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Computing with Real Numbers  I. The LFT Approach to Real Number Computation  II. A Domain Framework for Computational Geometry
 PROC APPSEM SUMMER SCHOOL IN PORTUGAL
, 2002
"... We introduce, in Part I, a number representation suitable for exact real number computation, consisting of an exponent and a mantissa, which is an in nite stream of signed digits, based on the interval [ 1; 1]. Numerical operations are implemented in terms of linear fractional transformations ( ..."
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We introduce, in Part I, a number representation suitable for exact real number computation, consisting of an exponent and a mantissa, which is an in nite stream of signed digits, based on the interval [ 1; 1]. Numerical operations are implemented in terms of linear fractional transformations (LFT's). We derive lower and upper bounds for the number of argument digits that are needed to obtain a desired number of result digits of a computation, which imply that the complexity of LFT application is that of multiplying nbit integers. In Part II, we present an accessible account of a domaintheoretic approach to computational geometry and solid modelling which provides a datatype for designing robust geometric algorithms, illustrated here by the convex hull algorithm.
Formalising exact arithmetic in type theory
 New Computational Paradigms: First Conference on Computability in Europe, CiE 2005
, 2005
"... Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalis ..."
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Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalise and reason about the infinite objects. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output. 1
Two Algorithms for Root Finding in Exact Real Arithmetic
, 1998
"... We present two algorithms for computing the root, or equivalently the fixed point, of a function in exact real arithmetic. The first algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact floating point. T ..."
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We present two algorithms for computing the root, or equivalently the fixed point, of a function in exact real arithmetic. The first algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact floating point. The second and more general algorithm is based on a trisection of intervals and can be compared with the wellknown bisection method in numerical analysis. It can be applied to any representation for exact real numbers; here it is described for the sign binary system in [\Gamma1; 1] which is equivalent to the exact floating point with linear fractional transformations. Keywords : Shrinking intervals, Normal products, Exact floating point, Expression trees, Sign Binary System, Iterative method, Trisection. 1 Introduction In the past few years, continued fractions and linear fractional transformations (lft), also called homographies or Mobius transformations, have been used to develop various...
Semantics of a Sequential Language for Exact RealNumber Computation
"... We study a programming language with a builtin ground type for real numbers. In order for the language to be sufficiently expressive but still sequential, we consider a construction proposed by Boehm and Cartwright. The nondeterministic nature of the construction suggests the use of powerdomains in ..."
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We study a programming language with a builtin ground type for real numbers. In order for the language to be sufficiently expressive but still sequential, we consider a construction proposed by Boehm and Cartwright. The nondeterministic nature of the construction suggests the use of powerdomains in order to obtain a denotational semantics for the language. We show that the construction cannot be modelled by the Plotkin or Smyth powerdomains, but that the Hoare powerdomain gives a computationally adequate semantics. As is well known, Hoare semantics can be used in order to establish partial correctness only. Since computations on the reals are infinite, one cannot decompose total correctness into the conjunction of partial correctness and termination as it is traditionally done. We instead introduce a suitable operational notion of strong convergence and show that total correctness can be proved by establishing partial correctness (using denotational methods) and strong convergence (using operational methods). We illustrate the technique with a representative example. 1.
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 Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "dig ..."
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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 Möbius transformations. We regard certain sets of Möbius 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 SternBrocot tree. We show how we can modify the usual SternBrocot representation to yield a ternary admissible digit set.
Computation with Real Numbers  Exact Arithmetic, Computational Geometry and Solid Modelling
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Admissible Digit Sets and a Modified Stern–Brocot Representation
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “dig ..."
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We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set ” yields an admissible representation of [0, +∞]. 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.
Exact Real Number Computation Using Linear Fractional Transformations
"... which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy. ..."
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which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy.
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