On some topological properties of numerical algorithms |
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Authors: | Martti Tienari |
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Institution: | (1) Computer Science Department, University of Helsinki, Töölönkatu 11, SF-00100 Helsinki 10, Finland |
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Abstract: | A result quantity in a numerical algorithm is considered as a function of the input data, roundoff and truncation errors. In order to investigate this functional relationship using the methods of mathematical analysis a structural model of the numerical algorithm calledR-automaton is introduced. It is shown that the functional dependence defined by anR-automaton is a continuous rational function in a neighborhood of any data point except in a point set, the Lebesgue measure of which is zero. An effective general-purpose algorithm is presented to compute the derivative of any result quantity with respect to the individual roundoff and truncation errors. Some ways of generalizing theR-automation model without losing the results achieved are finally suggested. |
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