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14
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 51 (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.
A DomainTheoretic Approach to Computability on the Real Line
, 1997
"... In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and ..."
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Cited by 47 (12 self)
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In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of StoltenbergHansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turingmachine based approach which dates back to Grzegorczyk and Lacombe, is used by PourEl & Richards in their found...
Semantics of Exact Real Arithmetic
, 1997
"... In this paper, we incorporate a representation of the nonnegative 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 exten ..."
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Cited by 30 (9 self)
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In this paper, we incorporate a representation of the nonnegative 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.
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 ..."
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Cited by 11 (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 ...
Computable Banach Spaces via Domain Theory
 Theoretical Computer Science
, 1998
"... This paper extends the ordertheoretic approach to computable analysis via continuous domains to complete metric spaces and Banach spaces. We employ the domain of formal balls to define a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of clos ..."
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Cited by 10 (2 self)
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This paper extends the ordertheoretic approach to computable analysis via continuous domains to complete metric spaces and Banach spaces. We employ the domain of formal balls to define a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded. We show that the domaintheoretic computability theory is equivalent to the wellestablished approach by PourEl and Richards. 1 Introduction This paper is part of a programme to introduce the theory of continuous domains as a new approach to computable analysis. Initiated by the various applications of continuous domain theory to modelling classical mathematical spaces and performing computations as outlined in the recent survey paper by Edalat [6], the authors started this work with [9] which was concerned with co...
Static Analyses of FloatingPoint Operations
 In SAS’01, volume 2126 of LNCS
, 2001
"... Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint operations lose so much precision that they quickly become irrelevant. In thi ..."
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Cited by 9 (0 self)
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Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint operations lose so much precision that they quickly become irrelevant. In this article, we review some of the problems one can encounter, focussing on the IEEE7541985 norm. We give a (sketch of a) semantics of its basic operations then abstract them (in the sense of abstract interpretation) to extract information about the possible loss of precision. The expected application is abstract debugging of software ranging from simple onboard systems (which use more and more ontheshelf microprocessors with floatingpoint units) to scientific codes. The abstract analysis is demonstrated on simple examples and compared with related work. 1
Lazy Computation with Exact Real Numbers
 PROCEEDINGS OF THE THIRD ACM SIGPLAN INTERNATIONAL CONFERENCE ON FUNCTIONAL PROGRAMMING (ICFP98), VOLUME 34, 1 OF ACM SIGPLAN NOTICES
, 1997
"... We extend the framework for exact real arithmetic using linear fractional transformations from the nonnegative numbers to the extended real line. We then present an extension of PCF with a real type which introduces an eventually breadthfirst strategy for lazy evaluation of exact real numbers. In ..."
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Cited by 9 (4 self)
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We extend the framework for exact real arithmetic using linear fractional transformations from the nonnegative numbers to the extended real line. We then present an extension of PCF with a real type which introduces an eventually breadthfirst 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 onepoint 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 singlevalued elementary functions.
The Appearance of Big Integers in Exact Real Arithmetic based on Linear Fractional Transformations
 In Proc. Foundations of Software Science and Computation Structures (FoSSaCS '98), volume 1378 of LNCS
, 1997
"... . 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 nu ..."
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Cited by 7 (4 self)
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. 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]. Onedimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic 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 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...
Big Integers and Complexity Issues in Exact Real Arithmetic
 In Third Comprox workshop
, 1998
"... One possible approach to exact real arithmetic is to use linear fractional transformations to represent real numbers and computations on real numbers. We show how to determine the digits that can be emitted from a transformation, and present a criterion which ensures that it is possible to emit a di ..."
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Cited by 6 (3 self)
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One possible approach to exact real arithmetic is to use linear fractional transformations to represent real numbers and computations on real numbers. We show how to determine the digits that can be emitted from a transformation, and present a criterion which ensures that it is possible to emit a digit. Using these results, we prove that the obvious algorithm to compute n digits from the application of a transformation to a real number has complexity O(n 2 ), and present a method to reduce this complexity to that of multiplying two n bit integers. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic [5,14,9,12,10,4]. Onedimensional LFT's x 7! ax+c bx+d are used as digits 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 denoting transcendental functions...
How Many Argument Digits are Needed to Produce n Result Digits?
 In RealComp '98 Workshop (June 1998 in Indianapolis), volume 24 of Electronic Notes in Theoretical Computer Science
, 1999
"... In the LFT approach to Exact Real Arithmetic, we study the question how many argument digits are needed to produce a certain number of result digits. We present upper and lower bounds for many simple functions and operations, and for exponential and square root. 1 Introduction In this paper, we wor ..."
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Cited by 6 (2 self)
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In the LFT approach to Exact Real Arithmetic, we study the question how many argument digits are needed to produce a certain number of result digits. We present upper and lower bounds for many simple functions and operations, and for exponential and square root. 1 Introduction In this paper, we work in an approach to Exact Real Arithmetic where real numbers are represented as potentially infinite streams of information units, called digits. Hence, an algorithm to compute a certain expression over real numbers is a device that reads some input streams and produces an output stream. Algorithms like this never terminate, but are considered as satisfactory if they produce any desired number of output digits in finite time, i.e., from a finite number of input digits by a finite number of internal operations. The (time) efficiency of a real number algorithm indicates how much time T (n) it takes to produce n result digits. It clearly depends on the number of input digits needed to produce ...