@TECHREPORT{CL2006,
  author = 	 {Chevillard, Sylvain and Lauter, Christoph},
  title = 	 {{A certified infinite norm for the validation of numerical algorithms}},
  institution =  {{Laboratoire de l'informatique du parallélisme (LIP), École normale supérieure de Lyon}},
  year = 	 {2006},
  type = 	 {Rapport de recherche},
  number = 	 {RR2006-49},
  month = 	 {Dec},
  hal = {ensl-00119810},
  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.},
  keywords = {infinite norm, optimization, interval arithmetic, certified algorithm, error analysis, approximation error}
}