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...

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 non-negative 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 so-called sign matrix determines an interval on which the real number lies. The subsequent so-called digit matrices have non-negative 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.

In this paper, we incorporate a representation of the non-negative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.

. We show how functional languages can be used to write programs for real-valued 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 [4--6, 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 floating-point algorithms can lead to completely erroneous results [1, 14]. In such situations, exact real number computation provides guaranteed correctness, although at the (probably...

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We extend the framework for exact real arithmetic using linear fractional transformations from the non-negative numbers to the extended real line. We then present an extension of PCF with a real type which introduces an eventually breadth-first strategy for lazy evaluation of exact real numbers. In this language, we present the constant redundant if, rif, for defining functions by cases which, in contrast to parallel if (pif), overcomes the problem of undecidability of comparison of real numbers in finite time. We use the upper space of the one-point compactification of the real line to develop a denotational semantics for the lazy evaluation of real programs. Finally two adequacy results are proved, one for programs containing rif and one for those not containing it. Our adequacy results in particular provide the proof of correctness of algorithms for computation of single-valued elementary functions. 1 Introduction It is well known that the accumulation of roundoff errors in floati...

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 well-defined 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]. One-dimensional LFT's x 7! ax+c bx+d are used in the representation of real numbers and to implement basic unary functions, while two-dimensional 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 ...

. One possible approach to exact real arithmetic is to use linear fractional transformations to represent real numbers and computations on real numbers. In this paper, we show that the bit sizes of the (integer) parameters of nearly all transformations used in computations are proportional to the number of basic computational steps executed so far. Here, a basic step means consuming one digit of the argument(s) or producing one digit of the result. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic [8, 16, 11, 14, 12, 6]. One-dimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic functions, while two-dimensional 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 infinite expression trees denoting transcendental functions. In Section 2, we present the details of the LFT approach. This provides the background for understanding the r...

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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 re-used 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...