All Title Author
Keywords Abstract

Publish in OALib Journal
ISSN: 2333-9721
APC: Only $99

ViewsDownloads

Relative Articles

More...

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

DOI: 10.1155/2012/346420

Full-Text   Cite this paper   Add to My Lib

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 . Thus, the family agrees with Kung-Traub conjecture for the case . 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 of the nonlinear equation , where 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 closer to the root and generates a sequence of successive iterates converging quadratically to the root. It is given by the following: 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 where is the Newton point and is a constant. This family requires two evaluations of the function and one evaluation of first derivative per iteration. The famous Ostrowski’s method [4, 5] is a member of this family for the case . 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

References

[1]  J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970.
[2]  S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, McGraw-Hill Book Company, New York, NY, USA, 1988.
[3]  R. F. King, “A family of fourth order methods for nonlinear equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 876–879, 1973.
[4]  A. M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, New York, NY, USA, 1960.
[5]  J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
[6]  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.
[7]  P. Jarratt, “Some efficient fourth order multipoint methods for solving equations,” BIT, vol. 9, pp. 119–124, 1969.
[8]  J. A. Ezquerro, M. A. Hernández, and M. A. Salanova, “Construction of iterative processes with high order of convergence,” International Journal of Computer Mathematics, vol. 69, no. 1-2, pp. 191–201, 1998.
[9]  J. M. Gutiérrez and M. A. Hernández, “An acceleration of Newton's method: super-Halley method,” Applied Mathematics and Computation, vol. 117, no. 2-3, pp. 223–239, 2001.
[10]  M. Grau and J. L. Díaz-Barrero, “An improvement to Ostrowski root-finding method,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 450–456, 2006.
[11]  J. R. Sharma and R. K. Guha, “A family of modified Ostrowski methods with accelerated sixth order convergence,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 111–115, 2007.
[12]  C. Chun and Y. Ham, “Some sixth-order variants of Ostrowski root-finding methods,” Applied Mathematics and Computation, vol. 193, no. 2, pp. 389–394, 2007.
[13]  J. Kou, “The improvements of modified Newton's method,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 602–609, 2007.
[14]  J. Kou and Y. Li, “An improvement of the Jarratt method,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1816–1821, 2007.
[15]  J. Kou, Y. Li, and X. Wang, “Some variants of Ostrowski's method with seventh-order convergence,” Journal of Computational and Applied Mathematics, vol. 209, no. 2, pp. 153–159, 2007.
[16]  C. Chun, “Some improvements of Jarratt's method with sixth-order convergence,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1432–1437, 2007.
[17]  S. K. Parhi and D. K. Gupta, “A sixth order method for nonlinear equations,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 50–55, 2008.
[18]  W. Bi, H. Ren, and Q. Wu, “New family of seventh-order methods for nonlinear equations,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 408–412, 2008.
[19]  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.
[20]  L. D. Petkovi?, M. S. Petkovi?, and J. D?uni?, “A class of three-point root-solvers of optimal order of convergence,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 671–676, 2010.
[21]  R. Thukral and M. S. Petkovi?, “A family of three-point methods of optimal order for solving nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 233, no. 9, pp. 2278–2284, 2010.
[22]  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.
[23]  A. Cordero, J. R. Torregrosa, and M. P. Vassileva, “Three-step iterative methods with optimal eighth-order convergence,” Journal of Computational and Applied Mathematics, vol. 235, no. 10, pp. 3189–3194, 2011.
[24]  S. K. Khattri and T. Log, “Derivative free algorithm for solving nonlinear equations,” Computing, vol. 92, no. 2, pp. 169–179, 2011.
[25]  S. K. Khattri and I. K. Argyros, “Sixth order derivative free family of iterative methods,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5500–5507, 2011.
[26]  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.
[27]  W. Gautschi, Numerical Analysis, Birkh?user, Boston, Mass, USA, 1997.
[28]  S. Weerakoon and T. G. I. Fernando, “A variant of Newton's method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87–93, 2000.

Full-Text

Contact Us

[email protected]

QQ:3279437679

WhatsApp +8615387084133