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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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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.
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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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 ...
Semantics of QueryDriven Communication of Exact Values1
"... Abstract: We address the question of how to communicate among distributed processes values such as real numbers, continuous functions and geometrical solids with arbitrary precision, yet efficiently. We extend the established concept of lazy communication using streams of approximants by introducin ..."
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Abstract: We address the question of how to communicate among distributed processes values such as real numbers, continuous functions and geometrical solids with arbitrary precision, yet efficiently. We extend the established concept of lazy communication using streams of approximants by introducing explicit queries. We formalise this approach using protocols of a queryanswer nature. Such protocols enable processes to provide valid approximations with certain accuracy and focusing on certain locality as demanded by the receiving processes through queries. A latticetheoretic denotational semantics of channel and process behaviour is developed. The query space is modelled as a continuous lattice in which the top element denotes the query demanding all the information, whereas other elements denote queries demanding partial and/or local information. Answers are interpreted as elements of lattices constructed over suitable domains of approximations to the exact objects. An unanswered query is treated as an error and denoted using the top element. The major novel characteristic of our semantic model is that it reflects the dependency of answers on queries. This enables the definition and analysis of an appropriate concept of convergence rate, by assigning an effort indicator to each query and a measure of information content to each answer. Thus we capture not only what function a process computes, but also how a process transforms the convergence rates from its inputs to its outputs. In future work these indicators can be used to capture further computational complexity measures. A robust prototype implementation of our model is available.
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. ..."
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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
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