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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
QArith: Coq formalisation of lazy rational arithmetic
 Types for Proofs and Programs, volume 3085 of LNCS
, 2003
"... Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation use ..."
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Cited by 9 (2 self)
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Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation uses advanced machinery of the Coq theorem prover and applies recent developments in formalising general recursive functions. This formalisation highlights the rôle of type theory both as a tool to verify handwritten programs and as a tool to generate verified programs. 1
Exact Arithmetic on the SternBrocot Tree
 Nijmeegs Instituut voor Informatica en Informateikunde, 2003. http://www.cs.ru.nl/research/reports/full/NIIIR0325.pdf
, 2003
"... In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms ..."
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Cited by 8 (2 self)
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In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simpli ed version of the homographic and the quadratic algorithms [19, 12]. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant representation for real numbers.
The constructive reals as a Java Library
 J. Log. Algebr. Program
, 2004
"... We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the ..."
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We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the implementation could be easily understood, and to allow simple informal correctness arguments. We hope to demonstrate that even such a basic implementation of constructive real arithmetic can be useful in a number of contexts, including in a desk calculator utility distributed with the package. A secondary goal was to demonstrate that some secondorder functions on the reals, such as restricted inverse and derivative operations, can be implemented with su#cient performance to be useful.
Admissible Digit Sets and a Modified SternBrocot Representation
, 2004
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Mobius transformations. We regard certain sets of Mobius transformations as a generalized notion of digits and introduce sucient conditions that such a \digit ..."
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We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Mobius transformations. We regard certain sets of Mobius transformations as a generalized notion of digits and introduce sucient conditions that such a \digit set" yields an admissible representation of [0; +1]. Furthermore we establish the productivity and correctness of the homographic algorithm for such \admissible" digit sets. In the second part of the paper we discuss representation of positive real numbers based on the Stern{Brocot tree. We show how we can modify the usual Stern{Brocot representation to yield a ternary admissible digit set.
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’.
Computer Algebra or Computer Mathematics?
, 2012
"... • We quote Maple, but do not intend to condemn it. • I do not ignore systems such as GAP, KANT, PARI and Magma: rather the thesis of this talk is (largely) irrelevant to them. 2 ..."
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• We quote Maple, but do not intend to condemn it. • I do not ignore systems such as GAP, KANT, PARI and Magma: rather the thesis of this talk is (largely) irrelevant to them. 2