An Optimal Class of Eighth-Order Iterative Methods Based on King’s Method

This paper based on King’s fourth order methods. A class of eighth-order methods is presented for solving simple roots of nonlinear equations. The class is developed by combining King’s fourth-order  method and Newton’s method as a third step using the forward divided difference and multiplication of  three weight function. Some numerical comparisons have been considered to show the performance of the proposed method

A biparametric family of four-step sixteenth-order root-finding methods with the optimal efficiency index

AbstractA biparametric family of four-step multipoint iterative methods of order sixteen to solve nonlinear equations are developed and their convergence properties are established. The optimal efficiency indices are all found to be 161/5≈1.741101. Numerical examples as well as comparison with existing methods are demonstrated to verify the developed theory

Three-step iterative methods with eighth-order convergence for solving nonlinear equations

A family of eighth-order iterative methods for solution of nonlinear equations is presented. We propose an optimal three-step method with eight-order convergence for finding the simple roots of nonlinear equations by Hermite interpolation method. Per iteration of this method requires two evaluations of the function and two evaluations of its first derivative, which implies that the efficiency index of the developed methods is 1.682. Some numerical examples illustrate that the algorithms are more efficient and performs better than the other methods

A Family of Three-Point Methods of Ostrowski's Type for Solving Nonlinear Equations

A class of three-point methods for solving nonlinear equations of eighth order is constructed. These methods are developed by combining two-point Ostrowski's fourth-order methods and a modified Newton's method in the third step, obtained by a suitable approximation of the first derivative using the product of three weight functions. The proposed three-step methods have order eight costing only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Two numerical examples for various weight functions are given to demonstrate very fast convergence and high computational efficiency of the proposed multipoint methods

On developing a higher-order family of double-Newton methods with a bivariate weighting function

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2014.12.130A high-order family of two-point methods costing two derivatives and two functions are developed by introducing a two-variable weighting function in the second step of the classical double-Newton method. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A variety of concrete numerical examples and relevant results are extensively treated to verify the underlying theoretical development. In addition, this paper investigates the dynamics of rational iterative maps associated with the proposed method and an existing method based on illustrated description of basins of attraction for various polynomials

New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented

Determination of multiple roots of nonlinear equations and applications

The final publication is available at Springer via https://dx.doi.org/10.1007/s10910-014-0460-8[EN] In this work we focus on the problem of approximating multiple roots of nonlinear equations. Multiple roots appear in some applications such as the compression of band-limited signals and the multipactor effect in electronic devices. We present a new family of iterative methods for multiple roots whose multiplicity is known. The methods are optimal in Kung-Traub's sense (Kung and Traub in J Assoc Comput Mach 21:643-651, [1]), because only three functional values per iteration are computed. By adding just one more function evaluation we make this family derivative free while preserving the convergence order. To check the theoretical results, we codify the new algorithms and apply them to different numerical examples.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-SP-2012-0474.Hueso Pagoaga, JL.; Martínez Molada, E.; Teruel Ferragud, C. (2015). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry. 53(3):880-892. https://doi.org/10.1007/s10910-014-0460-8S880892533H.T. Kung, J.F. Traub, Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)W. Bi, H. Ren, Q. Wu, Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 255, 105–112 (2009)W. Bi, Q. Wu, H. Ren, A new family of eighth-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 214, 236–245 (2009)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, New modifications of Potra-Pták’s method with optimal fourth and eighth order of convergence. J. Comput. Appl. Math. 234, 2969–2976 (2010)E. Schröder, Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)C. Chun, B. Neta, A third-order modification of Newtons method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)Y.I. Kim, S.D. Lee, A third-order variant of NewtonSecant method finding a multiple zero. J. Chungcheong Math. Soc. 23(4), 845–852 (2010)B. Neta, Extension of Murakamis high-order nonlinear solver to multiple roots. Int. J. Comput. Math. 8, 1023–1031 (2010)H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence. Appl. Math. Comput. 209, 206–210 (2009)Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants. Appl. Math. Comput. 214, 10–16 (2009)S. Amat, S. Busquier, On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177, 819–823 (2006)X. Feng, Y. He, High order iterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. doi: 10.1016/j.amc.2011.02.067F. Marvasti, A. Jain, Zero crossings, bandwidth compression, and restoration of nonlinearly distorted band-limited signals. J. Opt. Soc. Am. A 3, 651–654 (1986)S. Anza, C. Vicente, B. Gimeno, V.E. Boria, J. Armendáriz, Long-term multipactor discharge in multicarrier systems. Physics of Plasmas 14(8), 082–112 (2007)J.L. Hueso, E. Martínez, C. Teruel, New families of iterative methods with fourth and sixth order of convergence and their dynamics, in Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013, 24–27 June 2013A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. doi: 10.10016/j.cam.2014.01.024 (2014)J.R. Sharma, R. Sharma, Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010