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ABSTRACT
Over several decades, the computer has performed a significant number of arithmetic operations in the domain of sciences and techniques as well as in many of our daily activities. Despite the precious help provided, approximations are still an issue. Indeed, any numerical value can only be represented in the computer with a finite number of figures and must therefore be rounded without even taking into account the uncertainties brought about by measurement devices. This article presents computer arithmetics and its consequences for scientific calculation and then focuses on assessment methods for the propagation limits of rounding errors.
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Read the articleAUTHORS
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Jean VIGNES: Professor Emeritus, Pierre et Marie Curie University
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René ALT: Professor Emeritus, Pierre et Marie Curie University
INTRODUCTION
Today, computers are used in almost every scientific and technical discipline, as well as in many of our daily activities. However, we mustn't forget that the primary purpose of these machines was to be able to perform numerical calculations automatically. They are the successors of the abacus and mechanical, then electrical, calculating machines, and in this respect are the result of the combination of electronics and ancient, well-known calculation techniques. Thanks to the speed of execution provided by electronics, the very first computers were already able to carry out a large number of arithmetic operations in a reasonable time.
But on a computer, any numerical value can only be represented with a finite number of digits. As a result, any data or result provided by arithmetic operations must be rounded, i.e. replaced by a close value that can be represented exactly. Thus, at the level of each arithmetic operation, a rounding error is generated, which, although very small, will propagate throughout the calculations, affecting all results.
What's more, the data used in the calculation program often comes from measuring devices (sensors) and is therefore subject to uncertainties due to these devices. It is also essential to be able to assess the influence of these uncertainties on the results provided by the computer.
In chapter , computer arithmetic is presented and its consequences are illustrated with examples. Chapter is devoted to deterministic methods for estimating (majoring) bounds on the propagation of round-off errors. Regression analysis is particularly interesting for studying the stability of algorithms. However, it requires a detailed study of each algorithm studied.
Interval arithmetic can be used to calculate an interval that is certain to contain the exact solution to the problem under study, but generally requires a reformulation of the algorithm if you don't want to find an interval that is far too pessimistic.
Other aspects, notably the stochastic approach to error propagation, using the CESTAC method, and the contribution of CADNA software, will be studied in the dossier that follows,
Finally, the reader will find an impressive bibliography and recommended websites in the documentary section, the
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- Floating-point arithmetic - IEEE 754 - 01-08
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