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Keywords
(8)
Catalan Number
Dynamic Behavior
Iteration Method
Julia Set
Nonlinear Equation
Order of Convergence
Rational Function
Rational Map
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Dynamics of a new family of iterative processes for quadratic polynomials
Dynamics of a new family of iterative processes for quadratic polynomials,10.1016/j.cam.2009.11.017,Journal of Computational and Applied Mathematics,J
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Dynamics of a new family of iterative processes for quadratic polynomials
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Citations: 2
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J. M. Gutiérrez
,
M. A. Hernández
,
N. Romero
In this work we show the presence of the wellknown 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∈N∪{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.
Journal:
Journal of Computational and Applied Mathematics  J COMPUT APPL MATH
, vol. 233, no. 10, pp. 26882695, 2010
DOI:
10.1016/j.cam.2009.11.017
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(
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,
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,
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Journal of Mathematical Analysis and Applications  J MATH ANAL APPL
, vol. 341, no. 1, pp. 5261, 2008
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Citations
(2)
Study of the dynamics of thirdorder iterative methods on quadratic polynomials
(
Citations: 1
)
Alicia Cordero
,
Juan R. Torregrosa
,
Pura Vindel
Journal:
International Journal of Computer Mathematics  IJCM
, vol. aheadofp, no. aheadofp, pp. 111, 2012
Dynamics of a higherorder family of iterative methods
Gerardo Honorato
,
Sergio Plaza
,
Natalia Romero
Journal:
Journal of Complexity
, vol. 27, no. 2, pp. 221229, 2011