A Family of Iterative Methods with Accelerated Eighth-Order Convergence

Abstract

We propose a family of eighth-order iterative methods without memory for solving nonlinear equations. The new iterative methods are developed by using weight function method and using an approximation for the last derivative, which reduces the required number of functional evaluations per step. Their efficiency indices are all found to be 1.682. Several examples allow us to compare our algorithms with known ones and confirm the theoretical results.

1. Introduction

Finding the solution of nonlinear equations is one of the most important problems in numerical analysis. There are some texts that have become classic, as the one of Traub (see [1]) and Neta (see [2]) which include a vast collection of methods and their efficiency, or the paper by Neta and Johnson, [3]), Jarratt (see [4]), or Homeier (see [5]), among others.

As the order of an iterative method increases, so does the number of functional evaluations per step. The efficiency index (see [6]) gives a measure of the balance between those quantities, according to the formula 𝐼=𝑝^1/𝑑, where 𝑝 is the order of convergence of the method and 𝑑 is the number of functional evaluations per step. Kung and Traub conjectured in [7] that the order of convergence of any multipoint method without memory cannot exceed the bound 2^𝑑−1 (called the optimal order). Thus, the optimal order for a method with 3 functional evaluations per step would be 4. King's method [8], Chun's schemes (see [9, 10]), Chun et al. [11], Maheshwari's procedure (see [12]), and Jarratt's method [4] are some of optimal fourth-order methods, because they only perform three functional evaluations per step.

More recently, optimal eighth-order iterative methods have been investigated by many researchers (see, e.g., [13] where the authors show optimal methods of order eight and sixteen). Bi et al. in [14] developed a new family of eighth-order iterative methods for solving nonlinear equations, which is denoted by BRW8 and whose iterative expression is 𝑦_𝑚=𝑥_𝑚−𝑓𝑥^𝑚𝑧_𝑚𝑓^𝑥_𝑚,=𝑦_𝑚−𝑓𝑥^𝑚−(1/2)𝑓𝑦_𝑚𝑓𝑥_𝑚−(5/2)𝑓𝑦_𝑚𝑓𝑦^𝑚𝑥_𝑚+1𝑓^𝑥_𝑚,=𝑧_𝑚−𝑓𝑥^𝑚+(𝜃+2)𝑓𝑧_𝑚𝑓𝑥_𝑚+𝜃𝑓𝑧_𝑚𝑓𝑧^𝑚𝑓𝑧_𝑚,𝑦_𝑚+𝑓𝑧_𝑚,𝑥_𝑚,𝑥_𝑚𝑧_𝑚−𝑦_𝑚,(1.1) where 𝜃∈ℝ.

Also Cordero et al. in [15] developed a parametric family of eighth-order methods based on Ostrowski's scheme, that we will denote by M8. It can be expressed as 𝑦_𝑚=𝑥_𝑚−𝑓𝑥^𝑚𝑧_𝑚𝑓^𝑥_𝑚,=𝑥_𝑚−𝑓𝑥^𝑚𝑓^𝑥_𝑚𝑓𝑥^𝑚−𝑓𝑦_𝑚𝑢_𝑚𝑓𝑥_𝑚−2𝑓𝑦_𝑚,=𝑧_𝑚−𝑓𝑧^𝑚𝑓^𝑥_𝑚𝑓𝑥^𝑚−𝑓𝑦_𝑚𝑓𝑥_𝑚−2𝑓𝑦_𝑚+21𝑓𝑧^𝑚𝑓𝑦_𝑚−2𝑓𝑧_𝑚²𝑥_𝑚+1,=𝑢_𝑚−𝑓𝑧^𝑚𝑓^𝑥_𝑚3𝛽²+𝛽₃𝑢_𝑚−𝑧_𝑚𝛽₁𝑢_𝑚−𝑧_𝑚+𝛽₂𝑦_𝑚−𝑥_𝑚+𝛽₃𝑧_𝑚−𝑥_𝑚,(1.2) where 𝛽₂+𝛽₃≠0. Other family of variants of Ostrowski's method with eighth-order convergence was developed by Liu and Wang in [16]. We will denote it by LW8 and its iterative expression is 𝑦_𝑚=𝑥_𝑚−𝑓𝑥^𝑚𝑧_𝑚𝑓^𝑥_𝑚,=𝑥_𝑚−𝑓𝑥^𝑚𝑓𝑥_𝑚−𝑓𝑦_𝑚𝑥𝑓^𝑥_𝑚𝑓𝑥_𝑚−2𝑓𝑦_𝑚_𝑚+1,=𝑧_𝑚−𝑓𝑧^𝑚𝑓^𝑥_𝑚⎡⎢⎢⎣𝑓𝑥^𝑚−𝑓𝑦_𝑚𝑓𝑥_𝑚−2𝑓𝑦_𝑚²+𝑓𝑧^𝑚𝑓𝑥_𝑚−𝛼𝑓𝑦_𝑚+𝐺𝑓𝑧^𝑚𝑓𝑥_𝑚⎤⎥⎥⎦,(1.3) where 𝛼 is constant and 𝐺 denotes a real-valued function.

Let us note that all these methods are optimal in the sense of Kung-Traub's conjecture (see [7]) for methods without memory; that is, they reach eighth-order convergence with only four functional evaluations.

It is usual to design high-order methods from Ostrowski-type schemes, as they are optimal and use few operations per step, trying to obtain procedures as efficient as possible by using different techniques (see, for instance [13, 15, 16]). If we compose Newton and Ostrowski's methods, and estimate the last derivative by divided differences, it is necessary to use divided differences of second order to reach eighth order of convergence (see [14]). Nevertheless, as we will see in the next section, it is possible to obtain an optimal eighth-order scheme by composing King and Newton's methods by using only divided differences of first order.

In this paper, we design a family of eighth-order iterative methods to find a simple root 𝑥^∗ of the nonlinear equation 𝑓(𝑥)=0, where 𝑓∶𝐷→ℝ is a smooth function, and 𝐷 is an open interval. We present in Section 2 a family of eighth-order iterative methods, based on Chun's method. In Section 3, different numerical examples confirm the theoretical results and allow us to compare the new methods with other known methods mentioned in the introduction. Finally, some conclusions are presented in Section 4.

2. Development of Eighth-Order Algorithm

Let us consider the family of optimal methods proposed by Chun in [9], that is a variant of King's family, 𝑦_𝑚=𝑥_𝑚−𝑓𝑥^𝑚𝑧_𝑚𝑓^𝑥_𝑚,=𝑦_𝑚−𝑓²𝑥_𝑚𝑓²𝑥_𝑚−2𝑓𝑥_𝑚𝑓𝑦_𝑚+2𝛽𝑓²𝑦_𝑚𝑓𝑦^𝑚𝑓^𝑥_𝑚,(2.1) where 𝛽∈ℝ. We consider now a three-step iterative scheme composed by Chun's scheme (2.1) and a third step designed by using Newton's method and weight functions. 𝑥_𝑚+1=𝑧_𝑚−𝐻𝜇_𝑚𝑓𝑧^𝑚𝑓^𝑧_𝑚,(2.2) where 𝜇_𝑚=𝑓(𝑧_𝑚)/𝑓(𝑥_𝑚) and 𝐻(𝑡) represents a real-valued function.

However, this procedure is not optimal, as it uses two new functional evaluations in the last step. So, we express 𝑓^(𝑧_𝑚) as a linear combination of 𝑓[𝑦_𝑚,𝑥_𝑚], 𝑓[𝑧_𝑚,𝑦_𝑚], and 𝑓[𝑧_𝑚,𝑥_𝑚]. 𝑓^𝑧_𝑚=𝜃₁𝑓𝑦_𝑚,𝑥_𝑚+𝜃₂𝑓𝑧_𝑚,𝑦_𝑚+1−𝜃₁−𝜃₂𝑓𝑧_𝑚,𝑥_𝑚,(2.3) where 𝜃₁ and 𝜃₂ are real numbers.

By using (2.3), we propose the following iterative scheme: 𝑦_𝑚=𝑥_𝑚−𝑓𝑥^𝑚𝑧_𝑚𝑓^𝑥_𝑚,=𝑦_𝑚−𝑓²𝑥_𝑚𝑓²𝑥_𝑚−2𝑓𝑥_𝑚𝑓𝑦_𝑚+2𝛽𝑓²𝑦_𝑚𝑓𝑦^𝑚𝑥_𝑚+1𝑓^𝑥_𝑚,=𝑧_𝑚−𝐻𝜇_𝑚𝑓𝑧^𝑚𝜃₁𝑓𝑦_𝑚,𝑥_𝑚+𝜃₂𝑓𝑧_𝑚,𝑦_𝑚+1−𝜃₁−𝜃₂𝑓𝑧_𝑚,𝑥_𝑚,(2.4) whose convergence analysis will be made in the following result.

Theorem 2.1. Let 𝑥^∗ be a simple zero of a sufficiently differentiable function 𝑓∶𝐷⊆ℝ→ℝ. If the initial point 𝑥₀ is sufficiently close to 𝑥^∗, then the sequence {𝑥_𝑚}_𝑚≥0 generated by any method of the family (2.4) converges to 𝑥^∗. If 𝐻(𝑡) is any function with 𝐻(0)=1, 𝐻′(0)=1, 𝐻^(0)<∞, and 𝛽=1/2, then the convergence order of any method of the family (2.4) is eight if and only if 𝜃₁=−1, 𝜃₂=1.

Proof. Let 𝑒_𝑚=𝑥_𝑚−𝑥^∗ be the error at the mth iteration and 𝑐_𝑚=(1/𝑚!)(𝑓^(𝑚)(𝑥^∗)/𝑓^(𝑥^∗)),𝑚=2,3,…. By using Taylor expansions, we have 𝑓𝑓𝑥_𝑚=𝑓^𝑥^∗𝑒_𝑚+𝑐₂𝑒^2𝑚+𝑐₃𝑒^3𝑚+𝑐₄𝑒^4𝑚+𝑐₅𝑒^5𝑚+𝑐₆𝑒^6𝑚+𝑐₇𝑒^7𝑚+𝑐₈𝑒^8𝑚+𝑂𝑒^9𝑚_,𝑥_𝑚=𝑓^𝑥^∗1+2𝑐₂𝑒_𝑚+3𝑐₃𝑒^2𝑚+4𝑐₄𝑒^3𝑚+5𝑐₅𝑒^4𝑚+6𝑐₆𝑒^5𝑚+7𝑐₇𝑒^6𝑚+8𝑐₈𝑒^7𝑚+𝑂𝑒^8𝑚.(2.5) Now, from (2.5), we have 𝑦_𝑚=𝑥^∗+𝑐₂𝑒^2𝑚+2𝑐₃−2𝑐²²𝑒^3𝑚+3𝑐₄−3𝑐₂𝑐₃−22𝑐₃−2𝑐²²𝑐₂𝑒^4𝑚+4𝑐₅−10𝑐₂𝑐₄−6𝑐²³+20𝑐₃𝑐²²−8𝑐⁴²𝑒^5𝑚+−17𝑐₄𝑐₃+28𝑐₄𝑐²²−13𝑐₂𝑐₅+33𝑐₂𝑐²³+5𝑐₆−52𝑐₃𝑐³²+16𝑐⁵²𝑒^6𝑚+𝑂𝑒^7𝑚,(2.6) and then, we get 𝑓𝑦_𝑚=𝑓^𝑥^∗𝑐₂𝑒²+2𝑐₃−2𝑐²²𝑒^3𝑚+3𝑐₄−7𝑐₂𝑐₃+5𝑐³²𝑒^4𝑚+−6𝑐²³+24𝑐₃𝑐²²−10𝑐₂𝑐₄+4𝑐₅−12𝑐⁴²𝑒⁵+−17𝑐₄𝑐₃+34𝑐₄𝑐²²−13𝑐₂𝑐₅+5𝑐₆+37𝑐₂𝑐²³−73𝑐₃𝑐³²+28𝑐⁵²𝑒^6𝑚+𝑂𝑒^7𝑚.(2.7) Combining (2.5), (2.6) and (2.7), we obtain 𝑧_𝑚=𝑥^∗+−𝑐₂𝑐₃+𝑐³²+2𝛽𝑐³²𝑒^4𝑚−2(2+6𝛽)𝑐⁴²−2(2+3𝛽)𝑐²²𝑐₃+𝑐²³+𝑐₂𝑐₄𝑒^5𝑚+10+44𝛽−4𝛽²𝑐⁵²−6(5+14𝛽)𝑐³²𝑐₃+6(2+3𝛽)𝑐²²𝑐₄−7𝑐₃𝑐₄+3𝑐₂(6+8𝛽)𝑐²³−𝑐₅𝑒^6𝑚+𝑂𝑒^7𝑚.(2.8) So, from (2.8), we get 𝑓𝑧_𝑚=𝑓^𝑥^∗(1+2𝛽)𝑐³²−𝑐₂𝑐₃𝑒^4𝑚−2(2+6𝛽)𝑐⁴²−2(2+3𝛽)𝑐²²𝑐₃+𝑐²³+𝑐₂𝑐₄𝑒^5𝑚+10+44𝛽−4𝛽²𝑐⁵²−6(5+14𝛽)𝑐³²𝑐₃+6(2+3𝛽)𝑐²²𝑐₄−7𝑐₃𝑐₄+3𝑐₂(6+8𝛽)𝑐²³−𝑐₅𝑒^6𝑚+𝑂𝑒^7𝑚.(2.9) Using the Taylor expansion of 𝐻 around 0 and considering 𝐻^(0)<∞, we get 𝐻𝜇_𝑚=𝐻(0)+𝐻^(0)𝑓𝑧^𝑚𝑓𝑥_𝑚+𝑂𝜇^𝑚=𝐻(0)+𝐻_²(0)𝑐₂−𝑐₃+𝑐²²+2𝛽𝑐²²𝑒^3𝑚+𝐻^(0)−2𝑐₂𝑐₄+9𝑐₃𝑐²²−5𝑐⁴²+12𝛽𝑐₃𝑐²²−14𝛽𝑐⁴⁴−2𝑐²³𝑒^4𝑚+𝑂𝑒^5𝑚.(2.10) In these terms, the error equation of the method can be expressed as 𝑒_𝑚+1=−(−1+𝐻(0))𝑐₂(1+2𝛽)𝑐²²−𝑐₃𝑒^4𝑚+−4+12𝛽+𝐻(0)−5+2𝛽−7+𝜃₂+𝜃₂𝑐⁴²+8+12𝛽+𝐻(0)−9−12𝛽+𝜃₂𝑐²²𝑐₃+2(−1+𝐻(0))𝑐²³+2(−1+𝐻(0))𝑐₂𝑐₄𝑒^5𝑚+𝑂𝑒^6𝑚,(2.11) which shows that the convergence order of any method of the family (2.4) is at least five if 𝐻(0)=1. Then, 𝑒_𝑚+1=−−1+𝜃₂𝑐²²(1+2𝛽)𝑐²²−𝑐₃𝑒^5𝑚+𝑐₂−5+𝜃₁+7𝜃₂−𝜃²²+2𝛽−7+𝜃₁+9𝜃₂−𝜃²²𝑐⁴²+10−𝜃₁−14𝛽−1+𝜃₂−12𝜃₂+𝜃²²𝑐²²𝑐₃+3−1+𝜃₂𝑐²³+2−1+𝜃₂𝑐₂𝑐₄𝑒^6𝑚+𝑂𝑒^7𝑚.(2.12) and it is necessary that 𝜃₁=−1 and 𝜃₂=1 in order to reach order of convergence seven. Then, the error equation is 𝑒_𝑚+1=−𝑐₂𝐻^(0)+2𝛽𝐻^(0)𝑐²²−−1+𝐻^(0)𝑐₃(1+2𝛽)𝑐³²−𝑐₂𝑐₃𝑒^7𝑚+𝑂𝑒^8𝑚,(2.13) and finally, if 𝛽=−1/2 and 𝐻′(0)=1, it can be concluded that the order is eight and 𝑒_𝑚+1=𝑐₃𝑐²²2𝑐₂𝑐₃−𝑐₄+2𝑐³²𝑒^8𝑚+𝑂𝑒^9𝑚.(2.14)

Remark 2.2. Any method of the family (2.4) has the efficiency index equals to 8^1/4≈1.682, which is better than the Newton's method with efficiency index equals to 2^1/2≈1.414and equal to BRW8, M8, and LW8.

In what follows, we give some concrete optimal iterative methods of family (2.4) for different functions 𝐻. (F1)𝐻(𝑡)=(1+𝛽𝑡)^𝛾,𝛽⋅𝛾=1, where 𝛽,𝛾∈ℝ. Hence we get a new eighth-order method whose last step is 𝑥_𝑚+1=𝑧_𝑚−1+𝛽𝜇_𝑚^𝛾𝑓𝑧_𝑚−𝑓𝑦_𝑚,𝑥_𝑚+𝑓𝑧_𝑚,𝑦_𝑚+𝑓𝑧_𝑚,𝑥_𝑚.(2.15)(F2)𝐻(𝑡)=1+(𝑡/(1+𝜆𝑡)), where 𝜆∈ℝ. So, 𝑥_𝑚+1=𝑧_𝑚−1+𝜇^𝑚1+𝜆𝜇_𝑚𝑓𝑧^𝑚−𝑓𝑦_𝑚,𝑥_𝑚+𝑓𝑧_𝑚,𝑦_𝑚+𝑓𝑧_𝑚,𝑥_𝑚.(2.16)(F3)𝐻(𝑡)=1/(1−𝑡+𝜔𝑡²), where 𝜔∈ℝ. Then we obtain a new scheme 𝑥_𝑚+1=𝑧_𝑚−11−𝜇_𝑚+𝜔𝜇^2𝑚𝑓𝑧^𝑚−𝑓𝑦_𝑚,𝑥_𝑚+𝑓𝑧_𝑚,𝑦_𝑚+𝑓𝑧_𝑚,𝑥_𝑚.(2.17)

3. Numerical Results

We present some examples to illustrate the efficiency of the iterative algorithm. All computations were done using MAPLE. We have used as stopping criteria that |𝑥_𝑚+1−𝑥_𝑚|≤10⁻²⁰⁰ or |𝑓(𝑥_𝑚)|≤10⁻²⁰⁰. The test functions are listed below: (a)𝑓₁(𝑥)=𝑥³+4𝑥²−15;𝑥^∗≈1.6319808055661; (b)𝑓₂(𝑥)=𝑥𝑒^𝑥2−sin²(𝑥)+3cos(𝑥)+5;𝑥^∗≈−1.2076478271309; (c)𝑓₃(𝑥)=sin(𝑥)−(𝑥/2);𝑥^∗≈1.8954942670339; (d)𝑓₄(𝑥)=10𝑥𝑒^−𝑥2−1;𝑥^∗≈1.6796306104285; (e)𝑓₅(𝑥)=cos(𝑥)−𝑥;𝑥^∗≈0.73908513321516; (f)𝑓₆(𝑥)=sin²(𝑥)−𝑥²+1;𝑥^∗≈1.4044916482153. (g)𝑓₇(𝑥)=𝑒^−𝑥+cos(𝑥);𝑥^∗≈1.7461395304080.

We compare the classical Newtons (CN), BRW8 method with 𝜃=1; M8 scheme with 𝛽₁=0, 𝛽₃=0, and 𝛽₂=1; LW8 method with 𝛼=1 and 𝐺(𝑡)=4𝑡, and our methods with 𝜔=1, 𝜆=1, 𝛽=1, and 𝛾=1.

In Table 1, the following elements appear for each test function and each iterative method: the value of the elements involved in the stopping criterium, |𝑥_𝑚+1−𝑥_𝑚| and |𝑓(𝑥_𝑚)|, the number of iterations, iter, needed to converge to the solution, and the approximated computational order of convergence 𝜌, that can be calculated by using the formula (see [17]) 𝜌=ln||𝑥^𝑚+1−𝑥_𝑚||/||𝑥_𝑚−𝑥_𝑚−1||ln||𝑥_𝑚−𝑥_𝑚−1||/||𝑥_𝑚−1−𝑥_𝑚−2||,(3.1) where 𝑥_𝑚+1,𝑥_𝑚,𝑥_𝑚−1, and 𝑥_𝑚−2 are iterations close to a zero of the nonlinear equation.

Table 1 

Comparison of various iterative methods.

4. Conclusions

In this work, we have constructed a new general eighth-order iterative family of methods without memory for solving nonlinear equations. Convergence analysis shows that the order of convergence of the methods is eight. Per iteration the present methods require three evaluations of the function and one evaluation of its first derivative and therefore have the efficiency index equal to 8^1/4=1.682. Some of the obtained methods were also compared in their performance and efficiency to various other iteration methods of the same order, and it was observed that they demonstrate at least equal behavior.

Acknowledgments

The authors would like to thank the referee for the valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02 and by Vicerrectorado de Investigación, Universitat Politècnica de València PAID-06-2010-2285.