A Study of Iteration Formulas for Root Finding, Where Mathematics, Computer Algebra and Software Engineering Meet
For various reasons, including speed code simplicity and symbolic approximation, it is still very interesting to analyze simple iteration formulas for root finding. The classical analysis of iteration formulas concentrates on their convergence near a root. We find experimentally, that this information is almost useless. The (apparently) random walk followed by iteration formulas before reaching convergence is the dominating factor in their performance. We study a set of 29 iteration formulas from a theoretical and a practical point of view. We define a new property of the formulas, their far-convergence, in an effort to explain their behaviours. Extensive experimentation finding polynomial roots, shows that there are extreme differences in performance of seemingly similar iterators. This is a surprising result. We use this experimental approach to select the most effective performer, which is La-guerre’s method. The best companion (second method) to handle the failures of Laguerre’s is a new method which is an adaptation of Halley’s method to multipoint computation. The little-known Ostrowski’s method comes out with one of the best performances. We also find that an unknown simple variant of Newton’s method behaves much better than Newton’s method itself, which behaves very poorly. This shows that sometimes it pays to modify a method to improve its far-convergence. Various performance curiosities cannot be explained in terms of neither order of convergence and are probably caused by the paths that the methods force on the iteration values. The study of these random paths is an open problem, probably beyond our present tools.
KeywordsPractical Study Quadratic Interpolation Iteration Formula Root Finding Symbolic Approximation
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- J. F. Traub. Iterative Methods for Solution of Equations. Prentice-Hall, 1964.Google Scholar