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑥_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛−2𝑓𝑦_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓𝑧_𝑛𝑓^′𝑥_𝑛12+5𝑓𝑥_𝑛²+8𝑓𝑥_𝑛𝑓𝑦_𝑛+2𝑓𝑦_𝑛²5𝑓𝑥_𝑛²−12𝑓𝑥_𝑛𝑓𝑦_𝑛12+𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑥_𝑛+1=𝑤_𝑛−1+𝑏₅(𝑤_𝑛−𝑥_𝑛)²𝑓𝑤_𝑛𝑓^′𝑥_𝑛+2𝑏₃𝑤_𝑛−𝑥_𝑛+3𝑏₄+𝑏₃𝑏₅𝑤_𝑛−𝑥_𝑛²+2𝑏₄𝑏₅𝑤_𝑛−𝑥_𝑛³.(2.7)

Theorem 2.1. Assume that the scalar function 𝑓 be sufficiently smooth in the real open domain 𝐼⊆ℝ. Furthermore let 𝑒_𝑛=𝑥_𝑛−𝑟 and 𝑐_𝑘=(𝑓^(𝑘)(𝑟))/(𝑘!𝑓^′(𝑟)), for 𝑘∈𝑁, 𝑁 is the set of natural numbers. Then, the iterative method (2.7) has the optimal order of convergence 16 and satisfies the following error equation: 𝑒_𝑛+1=125𝑐₃𝑐³₂𝑐²₂−𝑐₃²11𝑐⁴₂−25𝑐²₂𝑐₃+5𝑐²₃+5𝑐₂𝑐₄×11𝑐⁴₂𝑐₃−25𝑐²₂𝑐²₃+5𝑐₂𝑐₃𝑐₄+5𝑐³₃+𝑐²₄−𝑐₃𝑐₅𝑒_𝑛¹⁶+𝑂𝑒_𝑛¹⁷.(2.8)

Proof. Using Taylor series expansion around the simple root 𝑟 in the 𝑛th iterate, [12], results in 𝑓(𝑥_𝑛)=𝑓^′(𝑟)[𝑒_𝑛+𝑐₂𝑒²_𝑛+𝑐₃𝑒³_𝑛+𝑐₄𝑒⁴_𝑛+⋯+𝑂(𝑒_𝑛¹⁷)], and also 𝑓^′(𝑥_𝑛)=𝑓^′(𝑟)[1+2𝑐₂𝑒_𝑛+3𝑐₃𝑒²_𝑛+4𝑐₄𝑒³_𝑛+⋯+𝑂(𝑒_𝑛¹⁶)]. Applying these expansions in the first step of (2.7), we have 𝑦_𝑛−𝑟=𝑐₂𝑒²_𝑛+−2𝑐²₂+2𝑐₃𝑒³_𝑛+4𝑐³₂−7𝑐₂𝑐₃+3𝑐₄𝑒⁴_𝑛+⋯+𝑂𝑒_𝑛¹⁷.(2.9) Note that to keep the prerequisites at the minimum, we only mention the simplified error equations at the end of first, second, and third steps. Because we also know that the first three steps are an eighth-order technique. Taylor series expanding in the second step of (2.7) by applying (2.9) yields in 𝑧_𝑛−𝑟=𝑐³₂−𝑐₂𝑐₃𝑒⁴_𝑛−22𝑐⁴₂−4𝑐²₂𝑐₃+𝑐²₃+𝑐₂𝑐₄𝑒⁵_𝑛+⋯+𝑂𝑒_𝑛¹⁷.(2.10) Using (2.10) and the third step of (2.7) gives us 𝑤_𝑛1−𝑟=5𝑐₂𝑐²₂−𝑐₃11𝑐⁴₂−25𝑐²₂𝑐₃+5𝑐²₃+5𝑐₂𝑐₄𝑒⁸_𝑛+⋯+𝑂𝑒_𝑛¹⁷.(2.11) Now, we expand 𝑓(𝑤_𝑛), we obtain 𝑓𝑤_𝑛=15𝑐₂𝑐²₂−𝑐₃11𝑐⁴₂−25𝑐²₂𝑐₃+5𝑐²₃+5𝑐₂𝑐₄𝑓^′(𝑟)𝑒⁸_𝑛+⋯+𝑂𝑒_𝑛¹⁷.(2.12) Subsequently, in the last step we have 1+𝑏₅𝑤_𝑛−𝑥_𝑛²𝑓𝑤_𝑛𝑓^′𝑥_𝑛+2𝑏₃𝑤_𝑛−𝑥_𝑛+3𝑏₄+𝑏₃𝑏₅𝑤_𝑛−𝑥_𝑛²+2𝑏₄𝑏₅𝑤_𝑛−𝑥_𝑛³=15𝑐₂𝑐²₂−𝑐₃11𝑐⁴₂−25𝑐²₂𝑐₃+5𝑐²₃+5𝑐₂𝑐₄𝑒⁸_𝑛+⋯+𝑂𝑒_𝑛¹⁷.(2.13) Finally, using (2.12) and (2.13) in the last step of (2.7) ends in 𝑒_𝑛+1=𝑥_𝑛+11−𝑟=25𝑐₃𝑐³₂𝑐²₂−𝑐₃²11𝑐⁴₂−25𝑐²₂𝑐₃+5𝑐²₃+5𝑐₂𝑐₄×11𝑐⁴₂𝑐₃−25𝑐²₂𝑐²₃+5𝑐₂𝑐₃𝑐₄+5𝑐³₃+𝑐²₄−𝑐₃𝑐₅𝑒_𝑛¹⁶+𝑂𝑒_𝑛¹⁷,(2.14) which shows that (2.7) is a 16th order method consuming only five evaluations per iteration. This completes the proof.

Remark 2.2. If we choose any of the other optimal 8th order derivative-involved methods in the first three steps of (2.6), then a novel optimal 16th order technique will be obtained. For instance, using the optimal 8th order method given by J. R. Sharma and R. Sharma in [13] results in the following four-step optimal method: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−𝑓𝑥_𝑛𝑓𝑥_𝑛−2𝑓𝑦_𝑛𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓𝑧1+^𝑛𝑓𝑥_𝑛𝑓𝑥_𝑛,𝑦_𝑛𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑧_𝑛𝑓𝑥_𝑛,𝑧_𝑛,𝑥_𝑛+1=𝑤_𝑛−1+𝑏₅𝑤_𝑛−𝑥_𝑛²𝑓𝑤_𝑛𝑓^′𝑥_𝑛+2𝑏₃𝑤_𝑛−𝑥_𝑛+3𝑏₄+𝑏₃𝑏₅𝑤_𝑛−𝑥_𝑛²+2𝑏₄𝑏₅𝑤_𝑛−𝑥_𝑛³,(2.15) which satisfies the following error equation: 𝑒_𝑛+1=𝑐⁴₂𝑐₃𝑐²₂−𝑐₃²3𝑐³₂−4𝑐₂𝑐₄+𝑐₄3𝑐⁴₂𝑐₃−4𝑐²₂𝑐²₃+𝑐₂𝑐₃𝑐₄+𝑐²₄−𝑐₃𝑐₅𝑒_𝑛¹⁶+𝑂𝑒_𝑛¹⁷.(2.16)

Remark 2.3. Any method of the developed class carries out five evaluations per full cycle to reach the optimal order 16. Hence, the index of efficiency for our class is √⁵16≈1.741, which is greater than that of optimal 4th order techniques √³4≈1.587, and optimal 8th order techniques √⁴8≈1.682.

3. Computational Experiments

The analytical outcomes given in the last section are fully supported by numerical experiments here. Two methods of the class (2.6), that is, methods (2.7) (S-S 16 (I)) and (2.15) (S-S 16 (II)), are compared with some of high-order methods.

For numerical comparisons, we have the 16th-order method of Geum and Kim [14] given as follows: 𝑦_𝑛=𝑥_𝑛−𝑓^′𝑥_𝑛⁻¹𝑓𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−𝐾_𝑓𝑓^′𝑥_𝑛⁻¹𝑓𝑦_𝑛,𝑤_𝑛=𝑧_𝑛−𝐻_𝑓𝑓^′𝑥_𝑛⁻¹𝑓𝑧_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝑊_𝑓𝑓^′𝑥_𝑛⁻¹𝑓𝑤_𝑛,(3.1) wherein 𝐾_𝑓=(1+2𝑢−4𝑢²)/(1−3𝑢²), 𝐻_𝑓=(1+2𝑢)/(1−𝑣−2𝑞), 𝑊_𝑓=𝑢(1−6𝑢−𝑢²−24𝑢³)𝑞+ 2(1−𝑢)𝑞²+(1+2𝑢)/(1−𝑡−𝑣−2𝑞−2𝑣𝑞), and 𝑢=𝑓(𝑦)/𝑓(𝑥), 𝑣=𝑓(𝑧)/𝑓(𝑦), 𝑞=𝑓(𝑧)/𝑓(𝑥), 𝑡=𝑓(𝑤)/𝑓(𝑧) (without the index 𝑛). This method is denoted by (G-K 16). We also used the 16th order method obtained by Neta and Petković (1.5) with 𝑡=0 (N-P 16). We do not include other schemes with other orders, and one may refer to [15–21] for having further information about the other recent methods. Among many numerical examples, we have selected ten instances, with the use of multiprecision arithmetic. The test functions are displayed in Table 1.

Table 1 

Test functions and their zeros.

Table 2 

Results of comparisons for different methods.

For comparisons, 4000 digits floating points have been considered. As it shows, any method of the presented class is more efficient than the existing high order methods in the literature. Results in Table 2 manifest the applicability of the new scheme in the test problems. Also notice that F. stands for failure, that is, the starting value is not sufficiently close to the zero to make the iterative method converge.

We have checked that the sequence of iterations converge to an approximation of the solution for nonlinear equations in our numerical works. We note that an important problem of determining good starting points appears, when applying iterative methods for solving nonlinear equations. A quick convergence, one of the advantages of multipoint methods, can be attained only if initial approximations are sufficiently close to the sought roots; otherwise, it is not possible to realize the expected convergence speed in practice. For achieving such a goal, we remind the well-known technique of interval Newton’s method as a tool to obtain robust initial approximations in the next section.

4. All the Zeros

In this section, we pay close attention to the matter of finding all the (simple) zeros of nonlinear functions in an interval using a hybrid algorithm. In fact, in practice one wish to find all the zeros at the same time. On the other hand, the iterative scheme, such as (2.7), is so sensitive upon the choice of the initial guess. Generally speaking, no method is the best iterative scheme in all test problems. Mostly, this is due to the test function and the accuracy of the initial approximation.

Having a sharp enough initial estimation, one may start the process as efficiently as possible and to see the computational order of convergence correctly, that is, the rate of correcting the number decimal places to the true solution. However, having such an initial guess is not always an easy task for practical problems.

In what follows, we remind a well-known scheme based on [22], also known as interval Newton’s method for finding enough accurate initial approximations for all the zeros of a nonlinear function in an interval. Hopefully, Mathematica 8 gives the users a friendly environment for working with lists (sequences), that is to say, the obtained list then would be corrected using the new 16th order schemes of this paper.

This procedure was efficiently coded in [22] and with some changes we have provided in what follows:   intervalnewton::rec = "MaxRecursion exceeded.";   intnewt[f_,df_,x_,{a_,b_},eps_,n_]:=   Block[{xmid,int=Interval[{a,b}]},If[b-a<eps,Return[int]];     If[n==0,Message[intervalnewton::rec];      Return[int]];     xmid=Interval[SetAccuracy[(a+b)/2,16]];     int=IntervalIntersection[int,SetAccuracy[xmid-   N[f/.x->xmid]/N[df/.x->int],16]];    (intnewt[f,df,x,#,eps,n-1])&/@(List@@int)];   Options[intervalnewton]={MaxRecursion->2000};

The above piece of code takes the nonlinear function 𝑓, the lower and upper bounds of the working interval and applies the interval form of the Newton’s iteration with the maximum number of recursion considered as 2000 and also the use of the command SetAccuracy[exp,16]. Note that in this case, like the Newton’s method, the function should have first order derivative on the interval. Thus now, the call-out function to implement the above piece of code could be given by intervalnewton[f_,x_,int_Interval,eps_,opts___] :=Block[{df,n},n=MaxRecursion/.{opts}/.Options[intervalnewton];   df=D[f,x]; IntervalUnion@@Select[Flatten[(intnewt[f,df,x,#,eps,n])&/@(List@@int)], IntervalMemberQ [f/.x->#,0]&]];

The tolerance should be chosen to be arbitrary in machine precision. To illustrate the procedure further, we consider the oscillatory nonlinear function 𝑓(𝑥)=𝑒^{sin(log(𝑥)cos(20𝑥))}−2, where its plot has been given in Figure 1, in the interval [𝑎,𝑏]=[2.,10.] while the tolerance is set to 0.0001 as follows:   f[x_]:=Exp[Sin[Log[x]*Cos[20x]]]-2;   IntervalSol=intervalnewton[f[x],x,Interval[{2.,10.}],.0001];   setInitial=N[Mean/@List@@IntervalSol]   NumberOfGuesses=Length[setInitial];

Figure 1 

The plot of the function 𝑓 with finitely many zeros in the interval [2, 10].

The implementation of the above part of code shows that the number of zeros is 51 while the list of initial approximations is {2.18856, 2.2125, 2.48473, 2.54372, 2.791, 2.86524, 3.10023, 3.18401, 3.41091, 3.50142, 3.72247, 3.81803, 4.03459, 4.13411, 4.3471, 4.44984, 4.65989, 4.7653, 4.9729, 5.08056, 5.28606, 5.39566, 5.59937, 5.71065, 5.91277, 6.02554, 6.22625, 6.34035, 6.53981, 6.65509, 6.85342, 6.96977, 7.16709, 7.28441, 7.4808, 7.599, 7.79455, 7.91356, 8.10833, 8.22809, 8.42214, 8.54259, 8.73598, 8.85707, 9.04983, 9.17152, 9.36371, 9.48595, 9.67761, 9.80037, and 9.99152}.

We should remark that now by considering the new optimal 16th order methods from the class (2.6), one may enrich the accuracy of the initial guesses up to the desired tolerance when high precision computing is needed.

5. Conclusions

A class of four-point four-step iterative methods has been developed for solving nonlinear equations. The analytical proof for one method of the class was written to clarify the 16th order convergence. The class was attained by approximating the first derivative of the function in the fourth step of a cycle in which the first three steps are any of the optimal eighth-order derivative-involved methods.

Per full cycle, the methods of the class consist of four evaluations of the function and one evaluation of the first derivative which results in 1.741 as the efficiency index. The presented class satisfies the still unproved conjecture of Kung-Traub on multi-point iterations without memory.

The accuracy and efficiency of two obtained methods of the class were illustrated by solving a lot of numerical examples. Numerical works also have attested the theoretical results given in the paper and have shown the fast rate of convergence.

Acknowledgment

The authors cheerfully thank the remarks of the two reviewers on an earlier version of this paper.