Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

Abstract

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

1. Introduction

One of the most important problems in numerical analysis is solving nonlinear equations. To solve these equations, we can use iterative methods such as Newton's method and its variants. Newton's classical method for a single nonlinear equation 𝑓(𝑥)=0, where 𝑟 is a single root, is written as 𝑥_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (1) which converges quadratically in some neighborhood of 𝑟.

Taking 𝑦_𝑛=𝑥_𝑛−𝑓(𝑥_𝑛)/𝑓^′(𝑥_𝑛), many modifications of Newton's method were recently published. In [1], Noor and Khan presented a fourth-order optimal method as defined by 𝑥_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛−𝑓𝑦_𝑛𝑓𝑥_𝑛−2𝑓𝑦_𝑛𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (2) which uses three functional evaluations.

In [2], Cordero et al. proposed a fourth-order optimal method as defined by 𝑧_𝑛=𝑥_𝑛−𝑓𝑥_𝑛+𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑤_𝑛=𝑧_𝑛−𝑓²𝑦_𝑛2𝑓𝑥_𝑛+𝑓𝑦_𝑛𝑓²𝑥_𝑛𝑓^′𝑥_𝑛, (3) which also uses three functional evaluations.

Chun presented a third-order iterative formula [3] as defined by 𝑥_𝑛+1=𝑥_𝑛−32𝑓𝑥_𝑛𝑓^′𝑥_𝑛+12𝑓𝑥_𝑛𝑓^′𝜙𝑥_𝑛𝑓^′2𝑥_𝑛, (4) which uses three functional evaluations, where 𝜙 is any iterative function of second order.

Li et al. presented a fifth-order iterative formula in [4] as defined by 𝑢_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛−𝑓𝑥_𝑛−𝑓𝑥_𝑛/𝑓^′𝑥_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑢_𝑛+1−𝑓𝑢_𝑛+1𝑓^′𝑥_𝑛−𝑓𝑥_𝑛/𝑓^′𝑥_𝑛, (5) which uses five functional evaluations.

The main goal and motivation in the development of new methods are to obtain a better computational efficiency. In other words, it is advantageous to obtain the highest possible convergence order with a fixed number of functional evaluations per iteration. In the case of multipoint methods without memory, this demand is closely connected with the optimal order considered in the Kung-Traub’s conjecture.

Kung-Traub's Conjecture (see [5]). Multipoint iterative methods (without memory) requiring 𝑛+1 functional evaluations per iteration have the order of convergence at most 2^𝑛.

Multipoint methods which satisfy Kung-Traub's conjecture (still unproved) are usually called optimal methods; consequently, 𝑝=2^𝑛 is the optimal order.

The computational efficiency of an iterative method of order 𝑝, requiring 𝑛 function evaluations per iteration, is most frequently calculated by Ostrowski-Traub's efficiency index [6] 𝐸=𝑝^1/𝑛.

On the case of multiple roots, the quadratically convergent modified Newton's method [7] is 𝑥_𝑛+1=𝑥_𝑛−𝑚𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (6) where 𝑚 is the multiplicity of the root.

For this case, there are several methods recently presented to approximate the root of the function. For example, the cubically convergent Halley's method [8] is a special case of the Hansen-Patrick's method [9] 𝑥_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛((𝑚+1)/2𝑚)𝑓^′𝑥_𝑛−𝑓𝑥_𝑛𝑓^′′𝑥_𝑛/2𝑓^′𝑥_𝑛. (7) Osada [10] has developed a third-order method using the second derivative: 𝑥_𝑛+1=𝑥_𝑛−12𝑚(𝑚+1)𝑢_𝑛+12(𝑚−1)²𝑓^′𝑥_𝑛𝑓^′′𝑥_𝑛, (8) where 𝑢_𝑛=𝑓(𝑥_𝑛)/𝑓^′(𝑥_𝑛).

Another third-order method [11] based on King's fifth-order method (for simple roots) [12] is the Euler-Chebyshev's method of order three 𝑥_𝑛+1=𝑥_𝑛−𝑚(3−𝑚)2𝑓𝑥_𝑛𝑓^′𝑥_𝑛+𝑚²2𝑓²𝑥_𝑛𝑓^′′𝑥_𝑛𝑓^′3𝑥_𝑛. (9)

Recently, Chun and Neta [13] have developed a third-order method using the second derivative: 𝑥_𝑛+1=𝑥_𝑛−2𝑚²𝑓²𝑥_𝑛𝑓^′′𝑥_𝑛×𝑚(3−𝑚)𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓^′′𝑥_𝑛+(𝑚−1)²𝑓^′3𝑥_𝑛⁻¹. (10)

All previous methods use the second derivative of the function to obtain a greater order of convergence. The objective of the new method is to avoid the use of the second derivative.

The new methods are based on a mixture of Lagrange's and Hermite's interpolations. That is to say not only Hermite’s interpolation. This is the novelty of the new methods. The interpolation process is a conventional tool for iterative methods; see [5, 7]. However, this tool has been applied recently in several ways. For example, in [14], Cordero and Torregrosa presented a family of Steffensen-type methods of fourth-order convergence for solving nonlinear smooth equations by using a linear combination of divided differences to achieve a better approximation to the derivative. Zheng et al. [15] proposed a general family of Steffensen-type methods with optimal order of convergence by using Newton's iteration for the direct Newtonian interpolation. In [16], Petković et al. investigated a general way to construct multipoint methods for solving nonlinear equations by using inverse interpolation. In [17], Džunić et al. presented a new family of three-point derivative free methods by using a self-correcting parameter that is calculated applying the secant-type method in three different ways and Newton's interpolatory polynomial of the second degree.

The three new methods (for simple roots) in this paper use three functional evaluations and have fourth-order convergence; thus, they are optimal methods and their efficiency index is 𝐸=1.587, which is greater than the efficiency index of Newton's method, which is 𝐸=1.414. In the case of multiple roots, the method developed here is cubically convergent and uses three functional evaluations without the use of second derivative of the function. Thus, the method has better performance than Newton's modified method and the above methods with efficiency index 𝐸=1.4422.

2. Development of the Methods

In this paper, we consider iterative methods to find a simple root 𝑟∈𝐼 of a nonlinear equation 𝑓(𝑥)=0, where 𝑓∶𝐼→ℜ is a scalar function for an open interval 𝐼. We suppose that 𝑓(𝑥) is sufficiently differentiable and 𝑓^′(𝑥)≠0 for 𝑥∈𝐼, and since 𝑟 is a simple root, we can define 𝑔=𝑓⁻¹ on 𝐼. Taking 𝑥₀∈𝐼 closer to 𝑟 and supposing that 𝑥_𝑛 has been chosen, we define 𝑧_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝐾=𝑓𝑥_𝑛,𝐿=𝑓𝑧_𝑛. (11)

2.1. First Method FAM1

Consider the polynomial 𝑝₂𝑦=𝐴𝑦−𝐾𝑦−𝐿+𝐵𝑦−𝐾+𝐶, (12) with the conditions 𝑝₂(𝐾)=𝑔(𝐾)=𝑥_𝑛,𝑝₂(𝐿)=𝑔(𝐿)=𝑧_𝑛,𝑝^′₂(𝐾)=𝑔^′(𝐾)=1𝑓^′𝑥_𝑛. (13) Solving simultaneously the conditions (13) and using the common representation of divided differences for Hermite's inverse interpolation, 𝑔𝑓𝑥_𝑛=𝑥_𝑛,𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛=1/𝑓^′𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛−𝑧_𝑛−𝑥_𝑛/𝑓𝑧_𝑛−𝑓𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛,𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛=1𝑓^′𝑥_𝑛,𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛=𝑧_𝑛−𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛. (14) Consequently, we find 𝐴=𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛,𝐵=𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛,𝐶=𝑔𝑓𝑥_𝑛. (15) and the polynomial (12) can be written as 𝑝₂𝑦=𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑦−𝑓𝑥_𝑛×𝑦−𝑓𝑧_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛×𝑦−𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛. (16) If we are making 𝑦=0 in (16) we have a new iterative method (FAM1) 𝑥_𝑛+1=𝑔𝑓𝑥_𝑛−𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑥_𝑛𝑓𝑧_𝑛. (17) It can be written as 𝑥_𝑛+1=𝑥_𝑛+𝑓𝑥_𝑛𝑓^′𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛−𝑓²𝑧_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛²,𝑧_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (18) which uses three functional evaluations and has fourth-order convergence.

2.2. Second Method FAM2

Consider the polynomial 𝑝₃𝑦=𝐴𝑦−𝐾²𝑦−𝐿+𝐵𝑦−𝐾𝑦−𝐿+𝐶𝑦−K+𝐷, (19) with the conditions 𝑝₃(𝐾)=𝑔(𝐾)=𝑥_𝑛,𝑝₃(𝐿)=𝑔(𝐿)=𝑧_𝑛,𝑝^′₃(𝐾)=𝑔^′(𝐾)=1𝑓^′𝑥_𝑛. (20) Taking 𝐵=𝐴(𝐿−2𝐾), we have 𝑝₃(𝑦)=𝐴^∗𝑦³+𝐵^∗𝑦+𝐶^∗. Solving simultaneously the conditions (20) and using the common representation of divided differences for Hermite's inverse interpolation, we find 𝑔𝑓𝑥𝐴=^𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑧_𝑛−2𝑓𝑥_𝑛,𝐵=𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛,𝐶=𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛,𝐷=𝑔𝑓𝑥_𝑛, (21)(22) and the polynomial (19) can be written as 𝑝₃𝑦=𝑔𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑧_𝑛−2𝑓𝑥_𝑛×𝑦−𝑓𝑥_𝑛²𝑦−𝑓𝑧_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑦−𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛×𝑦−𝑓𝑥_𝑛𝑦−𝑓𝑧_𝑛. (23) Then, if we are making 𝑦=0 in (23) we have our second iterative method (FAM2) 𝑥_𝑛+1=𝑔𝑓𝑥_𝑛−𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑥_𝑛𝑓𝑧_𝑛−𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑧_𝑛−2𝑓𝑥_𝑛𝑓²𝑥_𝑛𝑓𝑧_𝑛. (24) It can be written as 𝑥_𝑛+1=𝑥_𝑛+𝑓²𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛−𝑓²𝑧_𝑛𝑓𝑥_𝑛𝑓𝑧_𝑛−3𝑓𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛²𝑓𝑧_𝑛−2𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (25) which uses three functional evaluations and has fourth-order convergence.

2.3. Third Method FAM3

Consider the polynomial 𝑝₃𝑦=𝐴𝑦−𝐾²𝑦−𝐿+𝐵𝑦−𝐾𝑦−𝐿+𝐶𝑦−𝐾+𝐷, (26) with the conditions 𝑝₃(𝐾)=𝑔(𝐾)=𝑥_𝑛,𝑝₃(𝐿)=𝑔(𝐿)=𝑧_𝑛,𝑝^′₃(𝐾)=𝑔^′(𝐾)=1𝑓^′𝑥_𝑛,𝑝₃^′′(𝐾)𝑧=−2𝑓_𝑛𝑓²𝑥_𝑛𝑓^′𝑥_𝑛, (27)(28) where we have used an approximation of 𝑓^′′(𝑥_𝑛) in [2], 𝑓^′′(𝑥_𝑛)≈2𝑓(𝑧_𝑛)𝑓^′2(𝑥_𝑛)/𝑓²(𝑥_𝑛). Solving simultaneously the conditions (27) and (28) and using the common representation of divided differences for Hermite's inverse interpolation, we have 𝐴=𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛𝑓𝑧_𝑛𝑓²𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛×𝑓𝑧_𝑛−𝑓𝑥_𝑛⁻¹,𝐵=𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛,𝐶=𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛,𝐷=𝑔𝑓𝑥_𝑛. (29)(30) Thus, the polynomial (26) can be written as 𝑝₃𝑦=𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛𝑓𝑧_𝑛/𝑓²𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛×𝑦−𝑓𝑥_𝑛²𝑦−𝑓𝑧_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛×𝑦−𝑓𝑥_𝑛𝑦−𝑓𝑧_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑦−𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛. (31) Making 𝑦=0 in (31), we have 𝑥_𝑛+1=𝑔𝑓𝑥_𝑛−𝑔𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑥_𝑛𝑓𝑧_𝑛−𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛𝑓𝑧_𝑛/𝑓²𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛+𝑔𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑧_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛×𝑓²𝑥_𝑛𝑓𝑧_𝑛. (32) It can be written as 𝑥_𝑛+1=𝑥_𝑛+𝑓𝑧_𝑛𝑓𝑥_𝑛−𝑓²𝑥_𝑛−𝑓²𝑧_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛²𝑓𝑥_𝑛𝑓^′𝑥_𝑛−𝑓³𝑧_𝑛𝑓𝑧_𝑛−2𝑓𝑥_𝑛𝑓𝑧_𝑛−𝑓𝑥_𝑛³𝑓²𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛, (33) which uses three functional evaluations and has fourth-order convergence.

2.4. Method FAM4 (Multiple Roots)

Consider the polynomial 𝑝_𝑚(𝑥)=𝐴𝑥−𝑥_𝑛+𝑤^𝑚, (34) where 𝑚 is the multiplicity of the root and 𝑝_𝑚(𝑥) verify the conditions 𝑝_𝑚𝑥_𝑛=𝑓𝑥_𝑛,𝑝_𝑚𝑧_𝑛=𝑓𝑧_𝑛, (35) with 𝑧_𝑛=𝑥_𝑛−𝑚(𝑓(𝑥_𝑛)/𝑓^′(𝑥_𝑛)).

Solving the system, we obtain 𝑢𝑤=^𝑚𝑧_𝑛−𝑥_𝑛1−𝑢_𝑚, (36) where 𝑢_𝑚=𝑓𝑥_𝑛𝑓𝑦_𝑛^1/𝑚. (37) Thus, we have 𝑥_𝑛+1=𝑥_𝑛−𝑤, (38) that can be written as 𝑥_𝑛+1=𝑧_𝑛+𝑚𝑓𝑥_𝑛𝑓^′𝑥_𝑛11−𝑢_𝑚,𝑧_𝑛=𝑥_𝑛𝑓𝑥−𝑚^𝑛𝑓^′𝑥_𝑛,𝑢_𝑚=𝑓𝑥_𝑛𝑓𝑦_𝑛^1/𝑚, (39) which uses three functional evaluations and has third-order convergence.

3. Analysis of Convergence

Theorem 1. Let 𝑓∶𝐼⊆ℜ→ℜ be a sufficiently differentiable function, and let 𝑟∈𝐼 be a simple zero of 𝑓(𝑥)=0 in an open interval 𝐼, with 𝑓^′(𝑥)≠0 on 𝐼. If 𝑥₀∈𝐼 is sufficiently close to 𝑟, then the methods FAM1, FAM2, and FAM3, as defined by (18), (25), and (33), have fourth-order convergence.

Proof. Following an analogous procedure to find the error in Lagrange's and Hermite's interpolations, the polynomials (12), (19), and (26) in FAM1, FAM2, and FAM3, respectively, have the error 𝐸𝑦=𝑔𝑦−𝑝_𝑛𝑦=𝑔^′′′(𝜉)3!−𝛽𝐴𝑦−𝐾²𝑦−𝐿, (40) for some 𝜉∈𝐼. 𝐴 is the coefficient in (15), (21), (29) that appears in the polynomial 𝑝_𝑛(𝑦) in (12), (19), and (26), respectively.
Then, substituting 𝑦=0 in 𝐸(𝑦)𝑟−𝑥_𝑛+1𝑔=−^′′′(𝜉)𝐾3!−𝛽𝐴²𝐿𝑔=−^′′′(𝜉)𝑓3!−𝛽𝐴²𝑥_𝑛𝑓𝑧_𝑛,𝑥_𝑛+1𝑔−𝑟=^′′′(𝜉)×𝑓(𝑟)3!−𝛽𝐴+𝑓^′𝜉₁𝑥_𝑛−𝑟²×𝑓(𝑟)+𝑓^′𝜉₂𝑧_𝑛−𝑟, (41) with 𝑥_𝑛+1−𝑟=𝜖_𝑛+1,𝑥_𝑛−𝑟=𝜖_𝑛. Since 𝑧_𝑛 was taken from Newton's method, we know that 𝑧_𝑛−𝑟=(𝑓^′′(𝜉₃)/2𝑓^′(𝜉₄))𝜉²_𝑛+𝑂(𝜉²_𝑛+1). Then, we have 𝜖_𝑛+1≈𝑔^′′′(𝜉)𝑓3!−𝛽𝐴^′2𝜉₁𝜉²_𝑛𝑓^′𝜉₂𝑓^′′𝜉₃𝜉²_𝑛2𝑓^′𝜉₄,𝜖_𝑛+1≈𝑔^′′′(𝜉)𝑓3!−𝛽𝐴^′2𝜉₁𝑓^′𝜉₂𝑓^′′𝜉₃2𝑓^′𝜉₄𝜉⁴_𝑛. (42) Now, in FAM1, we take 𝛽=0, then, 𝜉_𝑛+1≈𝑔^′′′(𝜉)𝑓3!^′2𝜉₁𝑓^′𝜉₂𝑓^′′𝜉₃2𝑓^′𝜉₄𝜉⁴_𝑛. (43) In FAM2 and FAM3, we take 𝛽=1, then, 𝜉_𝑛+1≈𝑔^′′′(𝜉)𝑓3!−𝐴^′2𝜉₁𝑓^′𝜉₂𝑓^′′𝜉₃2𝑓^′𝜉₄𝜉⁴_𝑛,𝜉_𝑛+1𝜉⁴_𝑛≈𝑔^′′′(𝜉)𝑓3!−𝐴^′2𝜉₁𝑓^′𝜉₂𝑓^′′𝜉₃2𝑓^′𝜉₄. (44) Thus, FAM1, FAM2, and FAM3 have fourth-order convergence.

Theorem 2. Let 𝑓∶𝐼⊆ℜ→ℜ be a sufficiently differentiable function, and let 𝑟∈𝐼 be a zero of 𝑓(𝑥)=0 with multiplicity 𝑚 in an open interval 𝐼. If 𝑥₀∈𝐼 is sufficiently close to 𝑟, then, the method FAM4 defined by (12), (20) is cubically convergent.

Proof. The proof is based on the error of Lagrange's interpolation. Suppose that 𝑥_𝑛 has been chosen. We can see that 𝑓(𝑥)−𝑝_𝑚(𝑥)=𝑓^′′𝜉₁−𝑝_𝑚^′′𝜉₁2𝑥−𝑥_𝑛𝑥−𝑧_𝑛, (45) for some 𝜉₁∈𝐼.
Taking 𝑥=𝑟 and expanding 𝑝_𝑚(𝑟) around 𝑥_𝑘+1, we have −𝑝^′_𝑚𝜉₂𝑟−𝑥_𝑛+1=𝑓^′′𝜉₁−𝑝_𝑚^′′𝜉₁2𝑟−𝑥_𝑛𝑟−𝑧_𝑛, (46) with 𝜉₁,𝜉₂∈𝐼.
Since 𝑧_𝑛=𝑥_𝑛−𝑚(𝑓(𝑥_𝑛)/𝑓^′(𝑥_𝑛)), we know that 𝑧_𝑛𝑓−𝑟=^′′𝜉₃−𝑝_𝑚^′′𝜉₃2𝑝^′_𝑚𝜉₄𝜖²_𝑛, (47) for some 𝜉₃,𝜉₄∈𝐼.
Thus, 𝜖_𝑛+1=𝑓^′′𝜉₁−𝑝_𝑚^′′𝜉₁2𝑝^′_𝑚𝜉₂𝑓^′′𝜉₃−𝑝_𝑚^′′𝜉₃2𝑝^′𝜉₄𝜖³_𝑛. (48) Therefore, FAM4 has third-order convergence.

Note that 𝑝_𝑚^′′ is not zero for 𝑚≥2 and this fact allows the convergence of lim_𝑛→∞(𝜖_𝑛+1/𝜖³_𝑛).

4. Numerical Analysis

In this section, we use numerical examples to compare the new methods introduced in this paper with Newton's classical method (NM) and recent methods of fourth-order convergence, such as Noor's method (NOM) with 𝐸=1.587 in [1], Cordero's method (CM) with 𝐸=1.587 in [2], Chun's third-order method (CHM) with 𝐸=1.442 in [3], and Li's fifth-order method (ZM) with 𝐸=1.379 in [4] in the case of simple roots. For multiple roots, we compare the method developed here with the quadratically convergent Newton's modified method (NMM) and with the cubically convergent Halley's method (HM), Osada's method (OM), Euler-Chebyshev's method (ECM), and Chun-Neta's method (CNM). Tables 2 and 4 show the number of iterations (IT) and the number of functional evaluations (NOFE). The results obtained show that the methods presented in this paper are more efficient.

All computations were done using MATLAB 2010. We accept an approximate solution rather than the exact root, depending on the precision (𝜖) of the computer. We use the following stopping criteria for computer programs: (i) |𝑥_𝑛+1−𝑥_𝑛|<𝜖, (ii) |𝑓(𝑥_𝑛+1)|<𝜖. Thus, when the stopping criterion is satisfied, 𝑥_𝑛+1 is taken as the exact computed root 𝑟. For numerical illustrations in this section, we used the fixed stopping criterion 𝜖=10⁻¹⁸.

We used the functions in Tables 1 and 3.

Table 1 

List of functions for a single root.

Table 2 

Comparison of the methods for a single root.

Table 3 

List of functions for a multiple root.

Table 4 

Comparison of the methods for a multiple root.

The computational results presented in Tables 2 and 4 show that, in almost all of cases, the presented methods converge more rapidly than Newton’s method, Newton’s modified method, and those previously presented for the case of simple and multiple roots. The new methods require less number of functional evaluations. This means that the new methods have better efficiency in computing process than Newton’s method as compared to other methods, and furthermore, the method FAM3 produces the best results. For most of the functions we tested, the obtained methods behave at least with equal performance compared to the other known methods of the same order.

5. Conclusions

In this paper, we introduce three new optimal fourth-order iterative methods to solve nonlinear equations. The analysis of convergence shows that the three new methods have fourth-order convergence; they use three functional evaluations, and thus, according to Kung-Traub's conjecture, they are optimal methods. In the case of multiple roots, the method developed here is cubically convergent and uses three functional evaluations without the use of second derivative. Numerical analysis shows that these methods have better performance as compared with Newton's classical method, Newton's modified method, and other recent methods of third- (multiple roots) and fourth-order (simple roots) convergence.

Acknowledgments

The authors wishes to acknowledge the valuable participation of Professor Nicole Mercier and Professor Joelle Ann Labrecque in the proofreading of this paper. This paper is the result of the research project “Análisis Numérico de Métodos Iterativos Óptimos para la Solución de Ecuaciones No Lineales” developed at the Universidad del Istmo, Campus Tehuantepec, by Researcher-Professor Gustavo Fernández-Torres.