Results 11  20
of
20
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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...
Supervisor
, 1996
"... PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation ..."
Abstract
 Add to MetaCart
PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation
Report 5
"... The main goal of this research project is to develop a functional programming language to perform real number computation. This programming language uses the interval domain model to represent real numbers. The main difference with respect to other functional programming languages, which deal with r ..."
Abstract
 Add to MetaCart
The main goal of this research project is to develop a functional programming language to perform real number computation. This programming language uses the interval domain model to represent real numbers. The main difference with respect to other functional programming languages, which deal with real numbers, and hence the core of the theory developed is that it does not have any parallel construction among its sentences. The idea of not allowing parallel constructions comes from Real PCF due to the exponential increase in the size of computations. Once we fix the programming language, the aim of this work is to present the theory behind it. 3 Summary of work done so far My research done so far roughly speaking consist on: 1. Construction of the programming language. The language that we propose is based on Real PCF (hence calculus), the key point at this stage was the decision of what to delete and add to the programming language in order to have the required properties. After some search in the literature and analysis of the implications of the election, the language was established. 2. Search of the denotational semantics.
Part I Case for Support Computing with arbitrary precision curves
, 2004
"... Previous research track record and plans for the future The proposer has a longstanding interest in the theory and applications of computing with arbitrary precision (AP), which began with his PhD training (1996–2000) at the University of Birmingham under the supervision of Prof. Jung. PhD thesis. ..."
Abstract
 Add to MetaCart
Previous research track record and plans for the future The proposer has a longstanding interest in the theory and applications of computing with arbitrary precision (AP), which began with his PhD training (1996–2000) at the University of Birmingham under the supervision of Prof. Jung. PhD thesis. The thesis (Konečn´y 2000) built on the work of Wiedmer, Weihrauch, Edalat, Ko and others on representing real numbers as infinite streams of digits in an arbitrary precision computation. The thesis addressed, in this context, one of the first fundamental questions in computational complexity theory: Which realnumber functions can be computed to an arbitrary precision without an ever growing need for more memory? This question was answered for many different ways of representing the real numbers as infinite streams of symbols and for all reasonably wellbehaved functions (in some precise sense). This result is also described and proved in two journal articles (Konečn´y 2004, Konečn´y 2002), each for different types of realnumber representations. The thesis extends the articles in terms of the scope of representations and also generalises the theorem in another direction. It covers not only functions but also onetomany mappings. Such mappings arise naturally from the fact that each real number can be represented in many ways: for different representations of the same arguments different, but all correct results may be computed. More importantly, some frequently occurring practical problems, such as finding a zero of a polynomial, cannot be computed as a function but rather as a onetomany mapping. Work in Edinburgh. More recently, the proposer worked as a research fellow for the EPSRC funded project “Type
Exact Real Number Computation Using Linear Fractional Transformations
"... which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy. ..."
Abstract
 Add to MetaCart
which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy.
Computation with Real Numbers  Exact Arithmetic, Computational Geometry and Solid Modelling
"... ..."
THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING
, 2005
"... www.elsevier.com/locate/jlap Exact real arithmetic using centred intervals and bounded error terms ..."
Abstract
 Add to MetaCart
www.elsevier.com/locate/jlap Exact real arithmetic using centred intervals and bounded error terms
Effective real numbers in Mmxlib Joris van der Hoeven
"... Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. Based on earlier, theoretical work, the author has started to develop a systematic C++ library Mmxlib for mathematically correct computations with more analytic objects, like complex numbe ..."
Abstract
 Add to MetaCart
Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. Based on earlier, theoretical work, the author has started to develop a systematic C++ library Mmxlib for mathematically correct computations with more analytic objects, like complex numbers and analytic functions. While implementing the library, we found that several of our theoretical ideas had to be further improved or adapted. In this paper, we report on the current implementation, we present several new results and suggest directions for future improvements. Categories and Subject Descriptors F.2.1 [Theory of Computation]: Analysis of algorithms and problem complexity—Numerical algorithms and problems