A Priori Error Analysis and Spring Arithmetic

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 one 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.