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31
The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 107 (12 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
PCF extended with real numbers
, 1996
"... We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (singlepoint intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be ..."
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Cited by 54 (15 self)
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We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (singlepoint intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be considered as “continuous words”. Concatenation of continuous words corresponds to refinement of partial information. The usual basic operations cons, head and tail used to explicitly or recursively define functions on words generalize to partial real numbers. We use this fact to give an operational semantics to the above referred extension of PCF. We prove that the operational semantics is sound and complete with respect to the denotational semantics. A program of real number type evaluates to a headnormal form iff its value is different from ⊥; if its value is different from ⊥ then it successively evaluates to headnormal forms giving better and better partial results converging to its value.
A New Representation for Exact Real Numbers
, 1997
"... We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the rea ..."
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Cited by 52 (9 self)
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We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S¹, is expressed as the image of the base interval [0�1] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first socalled sign matrix determines an interval on which the real number lies. The subsequent socalled digit matrices have nonnegative integer coe cients and successively re ne that interval. Based on the classi cation of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S¹ by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.
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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Cited by 20 (0 self)
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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
A universal characterization of the closed euclidean interval (Extended Abstract)
 PROC. OF 16TH ANN. IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE, LICS'01
, 2001
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basi ..."
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Cited by 19 (1 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the
A golden ratio notation for the real numbers
, 1991
"... Several methods to perform exact computations on real numbers have been proposed in the literature. In some of these methods real numbers are represented by infinite (lazy) strings of digits. It is a well known fact that, when this approach is taken, the standard digit notation cannot be used. New f ..."
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Cited by 11 (0 self)
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Several methods to perform exact computations on real numbers have been proposed in the literature. In some of these methods real numbers are represented by infinite (lazy) strings of digits. It is a well known fact that, when this approach is taken, the standard digit notation cannot be used. New forms of digit notations are necessary. The usual solution to this representation problem consists in adding new digits in the notation, quite often negative digits. In this article we present an alternative solution. It consists in using non natural numbers as “base”, that is, in using a positional digit notation where the ratio between the weight of two consecutive digits is not necessarily a natural number, as in the standard case, but it can be a rational or even an irrational number. We discuss in full detail one particular example of this form of notation: namely the one having two digits (0 and 1) and the golden ratio as base. This choice is motivated by the pleasing properties enjoyed by the golden ratio notation. In particular, the algorithms for the arithmetic operations are quite simple when this notation is used.
Integration in real PCF
 Information and Computation
, 1996
"... Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number ..."
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Cited by 8 (4 self)
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Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number of steps. Based on a domaintheoretic approach to integration, we show how to define integration in Real PCF. We propose two approaches to integration in Real PCF. One consists in adding integration as primitive. The other consists in adding a primitive for function maximization and then recursively defining integration from maximization. In both cases we have a computational adequacy theorem for the corresponding extension of Real PCF. Moreover, based on previous work on Real PCF definability, we show that Real PCF extended with the maximization operator is universal. 1
The Robustness Issue
"... This article first recalls with some examples the damages that numerical inaccuracy of floating point arithmetic can cause during geometric computations, in methods from Computational Geometry, Computer Graphics or CADCAM. Then it surveys the various approaches proposed to overcome inaccuracy dif ..."
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Cited by 2 (1 self)
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This article first recalls with some examples the damages that numerical inaccuracy of floating point arithmetic can cause during geometric computations, in methods from Computational Geometry, Computer Graphics or CADCAM. Then it surveys the various approaches proposed to overcome inaccuracy difficulties. It seems that the only way to achieve robustness for existing methods from Computational Geometry is exact computation, it is the "Exact Computation Paradigm" of C.K. Yap and T. Dube. In Computer Graphics or CADCAM, people prefer to abandon methods and data structures not robust enough against inaccuracy, namely Boundary Representations and related methods, this may be called the "Approximate Computation Paradigm".
Incremental Addition in Exact Real Arithmetic
, 1998
"... Boehm et al. mention three different main approaches to exact real number arithmetic: Representation of reals via continued fractions, signed digit expansions, and as functions (Cauchy sequences). There exist prototype implementations of packages providing exact real arithmetic based on all three of ..."
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Cited by 2 (0 self)
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Boehm et al. mention three different main approaches to exact real number arithmetic: Representation of reals via continued fractions, signed digit expansions, and as functions (Cauchy sequences). There exist prototype implementations of packages providing exact real arithmetic based on all three of these approaches. A key property distinguishing the approaches is incrementality: If the accuracy of the result has to be increased in the function approach, computation starts from scratch and all previous calculations have to be disregarded. In contrast, the signed digit approach is incremental, i.e. the previous result is reused and some further digits are computed to increase precision. In this paper, we show how the function approach can be modified, resulting in a hybrid representation where signed digit expansions can be read as functions and vice versa. We develop an algorithm for addition in this setting combining advantages of both approaches. Keywords: Exact real arithmetic, in...
Computability of selfsimilar sets
 Mathematical Logic Quarterly
, 1999
"... Abstract. We investigate computability of a selfsimilar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a selfsimilar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractio ..."
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Cited by 2 (0 self)
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Abstract. We investigate computability of a selfsimilar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a selfsimilar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractions are computable, then the selfsimilar is a recursive compact set. A further result on the case that the selfsimilar set forms a curve is also discussed. 1.