hal-00428952 https://hal.science/hal-00428952 https://hal.science/hal-00428952/document https://hal.science/hal-00428952/file/69698.pdf doi:10.1137/070696982 [UNIV-NANTES] Université de Nantes [ENSTA-BRETAGNE] ENSTA Bretagne [MINES-NANTES] Ecole des Mines de Nantes [CNRS] CNRS - Centre national de la recherche scientifique [LINA] Laboratoire d'Informatique de Nantes Atlantique [ENSTA-BRETAGNE-STIC] Département STIC [INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions [LINA-TASC] LINA - TASC [ENSIETA-DTN] Développement Technologies Nouvelles [INFO] Département informatique [TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes [IMTA_DAPI] IMT Atlantique - Département automatique, productique et informatique [LS2N-IMTA] LS2N - IMT Atlantique [IMT-ATLANTIQUE] IMT-ATLANTIQUE [NANTES-UNIVERSITE] Nantes Université [UNIV-NANTES-AV2022] Université de Nantes A Priori Error Analysis and Spring Arithmetic Chabert, Gilles Jaulin, Luc [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA] ART Error analysis interval arithmetic global optimization Error analysis is defined by the following concern: bounding the output variation of a (nonlinear) function with respect to a given variation of the input variables. This paper investigates this issue in the framework of interval analysis. The classical way of analyzing the error is to linearize the function around the point corresponding to the actual input, but this method is local and not reliable. Both drawbacks can be easily circumvented by a combined use of interval arithmetic and domain splitting. However, because of the underlying linearization, a standard interval algorithm leads to a pessimistic bound, and even simply fails (i.e., returns an infinite error) in case of singularity. We propose an original nonlinear approach where intervals are used in a more sophisticated way through the so-called "springs". This new structure allows to represent an (infinite) set of intervals constrained by their midpoints and their radius. The output error is then calculated with a spring arithmetic in the same way as the image of a function is calculated with interval arithmetic. Our method is illustrated on two examples, including an application of geopositioning. 2009 2009-10-30 en SIAM Journal on Scientific Computing Society for Industrial and Applied Mathematics