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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 850365, 5 pages

http://dx.doi.org/10.1155/2013/850365

## Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

^{1}Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia^{2}Department de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, 08036 Barcelona, Spain

Received 10 April 2013; Accepted 27 July 2013

Academic Editor: Saeid Abbasbandy

Copyright © 2013 Malik Zaka Ullah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present an iterative method for solving nonlinear equations. The proposed iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture (Kung and Traub, 1974); it means that the iterative scheme uses five functional evaluations to achieve 16(=) order of convergence. The proposed iterative method utilizes one derivative and four function evaluations. Numerical experiments are made to demonstrate the convergence and validation of the iterative method.

#### 1. Introduction

According to Kung and Traub conjecture, a multipoint iterative method without memory could achieve optimal convergence order by performing evaluations of function or its derivatives [1]. In order to construct an optimal sixteenth-order convergent iterative method for solving nonlinear equations, we require four and eight optimal-order iterative schemes. Many authors have been developed the optimal eighth-order iterative methods, namely, Bi et al. [2], Bi et al. [3], Geum and Kim [4], Liu and Wang [5], Wang and Liu [6], and Soleymani et al. [7–9]. Some recent applications of nonlinear equation solvers in matrix inversion for regular or rectangular matrices have been introduced in [10–12].

For the proposed iterative method, we developed new optimal fourth- and eighth-orders iterative methods to construct optimal sixteenth-order iterative scheme. On the other hand, it is known that rational weight functions give a better convergence radius. By keeping this fact in mind, we introduced rational terms in weight functions to achieve optimal sixteenth order.

For the sake of completeness, we list some existing optimal sixteenth-order convergent methods. Babajee and Thukral [13] suggested 4-point sixteenth-order king family of iterative methods for solving nonlinear equations (BT): where In 2011, Geum and Kim [14] proposed a family of optimal sixteenth-order multipoint methods (GK2): where one of the choices for along with and : In the same year, Geum and Kim [15] presented a biparametric family of optimally convergent sixteenth-order multipoint methods (GK1): where one of the choices for along with and :

#### 2. A New Method and Convergence Analysis

The proposed sixteenth-order iterative method is described as follows (MA): where

Theorem 1. *Let be a sufficiently differentiable function, and is a simple root of , for an open interval . If is chosen sufficiently close to , then the iterative scheme (9) converges to and shows an order of convergence at least equal to sixteen.*

*Proof. *Let error at step be denoted by and and , . We provided Maple based computer assisted proof in Algorithm 1 and got the following error equation:

#### 3. Numerical Results

If the convergence order is defined as then the following expression approximates the computational order of convergence (COC) [16] as follows: where is the root of nonlinear equation. A set of five nonlinear equations are used for numerical computations in Table 1. Three iterations are performed to calculate the absolute error () and computational order of convergence. Table 2 shows absolute error and computational order of convergence, respectively.

#### 4. Conclusion

An optimal sixteenth-order iterative scheme has been developed for solving nonlinear equations. A Maple program is provided to calculate error equation, which actually shows the optimal order of convergence in the sense of Kung-Traub conjecture. The computational order of convergence also verifies our claimed order of convergence. The proposed scheme uses four functions and one derivative evaluation per full cycle, which gives 1.741 as the efficiency index. We also have shown the validity of our proposed iterative scheme by comparing it with other existing optimal sixteenth-order iterative methods. The numerical results show that the performance of iterative scheme is competitive as compared to other methods.

#### Acknowledgments

The first and the second authors were funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. They, therefore, acknowledge with thanks DSR technical and financial support while the third author is supported for this research under the Spanish MEC Grants AYA2010-15685.

#### References

- H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,”
*Journal of the Association for Computing Machinery*, vol. 21, pp. 643–651, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Bi, H. Ren, and Q. Wu, “Three-step iterative methods with eighth-order convergence for solving nonlinear equations,”
*Journal of Computational and Applied Mathematics*, vol. 225, no. 1, pp. 105–112, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Bi, Q. Wu, and H. Ren, “A new family of eighth-order iterative methods for solving nonlinear equations,”
*Applied Mathematics and Computation*, vol. 214, no. 1, pp. 236–245, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. H. Geum and Y. I. Kim, “A multi-parameter family of three-step eighth-order iterative methods locating a simple root,”
*Applied Mathematics and Computation*, vol. 215, no. 9, pp. 3375–3382, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Liu and X. Wang, “Eighth-order methods with high efficiency index for solving nonlinear equations,”
*Applied Mathematics and Computation*, vol. 215, no. 9, pp. 3449–3454, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Wang and L. Liu, “Modified Ostrowski's method with eighth-order convergence and high efficiency index,”
*Applied Mathematics Letters*, vol. 23, no. 5, pp. 549–554, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Soleymani, “On a new class of optimal eighth-order derivative-free methods,”
*Acta Universitatis Sapientiae. Mathematica*, vol. 3, no. 2, pp. 169–180, 2011. View at Zentralblatt MATH · View at MathSciNet - F. Soleymani, “Efficient optimal eighth-order derivative-free methods for nonlinear equations,”
*Japan Journal of Industrial and Applied Mathematics*, vol. 30, no. 2, pp. 287–306, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - F. Soleimani, F. Soleymani, and S. Shateyi, “Some iterative methods free from derivatives and their basins of attraction,”
*Discrete Dynamics in Nature and Society*, vol. 2013, Article ID 301718, 11 pages, 2013. View at Publisher · View at Google Scholar - F. Soleymani, “A new method for solving ill-conditioned linear systems,”
*Opuscula Mathematica*, vol. 33, no. 2, pp. 337–344, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - F. Soleymani, “A rapid numerical algorithm to compute matrix inversion,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2012, Article ID 134653, 11 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Soleymani, “On a fast iterative method for approximate inverse of matrices,”
*Communications of the Korean Mathematical Society*, vol. 28, pp. 407–418, 2013. - D. K. R. Babajee and R. Thukral, “On a 4-point sixteenth-order King family of iterative methods for solving nonlinear equations,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2012, Article ID 979245, 13 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Y. H. Geum and Y. I. Kim, “A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function,”
*Computers & Mathematics with Applications*, vol. 61, no. 11, pp. 3278–3287, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. H. Geum and Y. I. Kim, “A biparametric family of optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic two-variable function,”
*Journal of Computational and Applied Mathematics*, vol. 235, no. 10, pp. 3178–3188, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Chun, “Construction of Newton-like iteration methods for solving nonlinear equations,”
*Numerische Mathematik*, vol. 104, no. 3, pp. 297–315, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet