Results 1  10
of
27
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 ..."
Abstract

Cited by 43 (8 self)
 Add to MetaCart
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...
The Realizability Approach to Computable Analysis and Topology
, 2000
"... policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government. ..."
Abstract

Cited by 41 (19 self)
 Add to MetaCart
policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.
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 ( ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
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.
The iRRAM: Exact Arithmetic in C++
"... The iRRAM is a very efficient C++ package for errorfree real arithmetic based on the concept of a RealRAM. Its capabilities range from ordinary arithmetic over trigonometric functions to linear algebra even with sparse matrices. We discuss the concepts and some highlights of the implementation. ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
The iRRAM is a very efficient C++ package for errorfree real arithmetic based on the concept of a RealRAM. Its capabilities range from ordinary arithmetic over trigonometric functions to linear algebra even with sparse matrices. We discuss the concepts and some highlights of the implementation.
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 ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
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...
A Monadic Probabilistic Language
 In Proceedings of the 2003 ACM SIGPLAN international workshop on Types in languages design and implementation
, 2003
"... Motivated by many practical applications that have to compute in the presence of uncertainty, we propose a monadic probabilistic language based upon the mathematical notion of sampling function. Our language provides a unified representation scheme for probability distributions, enjoys rich expressi ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
Motivated by many practical applications that have to compute in the presence of uncertainty, we propose a monadic probabilistic language based upon the mathematical notion of sampling function. Our language provides a unified representation scheme for probability distributions, enjoys rich expressiveness, and o#ers high versatility in encoding probability distributions. We also develop a novel style of operational semantics called a horizontal operational semantics, under which an evaluation returns not a single outcome but multiple outcomes. We have preliminary evidence that the horizontal operational semantics improves the ordinary operational semantics with respect to both execution time and accuracy in representing probability distributions.
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 ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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. 1 Introduction It is well known that the accumulation of roundoff errors in floati...
A Constructive Formalization of the Fundamental Theorem of Calculus
"... We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. In this formalization, we have closely followed Bishop's work ([4]). In this paper, we describe the formalization i ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. In this formalization, we have closely followed Bishop's work ([4]). In this paper, we describe the formalization in some detail, focusing on how some of Bishop's original proofs had to be refined, adapted or redone from scratch.
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 ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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 ...
Exact Arithmetic on the SternBrocot Tree
 Nijmeegs Instituut voor Informatica en Informateikunde, 2003. http://www.cs.ru.nl/research/reports/full/NIIIR0325.pdf
, 2003
"... In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simpli ed version of the homographic and the quadratic algorithms [19, 12]. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant representation for real numbers.