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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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. 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...
On Coalgebra of Real Numbers
, 1999
"... We define the continuum up to order isomorphism (and hence homeomorphism) as the final coalgebra of the functor X \Delta !, ordinal product with !. This makes an attractive analogy with the definition of the ordinal ! itself as the initial algebra of the functor 1; X , prepend unity, with both defin ..."
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We define the continuum up to order isomorphism (and hence homeomorphism) as the final coalgebra of the functor X \Delta !, ordinal product with !. This makes an attractive analogy with the definition of the ordinal ! itself as the initial algebra of the functor 1; X , prepend unity, with both definitions made in the category of posets. The variants 1; (X \Delta !), X o \Delta !, and 1; (X o \Delta !) yield respectively Cantor space (surplus rationals), Baire space (no rationals), and again the continuum as their final coalgebras. 1 Introduction Coinduction has only relatively recently been recognized as a genuine logical principle [2]. Before that, it was introduced and used mostly in the semantics of concurrency [13]. It has by now been presented from many different angles: [1,8,12,1618], to name just a few contributors. Why would so foundational a principle wait for the late 20th century to be discovered? In [14,16] the idea was put forward that coinduction is new only by nam...
Numerical Integration with Exact Real Arithmetic
 Automata, Languages and Programming, 26th International Colloquium, ICALP’99, Prague, Czech 227 Republic, July 1115, 1999, Proceedings, volume 1644 of Lecture Notes in Computer Science
, 1999
"... . We show that the classical techniques in numerical integration (namely the Darboux sums method, the compound trapezoidal and Simpson's rules and the Gauss{Legendre formulae) can be implemented in an exact real arithmetic framework in which the numerical value of an integral of an elementary f ..."
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. We show that the classical techniques in numerical integration (namely the Darboux sums method, the compound trapezoidal and Simpson's rules and the Gauss{Legendre formulae) can be implemented in an exact real arithmetic framework in which the numerical value of an integral of an elementary function is obtained up to any desired accuracy without any round{o errors. Any exact framework which provides a library of algorithms for computing elementary functions with an arbitrary accuracy is suitable for such an implementation; we have used an exact real arithmetic framework based on linear fractional transformations and have thereby implemented these numerical integration techniques. We also show that Euler's and Runge{Kutta methods for solving the initial value problem of an ordinary dierential equation can be implemented using an exact framework which will guarantee the convergence of the approximation to the actual solution of the dierential equation as the step size in the partiti...
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 ...
Coinduction for Exact Real Number Computation
, 2007
"... This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the corr ..."
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This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio. Key words: Exact real number computation, coinduction, corecursion, signed digit streams. 1
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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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...
The Continuum as a Final Coalgebra
, 1999
"... We define the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the final coalgebra of either the functor N X, product with the set of natural numbers, or the functor 1 + N X. This makes an attractive analogy with the definition of N itself as the i ..."
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Cited by 4 (2 self)
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We define the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the final coalgebra of either the functor N X, product with the set of natural numbers, or the functor 1 + N X. This makes an attractive analogy with the definition of N itself as the initial algebra of the functor 1 + X, disjoint union with a singleton. We similarly specify Baire space and Cantor space in terms of these final coalgebras. We identify two variants of this approach, a coinductive definition based on final coalgebraic structure in the category of sets, and a direct definition as a final coalgebra in the category of posets. We conclude with some paradoxical discrepancies between continuity and constructiveness in this setting.
Real Number Computation through Gray Code Embedding
, 2000
"... We propose an embedding G of the unit open interval to the set f0; 1g ! ?;1 of infinite sequences of f0; 1g with at most one undefined element. This embedding is based on Gray code and it is a topological embedding with a natural topology on f0; 1g ! ?;1 . We also define a machine called an IM2 mach ..."
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Cited by 3 (1 self)
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We propose an embedding G of the unit open interval to the set f0; 1g ! ?;1 of infinite sequences of f0; 1g with at most one undefined element. This embedding is based on Gray code and it is a topological embedding with a natural topology on f0; 1g ! ?;1 . We also define a machine called an IM2 machine (indeterministic multihead type 2 machine) which input/output sequences in f0; 1g ! ?;1 , and show that the computability notion induced on real functions through the embedding G is equivalent to the one induced by the signed digit representation and Type2 machines. We also show that basic algorithms can be expressed naturally with respect to this embedding.
Coinductive Field of Exact Real Numbers and General Corecursion
, 2006
"... In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be se ..."
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In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general corecursive functions. We use the machinery of Coq proof assistant for coinductive types to present the formalisation.