Basins of attraction for several optimal fourth order methods for multiple roots
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Authors
Neta, Beny
Chun, Changbum
Subjects
Iterative methods
Order of convergence
Rational maps
Basin of attraction
Julia sets
Conjugacy classes
Order of convergence
Rational maps
Basin of attraction
Julia sets
Conjugacy classes
Advisors
Date of Issue
2014
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Abstract
There are very few optimal fourth order methods for solving nonlinear algebraic equations having roots of multiplicity m. Here we compare 5 such methods, two of which require the evaluation of the (m--1)st root. The methods are usually compared by evaluating the computational e ciency and the e ciency index. In this paper all the methods have the same e ciency, since they are of the same order and use the same information. Frequently, comparisons of the various schemes are based on the number of iterations required for convergence, number of function evaluations, and/or amount of CPU time. If a particular algorithm does not converge or if it converges to a di erent solution, then that particular algorithm is thought to be inferior to the others. The primary aw in this type of comparison is that the starting point represents only one of an in nite number of other choices. Here we use the basin of attraction idea to recommend the best fourth order method. The basin of attraction is a method to visually comprehend how an algorithm behaves as a function of the various starting points.
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Article
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Applied Mathematics
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.