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inria-00119254, version 2

## Floating-Point $L^2$-Approximations

Nicolas Brisebarre () a12, Guillaume Hanrot () b3

18th IEEE Symposium in Computer Arithmetic (2007) 177-186

Abstract: Computing good polynomial approximations to usual functions is an important topic for the computer evaluation of those functions. These approximations can be good under several criteria, the most desirable being probably that the relative error is as small as possible in the $L^{\infty}$ sense, i.e. everywhere on the interval under study. In the present paper, we investigate a simpler criterion, the $L^2$ case. Though finding a best polynomial $L^2$-approximation with real coefficients is quite easy, we show that if the coefficients are restricted to be floating point numbers to some precision, the problem becomes a general instance of the CVP problem, and hence is NP-hard. We investigate the practical behaviour of exact and approximate algorithms for this problem. The conclusion is that it is possible in a short amount of time to obtain a relative or absolute best $L^2$-approximation. The main applications are for large dimension, as a preliminary step of finding $L^{\infty}$-approximations and for functions with large variations, for which relative best approximation is by far more interesting than absolute.

• a –  Université Jean Monnet - Saint-Etienne
• b –  INRIA
• 1:  ARENAIRE (Inria Grenoble Rhône-Alpes / LIP Laboratoire de l'Informatique du Parallélisme)
• INRIA – CNRS : UMR5668 – Université Claude Bernard - Lyon I – École Normale Supérieure - Lyon
• 2:  Laboratoire de Mathématiques de l'Université de Saint-Etienne (LAMUSE)
• Université Jean Monnet - Saint-Etienne
• 3:  CACAO (INRIA Lorraine - LORIA)
• CNRS : UMR7503 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)

• inria-00119254, version 2
• oai:hal.inria.fr:inria-00119254
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• Submitted on: Monday, 11 December 2006 10:02:33
• Updated on: Friday, 21 December 2007 11:33:04