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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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Cited by 16 (2 self)
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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.
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 ...
Arithmetic Unit Based on Continued Fraction
 TO APPEAR IN PROCEEDINGS OF ECI’2006 CONFERENCE
, 2006
"... We introduce architecture of an arithmetic unit that is based on continued fractions and allows computing any linear rational function of two variables, including basic arithmetic operations like addition, subtraction, multiplication and division. Such a unit can easily exploit the parallel nature o ..."
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Cited by 2 (1 self)
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We introduce architecture of an arithmetic unit that is based on continued fractions and allows computing any linear rational function of two variables, including basic arithmetic operations like addition, subtraction, multiplication and division. Such a unit can easily exploit the parallel nature of continued fraction arithmetic and accelerate the otherwise low performance of its software implementation. The proposed architecture uses continued logarithms, which are an adapted variant of continued fractions and which suit hardware implementation more naturally through their bitlevel granularity. We have used FPGA to implement such a unit and we present here some experimental results, which demonstrate promising properties of the proposed architecture.