Numerical Quality and High Performance In Interval Linear Algebra on Multi-Core Processors

Philippe Theveny 1, 2
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : This work aims at determining suitable scopes for several algorithms of interval matrices multiplication.First, we quantify the numerical quality. Former error analyses of interval matrix products establish bounds on the radius overestimation by neglecting the roundoff error. We discuss here several possible measures for interval approximations. We then bound the roundoff error and compare experimentally this bound with the global error distribution on several random data sets. This approach enlightens the relative importance of the roundoff and arithmetic errors depending on the value and homogeneity of relative accuracies of inputs, on the matrix dimension, and on the working precision. This also leads to a new algorithm that is cheaper yet as accurate as previous ones under well-identified conditions.Second, we exploit the parallelism of linear algebra. Previous implementations use calls to BLAS routines on numerical matrices. We show that this may lead to wrong interval results and also restrict the scalability of the performance when the core count increases. To overcome these problems, we implement a blocking version with OpenMP threads executing block kernels with vector instructions. The timings on a 4-octo-core machine show that this implementation is more scalable than the BLAS one and that the cost of numerical and interval matrix products are comparable.
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Theses
Other [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2014. English. < NNT : 2014ENSL0941 >
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Philippe Theveny. Numerical Quality and High Performance In Interval Linear Algebra on Multi-Core Processors. Other [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2014. English. < NNT : 2014ENSL0941 >. <tel-01126973>

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