An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

Abstract

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2^𝑛−1. Thus, the family agrees with Kung-Traub conjecture for the case 𝑛=4. Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods.

1. Introduction

Solving nonlinear equations is one of the most important problems in science and engineering [1, 2]. The boundary value problems arising in kinetic theory of gases, vibration analysis, design of electric circuits, and many applied fields are reduced to solving such equations. In the present era of advance computers, this problem has gained much importance than ever before.

In this paper, we consider iterative methods to find a simple root 𝑟 of the nonlinear equation 𝑓(𝑥)=0, where 𝑓∶𝑅→𝑅 be the continuously differentiable real function. Newton’s method [1] is probably the most widely used algorithm for solving such equations, which starts with an initial approximation 𝑥₀ closer to the root 𝑟 and generates a sequence of successive iterates {𝑥_𝑖}^∞₀ converging quadratically to the root. It is given by the following: 𝑥_𝑖+1=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑖=0,1,2,3,…. (1.1)

In order to improve the local order of convergence, a number of ways are considered by many researchers, see [3–26] and references therein. In particular, King [3] developed a one-parameter family of fourth-order methods defined by 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑤_𝑖−𝑓𝑥_𝑖+𝛽𝑓𝑤_𝑖𝑓𝑥_𝑖+𝛽−2𝑓𝑤_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖, (1.2) where 𝑤_𝑖 is the Newton point and 𝛽 is a constant.

This family requires two evaluations of the function 𝑓 and one evaluation of first derivative 𝑓^′ per iteration. The famous Ostrowski’s method [4, 5] is a member of this family for the case 𝛽=0. From practical point of view, the methods (1.2) are important because of higher efficiency than Newton’s method (1.1).

Traub [5] has divided iterative methods into two classes, namely, one-point methods and multipoint methods. Each class is further divided into two subclasses, namely, one-point methods with and without memory, and multipoint methods with and without memory. The important aspects related to these classes of methods are order of convergence and computational efficiency. Order of convergence shows the speed with which a given sequence of iterates converges to the root while the computational efficiency concerns with the economy of the entire process. Investigation of one-point methods with and without memory, has demonstrated theoretical restrictions on the order and efficiency of these two categories (see [5]). However, Kung and Traub [6] have conjectured that multipoint iteration methods without memory based on 𝑛 evaluations have optimal order 2^𝑛−1. In particular, with three evaluations a method of fourth-order can be constructed. The King’s method (1.2) is a well-known example of fourth-order multipoint methods without memory.

Recently, based on Ostrowski’s or King’s methods some higher-order multipoint methods have been proposed and analyzed for solving nonlinear equations. For example, Grau and Díaz-Barrero [10], Sharma and Guha [11], and Chun and Ham [12] have developed sixth-order modified Ostrowski’s methods each requires three 𝑓 and one 𝑓^′ evaluations per iteration. Kou et al. [15] presented a family of variants of Ostrowski’s method with seventh-order convergence requiring three 𝑓and one 𝑓^′ evaluations. With same number of evaluations, Bi et al. [18] developed a seventh-order family of modified King’s methods. Bi et al. [19] also presented an eighth-order family of modified King’s methods requiring four evaluations which agrees with the Kung-Traub conjecture.

In this paper, we present a new family of eighth-order methods without using second and higher derivatives. In terms of computational cost, it requires the evaluations of three functions and one first derivative per iteration. Thus the present methods provide a new example of multipoint methods without memory that with four evaluations a method of optimum order eight can be achieved as conjectured by Kung and Traub. The performance and effectiveness of the developed family of methods is tested and compared through some test functions.

Contents of the paper are summarized as follows. Some basic definitions relevant to the present work are presented in Section 2. In Section 3, we obtain new methods. Convergence analysis, for establishing eighth-order convergence, is carried out in Section 4. In Section 5, we provide some particular cases of the family. In Section 6, the method is tested and compared with other well-known methods on a number of problems. Concluding remarks are given in Section 7.

2. Basic Definitions

Definition 2.1. Let 𝑓(𝑥) be a real function with a simple root 𝑟 and let {𝑥_𝑖}_𝑖∈𝑁 be a sequence of real numbers that converges towards 𝑟. Then, we say that the order of convergence of the sequence is 𝑝, if there exits a number 𝑝∈ℝ⁺ such that lim_𝑖→∞𝑥_𝑖+1−𝑟𝑥_𝑖−𝑟^𝑝=𝐶, (2.1) for some 𝐶≠0, 𝐶 is known as the asymptotic error constant.
If 𝑝=1, 2 or 3, the sequence is said to have linear convergence, quadratic convergence or cubic convergence, respectively.

Definition 2.2. Let 𝑒_𝑖=𝑥_𝑖−𝑟 be the error in the 𝑖th iteration, we call the relation 𝑒_𝑖+1=𝐶𝑒^𝑝_𝑖+𝑂𝑒_𝑖^𝑝+1, (2.2) the error equation.

Definition 2.3. Let 𝑛 be the number of new pieces of information required by a method. A “piece of information” typically is any evaluation of a function or one of its derivatives. The efficiency of the method is measured by the concept of efficiency index [27] and is defined by the following: 𝐸=𝑝^1/𝑛, (2.3) where 𝑝 is the order of the method.

Definition 2.4. Suppose that 𝑥_𝑖+1,𝑥_𝑖 and 𝑥_𝑖−1 are three successive iterations closer to the root 𝑟. Then, the computational order of convergence 𝜌 (see [24, 25, 28]) is approximated by using (2.2) as follows: 𝑥𝜌≅ln_𝑖+1−𝑟/𝑥_𝑖−𝑟ln𝑥_𝑖−𝑟/𝑥_𝑖−1.−𝑟 (2.4)

3. The Method

We consider the iteration scheme of the form 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝜔𝜆_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−𝑊𝜇_𝑖𝑓𝑧_𝑖𝑓^′𝑧_𝑖, (3.1) where 𝜆_𝑖=𝑓(𝑤_𝑖)/𝑓(𝑥_𝑖), 𝜇_𝑖=𝑓(𝑧_𝑖)/𝑓(𝑥_𝑖), and 𝜔(𝑡) and 𝑊(𝑡) represent the real-valued functions (here onwards called weight functions). This scheme consists of three steps in which the first step represents Newton’s method and last two are weighted-Newton steps. It is quite obvious that formula (3.1) requires five evaluations per iteration. However, we can reduce the number of evaluations to four by using some suitable approximation of the derivative 𝑓^′(𝑧_𝑖). We obtain this approximation by considering the approximation of 𝑓(𝑥) by a rational linear function of the form 𝑦(𝑥)−𝑦𝑥_𝑖=𝑥−𝑥_𝑖+𝑎𝑏𝑥−𝑥_𝑖+𝑐, (3.2) where the parameters 𝑎,𝑏, and 𝑐 are determined by the condition that 𝑓 and 𝑦 coincide at 𝑥_𝑖, 𝑤_𝑖 and 𝑧_𝑖. That means 𝑦(𝑥) satisfies the conditions 𝑦𝑥_𝑖=𝑓𝑥_𝑖,𝑦𝑤_𝑖=𝑓𝑤_𝑖,𝑦𝑧_𝑖=𝑓𝑧_𝑖. (3.3)

From (3.2) and first condition of (3.3), it is easy to show that 𝑎=0. (3.4)

Substituting the value of 𝑎 into (3.2) then using the last two conditions of (3.3), after some simple calculations we obtain 𝑏𝑤_𝑖−𝑥_𝑖1+𝑐=𝑓𝑥_𝑖,𝑤_𝑖,𝑏𝑧_𝑖−𝑥_𝑖1+𝑐=𝑓𝑥_𝑖,𝑧_𝑖, (3.5) where 𝑓[𝑥_𝑖,𝑤_𝑖]=(𝑓(𝑤_𝑖)−𝑓(𝑥_𝑖))/(𝑤_𝑖−𝑥_𝑖) and 𝑓[𝑥_𝑖,𝑧_𝑖]=(𝑓(𝑧_𝑖)−𝑓(𝑥_𝑖))/(𝑧_𝑖−𝑥_𝑖) are first-divided differences.

Solving these equations, we can obtain 𝑏 and 𝑐 as follows: 1𝑏=𝑤_𝑖−𝑧_𝑖1𝑓𝑥_𝑖,𝑤_𝑖−1𝑓𝑥_𝑖,𝑧_𝑖,1𝑐=𝑤_𝑖−𝑧_𝑖𝑥_𝑖−𝑧_𝑖𝑓𝑥_𝑖,𝑤_𝑖−𝑥_𝑖−𝑤_𝑖𝑓𝑥_𝑖,𝑧_𝑖. (3.6)

Differentiation of (3.2) gives 𝑦^′(𝑥)=𝑐𝑏𝑥−𝑥_𝑖+𝑐². (3.7)

We can now approximate the derivative 𝑓^′(𝑥) with the derivative 𝑦^′(𝑥) of rational function (3.2) and obtain 𝑓^′𝑧_𝑖≈𝑦^′𝑧_𝑖. (3.8)

Substituting the values of 𝑏 and 𝑐 obtained in (3.6) into (3.7) then using (3.8), we get after simplifications 𝑓^′𝑧_𝑖=𝑓𝑥_𝑖,𝑧_𝑖𝑓𝑤_𝑖,𝑧_𝑖𝑓𝑥_𝑖,𝑤_𝑖. (3.9)

Then the iteration scheme (3.1) in its final form is given by the following: 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝜔𝜆_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−𝑊𝜇_𝑖𝑓𝑥_𝑖,𝑤_𝑖𝑓𝑧_𝑖𝑓𝑥_𝑖,𝑧_𝑖𝑓𝑤_𝑖,𝑧_𝑖, (3.10) where 𝜆_𝑖=𝑓(𝑤_𝑖)/𝑓(𝑥_𝑖), 𝜇_𝑖=𝑓(𝑧_𝑖)/𝑓(𝑥_𝑖), and 𝜔(𝑡) and 𝑊(𝑡) are the weight functions.

Thus the scheme (3.10) defines a new family of multipoint methods with two weight functions 𝜔(𝑡) and 𝑊(𝑡). In the next section, we will see that both of these functions play an important role in establishing eighth-order convergence of the methods.

4. Convergence of the Method

In order to examine the convergence property of the family (3.10), we prove the following theorem.

Theorem 4.1. Let the function 𝑓∶𝑅→𝑅 be sufficiently smooth in 𝑅. If 𝑓(𝑥) has a simple root 𝑟 in 𝑅 and 𝑥₀ is sufficiently close to 𝑟, then the sequence {𝑥_𝑖} generated by any method of the family (3.10) converges to 𝑟 with convergence order eight, provided the weight functions 𝜔(𝑡) and 𝑊(𝑡) satisfy the conditions 𝜔(0)=1,𝜔^′(0)=2,𝜔^′′(0)=8,𝑊(0)=1, 𝑊^′(0)=1, and |𝜔^′′′(0)|<∞.

Proof. Let 𝑒_𝑖=𝑥_𝑖−𝑟 be the error in the iterate 𝑥_𝑖. Using Taylor’s series expansion, we get 𝑓𝑥_𝑖=𝑓^′(𝑟)𝑒_𝑖+𝐴₂𝑒²_𝑖+𝐴₃𝑒³_𝑖+𝐴₄𝑒⁴_𝑖+𝐴₅𝑒⁵_𝑖+𝐴₆𝑒⁶_𝑖+𝐴₇𝑒⁷_𝑖+𝐴₈𝑒⁸_𝑖+𝑂𝑒⁹_𝑖,𝑓^′𝑥_𝑖=𝑓^′(𝑟)1+2𝐴₂𝑒_𝑖+3𝐴₃𝑒²_𝑖+4𝐴₄𝑒³_𝑖+5𝐴₅𝑒⁴_𝑖+6𝐴₆𝑒⁵_𝑖+7𝐴₇𝑒⁶_𝑖+8𝐴₈𝑒⁷_𝑖+𝑂𝑒⁸_𝑖, (4.1) where 𝐴_𝑘=𝑓^(𝑘)(𝑟)/𝑘!𝑓^′(𝑟) for 𝑘∈𝑁, 𝑁 is the set of natural numbers.
Now, 𝑓𝑥_𝑖𝑓^′𝑥_𝑖=𝑒_𝑖−𝐴₂𝑒²_𝑖−2−𝐴²₂+𝐴₃𝑒³_𝑖−4𝐴³₂−7𝐴₂𝐴₃+3𝐴₄𝑒⁴_𝑖−𝐾₁𝑒⁵_𝑖−𝐾₂𝑒⁶_𝑖−𝐾₃𝑒⁷_𝑖−𝐾₄𝑒⁸_𝑖+𝑂𝑒⁹_𝑖, (4.2) Following are the expressions of 𝐾_𝑛(𝑛=1,2,3,4)𝐾₁=−8𝐴⁴₂+20𝐴²₂𝐴₃−6𝐴²₃−10𝐴₂𝐴₄+4𝐴₅,𝐾₂=16𝐴⁵₂−52𝐴³₂𝐴₃+33𝐴₂𝐴²₃+28𝐴²₂𝐴₄−17𝐴₃𝐴₄−13𝐴₂𝐴₅+5𝐴₆,𝐾₃=−32𝐴⁶₂+128𝐴⁴₂𝐴₃−126𝐴²₂𝐴²₃+18𝐴³₃−72𝐴³₂𝐴₄+92𝐴₂𝐴₃𝐴₄−12𝐴²₄+36𝐴²₂𝐴₅−22𝐴₃𝐴₅−16𝐴₂𝐴₆+6𝐴₇,𝐾₄=64𝐴⁷₂−304𝐴⁵₂𝐴₃+408𝐴³₂𝐴²₃−135𝐴₂𝐴³₃+176𝐴⁴₂𝐴₄−348𝐴²₂𝐴₃𝐴₄+75𝐴²₃𝐴₄+64𝐴₂𝐴²₄−92𝐴³₂𝐴₅+118𝐴₂𝐴₃𝐴₅−31𝐴₄𝐴₅+44𝐴²₂𝐴₆−27𝐴₃𝐴₆−19𝐴₂𝐴₇+7𝐴₈. (4.3) For the sake of brevity, we omit their specific forms. We will use the same means in the following.
For 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖=𝑟+𝐴₂𝑒²_𝑖+2−𝐴²₂+𝐴₃𝑒³_𝑖+4𝐴³₂−7𝐴₂𝐴₃+3𝐴₄𝑒⁴_𝑖+𝐾₁𝑒⁵_𝑖+𝐾₂𝑒⁶_𝑖+𝐾₃𝑒⁷_𝑖+𝐾₄𝑒⁸_𝑖+𝑂𝑒⁹_𝑖.𝑒_𝑖̃=𝐴₂𝑒²_𝑖+2−𝐴²₂+𝐴₃𝑒³_𝑖+4𝐴³₂−7𝐴₂𝐴₃+3𝐴₄𝑒⁴_𝑖+𝐾₁𝑒⁵_𝑖+𝐾₂𝑒⁶_𝑖+𝐾₃𝑒⁷_𝑖+𝐾₄𝑒⁸_𝑖+𝑂𝑒⁹_𝑖. (4.4) Using Taylor’s series expansion, we get 𝑓𝑤_𝑖=𝑓^′(𝑟)𝑒_𝑖̃+𝐴₂𝑒²_𝑖̃+𝐴₃𝑒³_𝑖̃+𝐴₄𝑒⁴_𝑖̃+𝑂𝑒⁹_𝑖, (4.5) therefore, 𝑓𝑤_𝑖𝑓^′𝑥_𝑖=𝑒_𝑖̃−2𝐴₂𝑒_𝑖̃𝑒_𝑖+4𝐴²₂−3𝐴₃𝑒_𝑖̃𝑒²_𝑖+𝐴₂𝑒²_𝑖̃+−2𝐴²₂𝑒²_𝑖̃𝑒_𝑖+−8𝐴³₂+12𝐴₂𝐴₃−4𝐴₄𝑒_𝑖̃𝑒³_𝑖+𝑀₁𝑒⁶_𝑖+𝑀₂𝑒⁷_𝑖+𝑀₃𝑒⁸_𝑖+𝑂𝑒⁹_𝑖,𝑀₁=4𝐴³₂−3𝐴₂𝐴₃𝑒²_𝑖̃𝑒⁴_𝑖+16𝐴⁴₂−36𝐴²₂𝐴₃+9𝐴²₃+16𝐴₂𝐴₄−5𝐴₅𝑒_𝑖̃𝑒²_𝑖+𝐴₃𝑒³_𝑖̃𝑒⁶_𝑖,𝑀₂=−2𝐴₂𝐴₃𝑒³_𝑖̃𝑒⁶_𝑖+−8𝐴⁴₂+12𝐴²₂𝐴₃−4𝐴₂𝐴₄𝑒²_𝑖̃𝑒⁴_𝑖+−32𝐴⁵₂+96𝐴³₂𝐴₃−54𝐴₂𝐴²₃−48𝐴²₂𝐴₄+24𝐴₃𝐴₄+20𝐴₂𝐴₅−6𝐴₆𝑒_𝑖̃𝑒²_𝑖,𝑀₃=4𝐴²₂𝐴₃−3𝐴²₃𝑒³_𝑖̃𝑒⁶_𝑖+16𝐴⁵₂−36𝐴³₂𝐴₃+9𝐴₂𝐴²₃+16𝐴²₂𝐴₄−5𝐴₂𝐴₅𝑒²_𝑖̃𝑒⁴_𝑖+64𝐴⁶₂−240𝐴⁴₂𝐴₃+216𝐴²₂𝐴²₃−27𝐴³₃+128𝐴³₂𝐴₄−144𝐴₂𝐴₃𝐴₄+16𝐴²₄−60𝐴²₂𝐴₅+30𝐴₃𝐴₅+24𝐴₂𝐴₆−7𝐴₇𝑒_𝑖̃𝑒²_𝑖+𝐴₃𝑒⁴_𝑖̃𝑒⁸_𝑖. (4.6) Also 𝜆_𝑖=𝑓𝑤_𝑖𝑓𝑥_𝑖=𝑒_𝑖̃𝑒_𝑖−𝐴₂𝑒_𝑖̃+𝐴₂𝑒²_𝑖̃𝑒_𝑖+𝐴²₂−𝐴₃𝑒_𝑖̃𝑒_𝑖+−𝐴³₂+2𝐴₂𝐴₃−𝐴₄𝑒_𝑖̃𝑒²_𝑖−𝐴²₂𝑒²_𝑖̃+𝐴⁴₂−3𝐴²₂𝐴₃+𝐴²₃+2𝐴₂𝐴₄−𝐴₅𝑒_𝑖̃𝑒³_𝑖+𝐴³₂−𝐴₂𝐴₃𝑒²_𝑖̃𝑒_𝑖+𝐴₃𝑒³_𝑖̃𝑒_𝑖+−𝐴⁵₂+4𝐴³₂𝐴₃−3𝐴₂𝐴²₃−3𝐴²₂𝐴₄+2𝐴₃𝐴₄+2𝐴₂𝐴₅−𝐴₆𝑒_𝑖̃𝑒⁴_𝑖+−𝐴⁴₂+2𝐴²₂𝐴₃−𝐴₂𝐴₄𝑒²_𝑖̃𝑒²_𝑖−𝐴₂𝐴₃𝑒³_𝑖̃+𝑂𝑒⁷_𝑖. (4.7) Thus, using the Taylor expansion, we get 𝜔𝜆_𝑖=𝜔(0)+𝜔^′(0)𝜆_𝑖+1𝜔2!^′′(0)𝜆²_𝑖+1𝜔3!^′′′(0)𝜆³_𝑖+𝑂𝜆⁴_𝑖=𝜔(0)+𝜔^′(0)𝑒_𝑖̃𝑒_𝑖+12𝜔^′′(0)𝑒²_𝑖̃𝑒²_𝑖−𝐴₂𝜔^′(0)𝑒_𝑖̃+16𝜔^′′′(0)𝑒³_𝑖̃𝑒³_𝑖+𝐴₂𝜔^′(0)−𝜔^′′(0)𝑒²_𝑖̃𝑒_𝑖+𝐴²₂−𝐴₃𝜔^′(0)𝑒_𝑖̃𝑒_𝑖+𝐿₁𝑒⁴_𝑖+𝐿₂𝑒⁵_𝑖+𝐿₃𝑒⁶_𝑖+𝑂𝑒⁷_𝑖, (4.8) where 𝐿₁=1𝜔24^𝑖𝜈(0)𝑒⁴_𝑖̃𝑒⁸_𝑖+12𝐴₂2𝜔^′′(0)−𝜔^′′′(0)𝑒³_𝑖̃𝑒⁶_𝑖+−𝐴₃𝜔^′′(0)+𝐴²₂−𝜔^′(0)+32𝜔^′′(0)𝑒²_𝑖̃𝑒⁴_𝑖−𝐴³₂+2𝐴₂𝐴₃−𝐴₄𝜔^′(0)𝑒_𝑖̃𝑒²_𝑖,𝐿₂=1𝜔120^𝜈(0)𝑒⁵_𝑖̃𝑒_𝑖¹⁰+16𝐴₂3𝜔^′′′(0)−𝜔^(𝑖𝜈)(0)×𝐴₃𝜔^′(0)−12𝜔^′′′(0)+𝐴²₂−2𝜔^′′(0)+𝜔^′′′(0)𝑒³_𝑖̃𝑒⁶_𝑖+−𝐴₂𝐴₃𝜔^′(0)−3𝜔^′′(0)+𝐴³₂𝜔^′(0)−2𝜔^′′(0)−𝐴₄𝜔^′′(0𝑒̃)^2𝑖𝑒⁴_𝑖+𝐴⁴₂−3𝐴²₂𝐴₃+𝐴²₃+2𝐴₂𝐴₄−𝐴₅𝜔^′(0)𝑒_𝑖̃𝑒²_𝑖,𝐿₃=1𝜔720^𝜈𝑖(0)𝑒⁶_𝑖̃𝑒_𝑖¹²+1𝐴24²4𝜔^𝑖𝜈(0)−𝜔^𝜈(0)𝑒⁵_𝑖̃𝑒_𝑖¹⁰+1122𝐴₃6𝜔^′′(0)−𝜔^𝑖𝜈(0)+𝐴²₂6𝜔^′′(0)−18𝜔^′′′(0)+5𝜔^𝑖𝜈(0)𝑒⁴_𝑖̃𝑒⁸_𝑖+−𝐴₂𝐴₃𝜔^′(0)+2𝜔^′′(0)−2𝜔^′′′(0)+𝐴³₂3𝜔^′′(0)−53𝜔^′′′(0)−12𝐴₄𝜔^′′′(0)𝑒³_𝑖̃+2𝐴²₂𝐴₃𝜔^′(0)−3𝜔^′′(0)−𝐴₂𝐴₄𝜔^′(0)−3𝜔^′′(0)+123𝐴²₃−2𝐴₅𝜔^′′(0)+𝐴⁴₂−𝜔^′(0)+52𝜔^′′(0)𝑒²_𝑖̃𝑒⁴_𝑖+−𝐴⁵₂+4𝐴³₂𝐴₃−3𝐴₂𝐴²₃−3𝐴²₂𝐴₄+2𝐴₃𝐴₄+2𝐴₂𝐴₅−𝐴₆𝜔^′(0)𝑒_𝑖̃𝑒²_𝑖. (4.9) Using (4.6) and (4.8), we have 𝑧_𝑖=𝑤_𝑖−𝜔𝜆_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖=𝑟+[1−𝜔(0)]𝑒_𝑖̃−𝜔^′(0)𝑒²_𝑖̃𝑒_𝑖−2𝜔(0)𝐴₂𝑒_𝑖̃𝑒_𝑖−12𝜔^′′(0)𝑒³_𝑖̃𝑒²_𝑖+𝐴₂𝜔(0)−3𝜔^′(0)𝑒²_𝑖̃+𝜔(0)4𝐴²₂−3𝐴₃𝑒_𝑖̃𝑒²_𝑖−12𝜔^′′(0)𝑒³_𝑖̃𝑒²_𝑖+𝐴₂𝜔(0)−3𝜔^′(0)𝑒²_𝑖̃+𝜔(0)4𝐴²₂−3𝐴₃𝑒_𝑖̃𝑒²_𝑖𝑒⁵_𝑖−𝑀₆𝑒⁶_𝑖−𝑀₇𝑒⁷_𝑖−𝑀₈𝑒⁸_𝑖+𝑂𝑒⁹_𝑖, (4.10) where 𝑀_𝑛(𝑛=6,7,8) are the expression about 𝐴_𝑛(𝑛=2,3,…,8).
If (0)=1, 𝜔^′(0)=2 and substituting the value of 𝑒_𝑖̃ from (4.4), we get 𝑒_𝑖̂=𝑧_𝑖=1−𝑟5−2𝜔^′′(0)𝐴³₂−𝐴₂𝐴₃𝑒⁴_𝑖+−36+5𝜔^′′(0)−16𝜔^′′′(0)𝐴⁴₂+32−3𝜔^′′(0)𝐴²₂𝐴₃−2𝐴²₃−2𝐴₂𝐴₄𝑒⁵_𝑖+𝑀₆𝑒⁶_𝑖+𝑀₇𝑒⁷_𝑖+𝑀₈𝑒⁸_𝑖+𝑂𝑒⁹_𝑖. (4.11) Using Taylor’s series expansion, we get 𝑓𝑧_𝑖=𝑓^′(𝑟)𝑒_𝑖̂+𝐴₂𝑒²_𝑖̂+𝑂𝑒⁹_𝑖, (4.12) furthermore, 𝑓𝑥_𝑖,𝑤_𝑖=𝑓𝑤_𝑖−𝑓𝑥_𝑖𝑤_𝑖−𝑥_𝑖=𝑓^′(𝑟)1+𝐴₂𝑒_𝑖+𝐴²₂+𝐴₃𝑒²_𝑖+−2𝐴³₂+3𝐴₂𝐴₃+𝐴₄𝑒³_𝑖+4𝐴⁴₂−8𝐴²₂𝐴₃+2𝐴²₃+4𝐴₂𝐴₄+𝐴₅𝑒⁴_𝑖+𝑂𝑒⁵_𝑖.𝑓𝑥_𝑖,𝑧_𝑖=𝑓𝑧_𝑖−𝑓𝑥_𝑖𝑧_𝑖−𝑥_𝑖=𝑓^′(𝑟)1+𝐴₂𝑒_𝑖+𝐴₃𝑒²_𝑖+𝐴₄𝑒³_𝑖+15−2𝜔^′′(0)𝐴⁴₂−𝐴²₂𝐴₃+𝐴₅𝑒⁴_𝑖+𝑂𝑒⁵_𝑖.𝑓𝑤_𝑖,𝑧_𝑖=𝑓𝑧_𝑖−𝑓𝑤_𝑖𝑧_𝑖−𝑤_𝑖=𝑓^′(𝑟)1+𝐴²₂𝑒²_𝑖−2𝐴³₂−𝐴₂𝐴₃𝑒³_𝑖+19−2𝜔^′′(0)𝐴⁴₂−7𝐴²₂𝐴₃+3𝐴₂𝐴₄𝑒⁴_𝑖+𝑂𝑒⁵_𝑖. (4.13)
Using the above results, we obtain 𝑓𝑥_𝑖,𝑤_𝑖𝑓𝑤_𝑖,𝑧_𝑖𝑓𝑥_𝑖,𝑧_𝑖=1𝑓^′(𝑟)1+−𝐴³₂+𝐴₂𝐴₃𝑒³_𝑖+−7+𝜔^′′(0)𝐴⁴₂−4𝐴²₂𝐴₃+2𝐴²₃+𝐴₂𝐴₄𝑒⁴_𝑖+𝑂𝑒⁵_𝑖. (4.14) Also 𝜇_𝑖=𝑓𝑧_𝑖𝑓𝑥_𝑖=𝑒_𝑖̂𝑒_𝑖−𝐴₂𝑒_𝑖̂+𝑂𝑒⁵_𝑖. (4.15) Thus, using the Taylor expansion and |𝑊^′′(0)|<∞,we get 𝑊𝜇_𝑖=𝑊(0)+𝑊^′(0)𝜇_𝑖+𝑂𝜇²_𝑖=𝑊(0)+𝑊^′(0)𝑒_𝑖̂𝑒_𝑖−𝐴₂𝑒_𝑖̂+𝑂𝑒⁵_𝑖. (4.16) Using these results in 𝑥_𝑖+1=𝑧_𝑖−𝑊𝜇_𝑖𝑓𝑥_𝑖,𝑤_𝑖𝑓𝑧_𝑖𝑓𝑤_𝑖,𝑧_𝑖𝑓𝑥_𝑖,𝑧_𝑖, (4.17) we obtain 𝑒_𝑖+1=𝑒_𝑖̂−𝑊(0)+𝑊^′(0)𝑒_𝑖̂𝑒_𝑖−𝑊^′(0)𝐴₂𝑒_𝑖̂+𝑂𝑒⁵_𝑖𝑒_𝑖̂+𝐴₂𝑒²_𝑖̂+𝑂𝑒³_𝑖̂×1+−𝐴³₂+𝐴₂𝐴₃𝑒³_𝑖+−7+𝜔^′′(0)𝐴⁴₂−4𝐴²₂𝐴₃+2𝐴²₃+𝐴₂𝐴₄𝑒⁴_𝑖+𝑂𝑒⁵_𝑖=𝑒_𝑖̂−𝑊(0)+𝑊^′(0)𝑒_𝑖̂𝑒_𝑖−𝑊^′(0)𝐴₂𝑒_𝑖̂𝑒_𝑖̂+𝐴₂𝑒²_𝑖̂×1+−𝐴³₂+𝐴₂𝐴₃𝑒³_𝑖+−7+𝜔^′′(0)𝐴⁴₂−4𝐴²₂𝐴₃+2𝐴²₃+𝐴₂𝐴₄𝑒⁴_𝑖+𝑂𝑒⁹_𝑖=[1−𝑊(0)]𝑒_𝑖̂−𝑊(0)−𝐴³₂+𝐴₂𝐴₃𝑒³_𝑖+𝑊^′(0)𝑒_𝑖̂𝑒_𝑖𝑒_𝑖̂−𝑊(0)−𝑊^′(0)𝐴₂𝑒²_𝑖̂−−7+𝜔^′′(0)𝐴⁴₂−4𝐴²₂𝐴₃+2𝐴²₃+𝐴₂𝐴₄𝑊(0)𝑒⁴_𝑖𝑒_𝑖̂+𝑂𝑒⁹_𝑖=[1−𝑊(0)]𝑒_𝑖̂−𝑊(0)−𝐴³₂+𝐴₂𝐴₃+𝑊^′(0)15−2𝜔^′′(0)𝐴³₂−𝐴₂𝐴₃𝑒³_𝑖𝑒_𝑖̂+𝑊^′(0)−𝑊(0)𝐴₂𝑒²_𝑖̂+𝑊(0)7−𝜔^′′(0)𝐴⁴₂+4𝐴²₂𝐴₃−2𝐴²₃−𝐴₂𝐴₄−𝑊^′(0)−36+5𝜔^′′(0)−16𝜔^′′′(0)𝐴⁴₂+32−3𝜔^′′(0)𝐴²₂𝐴₃−2𝐴²₃−2𝐴₂𝐴₄×𝑒⁴_𝑖𝑒_𝑖̂+𝑂𝑒⁹_𝑖. (4.18) This means that convergence order of the family (3.10) is seventh-order with 𝑊(0)=1 and the error equation is 𝑒_𝑖+1=𝐴³₂−𝐴₂𝐴₃15−2𝜔^′′(0)𝐴³₂−𝐴₂𝐴₃−𝑊^′(0)15−2𝜔^′′(0)𝐴³₂−𝐴₂𝐴₃²𝑒⁷_𝑖+𝑂𝑒⁸_𝑖, (4.19) and if 𝑊 is any function with 𝑊(0)=1,𝑊^′(0)=1, and 𝜔^′′(0)=8, then the convergence order of any method of the family (3.10) arrives to eight, and the error equation is 𝑒_𝑖+1=𝐴²₂𝐴²₂−𝐴₃1−5+6𝜔^′′′(0)𝐴³₂−4𝐴₂𝐴₃+𝐴₄𝑒⁸_𝑖+𝑂𝑒⁹_𝑖. (4.20) Thus if 𝜔 and 𝑊are any functions with (0)=1,𝜔^′(0)=2,𝜔^′′(0)=8,𝑊(0)=1, and 𝑊^′(0)=1, then the eighth-order convergence is established. This completes the proof of the theorem.

Note that per iteration every method of the family (3.10) uses four pieces of information, namely, 𝑓(𝑥_𝑖),𝑓^′(𝑥_𝑖),𝑓(𝑤_𝑖),𝑓(𝑧_𝑖) and has eighth-order convergence with the conditions 𝜔(0)=1,𝜔^′(0)=2,𝜔^′′(0)=8,𝑊(0)=1, and𝑊^′(0)=1, which is in accordance with Kung-Traub conjecture for 4 evaluations.

5. Some Particular Forms

Here, we consider some forms of the functions 𝜔(𝑡) and 𝑊(𝑡) satisfying the conditions of the Theorem 4.1. Based on these forms some methods of the family (3.10) are also presented.

5.1. Forms of 𝜔(𝑡)

Form 1. For the function 𝜔 given by the following: 𝜔₁(𝑡)=1+2𝑡+4𝑡²+𝛼𝑡³, (5.1) where 𝛼∈𝑅 is a constant, it is clear that the conditions of Theorem 4.1 are satisfied.

Form 2. For the function 𝜔 defined by the following: 𝜔₂(𝑡)=1+2𝑡1−2𝑡+𝛼𝑡², (5.2) where 𝛼∈𝑅, it can be easily seen that this function satisfies the conditions of Theorem 4.1.

Form 3. For the function 𝜔 defined by the following: 𝜔₃(𝑡)=1+2𝛼𝑡+𝛽𝑡^21/𝛼,𝛽=2𝛼(𝛼+1), (5.3) where 𝛼∈𝑅−{0}, it can be seen that this function also satisfies the conditions of Theorem 4.1.

5.2. Forms of 𝑊(𝑡)

Form 1. For the function 𝑊 given by the following: 𝑊₁(𝑡)=1+𝑡+𝛾𝑡², (5.4) where 𝛾∈𝑅, it can be seen the function 𝑊₁(𝑡) satisfies the conditions of Theorem 4.1.

Form 2. For the function 𝑊 defined by the following: 𝑊₂(𝑡)𝑡=1+,1+𝛾𝑡 (5.5) where 𝛾∈𝑅, it is simple to see that 𝑊₂(𝑡) satisfies the conditions of Theorem 4.1

Form 3. For the function 𝑊defined by the following: 𝑊₃(𝑡)=1+𝛾𝑡^1/𝛾, (5.6) where 𝛾∈𝑅−{0}, again it can be seen that 𝑊₃(𝑡) satisfies the conditions of Theorem 4.1.

5.3. Forms of Methods

To form a concrete method we can take any combination of the above defined 𝜔(𝑡) and 𝑊(𝑡). For simplicity, we consider only three such combinations. For example, by taking 𝜔₂(𝑡) with 𝑊_𝑖(𝑡),𝑖=1,2,3 the following methods can be formed

Method 1. Taking 𝜔₂(𝑡) and 𝑊₁(𝑡), we get a new two-parameter family of eighth-order methods 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝑓²𝑥_𝑖+𝛼𝑓²𝑤_𝑖𝑓²𝑥_𝑖−2𝑓𝑥_𝑖𝑓𝑤_𝑖+𝛼𝑓²𝑤_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−𝑓𝑧1+^𝑖𝑓𝑥_𝑖𝑓+𝛾²𝑧_𝑖𝑓²𝑥_𝑖𝑓𝑥_𝑖,𝑤_𝑖𝑓𝑧_𝑖𝑓𝑤_𝑖,𝑧_𝑖𝑓𝑥_𝑖,𝑧_𝑖. (5.7)

Method 2. Considering 𝜔₂(𝑡) and 𝑊₂(𝑡), we get another new two-parameter family of eighth-order methods 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝑓²𝑥_𝑖+𝛼𝑓²𝑤_𝑖𝑓²𝑥_𝑖−2𝑓𝑥_𝑖𝑓𝑤_𝑖+𝛼𝑓²𝑤_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−𝑓𝑥_𝑖+1+𝛾𝑓𝑧_𝑖𝑓𝑥_𝑖+𝛾𝑓𝑧_𝑖𝑓𝑥_𝑖,𝑤_𝑖𝑓𝑧_𝑖𝑓𝑤_𝑖,𝑧_𝑖𝑓𝑥_𝑖,𝑧_𝑖. (5.8)

Method 3. Considering now 𝜔₂(𝑡) and 𝑊₃(𝑡), we get another new two-parameter family of eighth-order methods 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝑓²𝑥_𝑖+𝛼𝑓²𝑤_𝑖𝑓²𝑥_𝑖−2𝑓𝑥_𝑖𝑓𝑤_𝑖+𝛼𝑓²𝑤_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−1+𝛾𝑓(𝑧_𝑖)𝑓(𝑥_𝑖)^1/𝛾𝑓𝑥_𝑖,𝑤_𝑖𝑓𝑧_𝑖𝑓𝑤_𝑖,𝑧_𝑖𝑓𝑥_𝑖,𝑧_𝑖. (5.9) The proposed families require three evaluations of the function 𝑓 and one evaluation of first derivative 𝑓^′per iteration, and achieve eighth-order convergence. Thus the efficiency index (𝐸) defined by (2.3) of the present methods (3.10) is 𝐸=√⁴8≈1.682 which is better than 𝐸=√2≈1.414 of Newton’s method, 𝐸=√³4≈1.587 of King’s [3] and Ostrowski’s [4] methods, 𝐸=√⁴6≈1.565 of sixth-order methods [10–12] and 𝐸=√⁴7≈1.627 of seventh-order methods [15, 18].

6. Numerical Examples

We employ the present methods (4.1), and (4.4) denoted by M81, M82 and M83, respectively to solve some nonlinear equations and compare with Newton’s method (NM) defined by (1.1), the eighth-order method developed by Cordero et al. [23] denoted by C8 and defined as follows: 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑥_𝑖−𝑓𝑥_𝑖−𝑓𝑤_𝑖𝑓𝑥_𝑖−2𝑓𝑤_𝑖𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑢_𝑖=𝑧_𝑖−𝑓𝑥_𝑖−𝑓𝑤_𝑖𝑓𝑥_𝑖−2𝑓𝑤_𝑖+12𝑓𝑧_𝑖𝑓𝑤_𝑖−2𝑓𝑧_𝑖𝑓𝑧_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑢_𝑖−3𝛽₂+𝛽₃𝑢_𝑖−𝑧_𝑖𝛽₁𝑢_𝑖−𝑧_𝑖+𝛽₂𝑤_𝑖−𝑥_𝑖+𝛽₃𝑧_𝑖−𝑥_𝑖𝑓𝑧_𝑖𝑓^′𝑥_𝑖, (6.1) where 𝛽₁,𝛽₂,𝛽₃∈𝑅 and 𝛽₂+𝛽₃≠0, eighth-order method developed by Liu and Wang [22] denoted by L8 and defined as follows: 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝑓𝑥_𝑖𝑓𝑥_𝑖−2𝑓𝑤_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−𝑓(𝑥_𝑖)−𝑓(𝑤_𝑖)𝑓(𝑥_𝑖)−2𝑓(𝑤_𝑖)²+𝑓𝑧_𝑖𝑓𝑤_𝑖−𝛼₁𝑓𝑧_𝑖+4𝑓𝑧_𝑖𝑓𝑥_𝑖+𝛼₂𝑓𝑧_𝑖𝑓𝑧_𝑖𝑓^′𝑥_𝑖, (6.2) where 𝛼₁,𝛼₂∈𝑅, eighth-order method developed by Petković et al. [20] denoted by P8 and defined as follows: 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝑓𝑥_𝑖𝑓𝑥_𝑖−2𝑓𝑤_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−1+𝑎₄𝑧_𝑖−𝑥_𝑖²𝑓𝑧_𝑖𝑎₂−𝑎₁𝑎₄+𝑎₃𝑧_𝑖−𝑥_𝑖2+𝑎₄𝑧_𝑖−𝑥_𝑖, (6.3) where 𝑎₁=𝑓(𝑥_𝑖), 𝑎₃=(𝑓^′(𝑥_𝑖)𝑓[𝑤_𝑖,𝑧_𝑖]−𝑓[𝑥_𝑖,𝑤_𝑖]𝑓[𝑥_𝑖,𝑧_𝑖])/((𝑥_𝑖𝑓[𝑤_𝑖,𝑧_𝑖]+(𝑤_𝑖𝑓(𝑧_𝑖)−𝑧_𝑖𝑓(𝑤_𝑖))/(𝑤_𝑖−𝑧_𝑖))−𝑓(𝑥_𝑖)), 𝑎₄=𝑎₃𝑓𝑥_𝑖,𝑤_𝑖+𝑓^′𝑥_𝑖−𝑓𝑥_𝑖,𝑤_𝑖𝑤_𝑖−𝑥_𝑖𝑓𝑥_𝑖,𝑤_𝑖,𝑎₂=𝑓^′𝑥_𝑖+𝑎₄𝑓𝑥_𝑖, (6.4) eighth-order method developed by Thukral and Petković [21] denoted by T8 and defined as follows: 𝑤_𝑖=𝑥_𝑖−𝑓𝑥_𝑖𝑓^′𝑥_𝑖,𝑧_𝑖=𝑤_𝑖−𝑓𝑥_𝑖𝑓𝑥_𝑖−2𝑓𝑤_𝑖𝑓𝑤_𝑖𝑓^′𝑥_𝑖,𝑥_𝑖+1=𝑧_𝑖−𝑓²𝑥_𝑖𝑓²𝑥_𝑖−2𝑓𝑥_𝑖𝑓𝑤_𝑖−𝑓²𝑤_𝑖+𝑓𝑧_𝑖𝑓𝑤_𝑖−𝛼𝑓𝑧_𝑖+4𝑓𝑧_𝑖𝑓𝑥_𝑖𝑓𝑧_𝑖𝑓^′𝑥_𝑖, (6.5) where 𝛼∈𝑅, eighth-order methods presented in Section  3 of [19] by Bi et al. denoted by B81, B82 and B83.

The test functions and root 𝑟 correct up to 16 decimal places are displayed in Table 1. The first eight functions we have selected are same as in [19]. The last function is selected from [18]. In Table 2, we exhibit the absolute values of the difference of root 𝑟 and its approximation 𝑥_𝑖, where 𝑟 is computed with 350 significant digits and 𝑥_𝑖 is calculated by costing the same total number of function evaluations (TFE) for each method. The TFE is counted as sum of the number of evaluations of the function itself plus the number of evaluations of the derivatives. In the calculations, 12 TFE are used by each method. That means 6 iterations are used for NM and 3 iterations for the remaining methods. The absolute values of the function |𝑓(𝑥_𝑖)| and the computational order of convergence (𝜌) are also displayed in Table 2. It can be observed that the computed results, displayed in Table 2, overwhelmingly support the theory of convergence and efficiency analyses discussed in the previous sections. From the results, it can be concluded that the proposed methods are competitive with existing methods and possess quick convergence for good initial approximations. Among the eighth-order methods, we are not able to select one as the best. For some initial guess one is better while for other initial guess the another one would be appropriate. Thus the present methods can be of practical interest.

Table 1 

Test functions.

Table 2 

Comparison of methods using same total number of function evaluations for all methods (TFE = 12).

7. Conclusions

In this work, we have obtained a new simple and elegant family of eighth-order multipoint methods for solving nonlinear equations. Thus, one requires three evaluations of the function 𝑓 and one of its first-derivative 𝑓^′ per full step and therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index of Newton method, fourth-order methods, sixth-order methods, and seventh-order methods.

Many numerical applications use higher precision in their computations. In these types of applications, numerical methods of higher-order are important. The numerical results show that the methods associated with a multiprecision arithmetic floating point are very useful, because these methods yield a clear reduction in number of iterations. Finally, we conclude that the methods presented in this paper are preferable to other recognized efficient methods, namely, Newton’s method, King’s methods, sixth-order methods [10–12], seventh-order methods [15, 18], etc.