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30
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 94 (10 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...
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
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 47 (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 42 (8 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.
Lazy Functional Algorithms for Exact Real Functionals
 Lec. Not. Comput. Sci
, 1998
"... . We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a ..."
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Cited by 23 (0 self)
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. We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a practical application of a method, due to Berger, for computing quantifiers over streams. Correctness proofs for the algorithms make essential use of domain theory. 1 Introduction In exact real number computation, infinite representations of reals are employed to avoid the usual rounding errors that are inherent in floating point computation [46, 17]. For certain real number computations that are highly sensitive to small variations in the input, such rounding errors become inordinately large and the use of floatingpoint algorithms can lead to completely erroneous results [1, 14]. In such situations, exact real number computation provides guaranteed correctness, although at the (probably...
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 19 (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
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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Cited by 16 (1 self)
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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.
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 11 (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
Contractivity of Linear Fractional Transformations
 Third Real Numbers and Computers Conference (RNC3
, 1998
"... One possible approach to exact real arithmetic is to use linear fractional transformations (LFT's) to represent real numbers and computations on real numbers. Recursive expressions built from LFT's are only convergent (i.e., denote a welldefined real number) if the involved LFT's are sufficiently c ..."
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Cited by 8 (3 self)
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One possible approach to exact real arithmetic is to use linear fractional transformations (LFT's) to represent real numbers and computations on real numbers. Recursive expressions built from LFT's are only convergent (i.e., denote a welldefined real number) if the involved LFT's are sufficiently contractive. In this paper, we define a notion of contractivity for LFT's. It is used for convergence theorems and for the analysis and improvement of algorithms for elementary functions. Keywords : Exact Real Arithmetic, Linear Fractional Transformations 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic [8, 17, 11, 14, 12, 6]. Onedimensional LFT's x 7! ax+c bx+d are used in the representation of real numbers and to implement basic unary functions, while twodimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as addition and multiplication, and can be combined to obtain infinite expression trees ...
A high radix online arithmetic for credible and accurate computing
 Journal of Universal Computer Science
, 1995
"... Abstract: The result of a simple oatingpoint computation can be in great error, even though no error is signaled, no coding mistakes are in the program, and the computer hardware is functioning correctly. This paper proposes a set of instructions appropriate for a general purpose microprocessor tha ..."
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Cited by 8 (1 self)
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Abstract: The result of a simple oatingpoint computation can be in great error, even though no error is signaled, no coding mistakes are in the program, and the computer hardware is functioning correctly. This paper proposes a set of instructions appropriate for a general purpose microprocessor that can be used to improve the credibility and accuracy of numerical computations. Such instructions provide direct hardware support for monitoring events which may threaten computational integrity, implementing oatingpoint data types of arbitrary precision, and repeating calculations with greater precision. These useful features are obtained by the e cient implementation of high radix online arithmetic. The prevalence of superscalar and VLIW processors makes this approach especially attractive.