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Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Arbitrary precision real arithmetic: design and algorithms Valerie MenissierMorain
"... We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe ..."
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We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe how to produce a sequence representing the result of the application of this function to its arguments, according to the sequences representing these arguments. For each algorithm we prove that the resulting sequence is a valid representation of the exact real result. This arithmetic is the rst abritrary precision real arithmetic with mathematically proved algorithms. Resume Nous proposons une representation des nombres reels calculables ainsi que des algorithmes pour les fonctions elementaires usuelles pour cette representation. Un nombre reel est represente par une suite de nombres Badiques nis et pour chaque fonction classique (rationnelle, algebrique ou transcendante), nous montrons comment produire une suite representant le resultat a partir de suites representant les parametres. Pour chacun de ces algorithmes nous demontrons que la suite qui en resulte represente bien le resultat reel exact. Cette arithmetique est la premiere arithmetique reelle en precision arbitraire dotee d'un jeu complet d'algorithmes
Complexity in the Real world
, 2005
"... Whereas Turing Machines lay a solid foundation for computation of functions on countable sets, a lot of realworld calculations require real numbers. The question arises naturally whether there is a satisfying extension to functions on uncountable sets. This thesis states and discusses such a genera ..."
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Whereas Turing Machines lay a solid foundation for computation of functions on countable sets, a lot of realworld calculations require real numbers. The question arises naturally whether there is a satisfying extension to functions on uncountable sets. This thesis states and discusses such a generalization, based on previous research. It also discusses higher order functions, e.g. differentiation. In contrast to preceding works, however, the focus is on complexity – after computability, of course. By giving a different perspective on Weihrauch’s excellent definition of computability in the uncountable case, we show that this theory indeed admits a useful notion of complexity. Various examples are given to demonstrate the theory, including an application to distributions, also called generalized functions, as a form of ‘stresstest’.