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A certified infinite norm for the validation of numerical algorithms

Abstract : The development of numerical algorithms requires the bounding image domain of functions, in particular functions eps(x) associated to an approximation error. This problem can often be reduced to computing the infinite norm ||eps(x)|| of the given function eps(x). For instance, the development of elementary function operators in hard- and software makes use of such algorithms. Implementations for computing in practice highly accurate floating-point approximations to infinite norms are known and available. Nevertheless, no highly precise, sufficiently fast and certified or self-validating algorithms are available. Their results could be seen as an element in the correctness proof of safety critical or provenly guaranteed implementations. We present an algorithm for computing infinite norms in interval arithmetic. The algorithm is optimized for functions representing absolute or relative approximation errors that are ill-conditioned because of high cancellation. It can handle even functions that are numerically unstable on floating-point points because they are defined there only by continuous extension. In addition the given algorithm is capable of generating a correctness proof for an infinite norm instance by retaining its computational tree.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-00119810
Contributor : Sylvain Chevillard <>
Submitted on : Tuesday, December 12, 2006 - 11:19:38 AM
Last modification on : Friday, April 20, 2018 - 3:44:24 PM
Document(s) archivé(s) le : Tuesday, April 6, 2010 - 7:43:17 PM

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  • HAL Id : ensl-00119810, version 1

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Sylvain Chevillard, Christoph Lauter. A certified infinite norm for the validation of numerical algorithms. 2006. ⟨ensl-00119810v1⟩

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