Dynamics of a new family of iterative processes for quadratic polynomials

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:233

Article

Gutierrez, J. M.¹  Hernandez, M. A.¹  Romero, N.¹ 

[1] Univ La Rioja, Dept Math & Computat, Logrono 26004, Spain

关键词: General convergence;    Iterative processes;    Julia sets;    Order of convergence;    Quadratic equation;   

DOI  :  10.1016/j.cam.2009.11.017

来源: Elsevier

【 摘 要 】

In this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter m is an element of N boolean OR {0}. These methods reach the order of convergence m + 2 when they are applied to quadratic polynomials with different roots. Newton's and Chebyshev's methods appear as particular choices of the family appear for m = 0 and m = 1, respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly. we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods. (C) 2009 Elsevier B.V. All rights reserved.

【 授权许可】

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