Algorithmes compensés en arithmétique flottante : précision, validation, performances. (Compensated algorithms in floating point arithmetic : accuracy, validation, performances)

Rounding error may totally corrupt the result of a oating point computation.How to improve and validate the accuracy of a oating point computation, without largecomputing time overheads ? We contribute to this question considering two examples:polynomial evaluation and linear triangular system solving. In both cases we use thecompensation of the rounding errors to improve the accuracy of the computed results.1) Improving the accuracy. We propose a compensated Horner scheme that computespolynomial evaluation with the same accuracy as the classic Horner algorithm performed intwice the working precision. Generalizing this algorithm, we present another compensatedversion of the Horner scheme simulating K times the working precision (K ≥ 2). We alsoshow how to compensate the rounding errors generated by the substitution algorithm fortriangular system solving.2) Validating the computed result. We show how to validate the quality of the com-pensated polynomial evaluation. We propose a method to compute an a posteriori errorbound together with the compensated result. This error bound is computed using onlybasic oating point operations to ensure portability and e ciency of the method.3) Performances of compensated algorithms. Our computing time measures show theinterest of compensated algorithms compared to other software solutions that provide thesame output accuracy. We also justify such good practical performances thanks to a de-tailed study of the instruction-level parallelism they contain.