Algorithmes compensés en arithmétique flottante : précision, validation, performances

Abstract : Rounding error may totally corrupt the result of a floating point computation. How to improve and validate the accuracy of a floating point computation, without large computing time overheads ? We consider two case studies: polynomial evaluation and linear triangular system solving. In both cases we use compensation of the rounding errors to improve the accuracy of the computed result. The contributions of this work are divided into three levels. 1) Improving the accuracy. We propose a compensated Horner scheme that computes polynomial evaluation with the same accuracy as the classic Horner algorithm performed in twice the working precision. Generalizing this algorithm, we present another compensated version of the Horner scheme simulating K times the working precision (K>1). We also show how to compensate the rounding errors generated by the substitution algorithm for triangular system solving. 2) Validating the computed result. We show how to validate the quality of the compensated polynomial evaluation. We propose a method to compute an "a posteriori" error bound together with the compensated result. This error bound is computed using only basic floating point operations, which ensures portability and efficiency of the method. 3) Performances of compensated algorithms. Our computing time measures show the interest of compensated algorithms compared to other software solutions that provide the same output accuracy. We also justify good practical performances of compensated algorithms thanks to a detailed study of the instruction-level parallelism they contain.
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Arithmétique des ordinateurs. Université de Perpignan Via Domitia, 2007. Français
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Submitted on : Friday, May 13, 2016 - 12:39:13 PM
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Nicolas Louvet. Algorithmes compensés en arithmétique flottante : précision, validation, performances. Arithmétique des ordinateurs. Université de Perpignan Via Domitia, 2007. Français. <tel-01315543>

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