J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, Aug. 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theoretical characterisation of termination, but doesn t suggest a method to infer levelmappings required, nor does he study the in uence of imprecision inherent to oating point computations. Much more attention to the oating point arithmetic was payed outside of the logic programming community [4, 14 16, 21, 26]. However, the works we are aware of concentrated on precision of numerical computations. Some of them suggest alternative exact representations of real arithmetic, for example using continued fraction expansions [26] sequences of linear maps [14] or linear fractional transformations with ....

....arithmetic was payed outside of the logic programming community [4, 14 16, 21, 26] However, the works we are aware of concentrated on precision of numerical computations. Some of them suggest alternative exact representations of real arithmetic, for example using continued fraction expansions [26], sequences of linear maps [14] or linear fractional transformations with non negative integer coecients [21] Other authors apply abstract interpretation to infer whether the precision might be lost [4, 15] These works tend, however, to ignore the use of denormalised numbers and the to the ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087-1105, August 1990.

....Report LBNL45991, Lawrence Berkeley National Lab, 2000. Submitted to ACM TOMS. 14] S. Linnainmaa. Analysis of some known methods of improving the accuracy of floating point sums. BIT, 14:167 202, 1974. W. Kahan: A thorough analysis, validation, and comparison of tricks like these. [15] Seppo Linnainmaa. Software for doubled precision floating point computations. ACM Transactions on Mathematical Software, 7(3) 272 283, 1981. Generalized Dekker s algorithms, proofs, etc. 16] Ole Mller. Note on quasi double precision. BIT, 5:251 255, 1965. 17] Ole Mller. Quasi ....

Jean Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105,

....never published, it has remained available throughout the intervening years, it was widely read, and sometimes it is even cited. But while the algorithms derived in it have been acknowledged as the source of inspiration for some of the most interesting modern approaches to exact real arithmetic [6,13,10,23], the underlying coalgebraic idea seems to have gone unnoticed. While hoping to point to this conceptual link, we must add that the constructions on the following pages should not be taken as a rational reconstruction of the Hakmem view of reals. In fact, they were obtained while we were trying ....

J. Vuillemin, Exact real computer arithmetic with continued fractions. IEEE Trans. Comp. 39(1990) 1087-1105 19

....numbers to approximate at best real number computations. Rational arithmetic [Bak75, Cle74, HCL 68, Kla93, KM83, KM88, KM89, Sch83] implements exact arithmetic for rational numbers, which can be used to approximate real numbers, for instance using continued fractions [KM85, RT73, Sei83, Vui90] These methods do not solve the problem in all cases; interval arithmetic might lead to very imprecise (but true) results 11 , multi precision arithmetic and rational arithmetic may be very costly to perform on real scientific codes 12 . A new and promising line of research uses domain ....

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8), 1990. 20

....where both digits and continuants intervene. Second, the continued fraction expansion of a rational number naturally provides an encoding for integer pairs that uses the digits of the continued fraction expansion. In computer systems that directly deal with continued fraction expansions [4] [39], it is important to analyse the average length of this continued fraction encoding. In this paper, we provide new analyses of the precise expected values of the main parameters in the discrete framework. We then obtain new results about the average bit complexity of classical Euclidean ....

J. Vuillemin, Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers 39, 8 (Aug. 1990), 1087--1105. 558 Brigitte Vall' ee

....to the value of the function at a given computable real number up to any threshold of accuracy. However, the computation is not incremental in the sense that to obtain any more accurate approximation one has to compute from scratch. 2 Edalat and Potts Kornerup and Matula [9] and Vuillemin [15], proposed a representation of computable real numbers by redundant continued fractions and presented various incremental algorithms for basic arithmetic operations using the earlier work of Gosper [7] and for some transcendental functions. Any continued fraction expansion of a real number can be ....

....n ) with r n = a n 1 b n 1 a n 2 b n 2 a n 3 . and OE i (x) a i b i x for 0 i n. One can therefore identify the original continued fraction for r with the infinite composition OE 0 OE 1 OE 2 Delta Delta Delta. Such a representation of real numbers was already present in [15]. Nielson and Kornerup [11] later developed a general framework for exact arithmetic by representing real numbers by redundant infinite composition of linear fractional transformations (lft) Escardo s extension of PCF [6] is based on the redundant representation of a real number in [0; 1] as an ....

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990.

.... of linear maps proposed by Avizienis [1] and appeared in the work of Watanuki et al. [23] Boehm an Cartwright [2] Di Gianantonio [5] Escardo [4] Nielsen et al. [18] and Menissier Morain [16] ii) Continued fraction expansions proposed by Gosper [7] developed by Peyton Jones [10] and Vuillemin [21] and advanced more recently by Kornerup et al. [15, 13, 12, 14] iii) Infinite composition of linear fractional transformations (also known as homographies or Mobius transformations) generalises the other two frameworks as demonstrated by Vuillemin [21] Nielsen et al. [18] showed that this ....

.... developed by Peyton Jones [10] and Vuillemin [21] and advanced more recently by Kornerup et al. [15, 13, 12, 14] iii) Infinite composition of linear fractional transformations (also known as homographies or Mobius transformations) generalises the other two frameworks as demonstrated by Vuillemin [21]. Nielsen et al. [18] showed that this framework can be used to represent quasi normalised floating point [23] We introduce here a new, feasible and incremental representation of the extended real numbers based on the composition of linear fractional transformations with either all non negative or ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on computers, 39(8):1087--1105, August 1990.

....abstract. Related work This paper has its origins in the first author s work on exact real number computation [10, 11] In this approach, real numbers are represented by concrete computational structures such as streams, allowing computations to be performed to any desired degree of accuracy [35, 6, 4, 5, 33]. Of particular relevance to our work is the issue of obtaining an abstract data type of real numbers, in which the underlying computational representation is hidden [5, 8, 10, 11] In the programming language Real PCF [10] the abstract data type is based on simple real number constructors and ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990.

....incremental in the sense that to obtain any more accurate approximation one has to compute from scratch. Furthermore, the algorithms are constructed using various di#erent techniques and therefore, except for the simplest arithmetic operations, it is di#cult to verify their correctness. Vuillemin [120], proposed a representation of computable real numbers by redundant continued fractions and, using the earlier work of Gosper [61] DOMAINS FOR COMPUTATION IN MATHEMATICS, PHYSICS AND EXACT . 439 presented various incremental algorithms for basic arithmetic operations and some ....

....# 1 # n (r n ) with r n = a n 1 b n 1 a n 2 b n 2 a n 3 . and # i (x) a i b i x for 0 # i # n. One can therefore identify the original continued fraction for r with the infinite composition # 0 # 1 # 2 . Such a representation of real numbers was already present in [120]. The set of all real lft s, denoted by M, consists of maps f given in Equation (4) with a, b, c, d # R and ad bc #= 0. An lft is a homeomorphism of R # ; it is orientation preserving if ad bc 0 and orientation reversing if ad bc 0. We will study the IFS (S 1 , M) First ....

J. E. Vuillemin, Exact real computer arithmetic with continued fractions, IEEE Transactions on Computers, vol. 39 (1990), no. 8, pp. 1087--1105.

....this method is not so di erent from the multi digit approach presented here, except that our transcendental operations are based on LFT s, which provide a general underlying framework that simpli es the nding of the algorithms and makes the proofs of their correctness automatic. Vuillemin [26] proposed a representation of computable real numbers by continued fractions and presented various incremental algorithms for basic arithmetic operations using the earlier work of Gosper [13] and for some transcendental functions. However, this representation is rather complicated and the ....

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087-1105, 1990.

....much faster than the first one. We studied this second representation and now we propose a complete and entirely proved set of algorithms for all elementary functions. This work leads to an implementation in the Caml implementation of the ML language. Finally, in (Vuillemin, 1987; Vuillemin, 1988; Vuillemin, 1990), Vuillemin interprets Bill Gosper s work on the continued fractions arithmetic (essentially rational operations) Gosper, 1972) and represents real numbers by continued fractions, with the underlying idea that continued fractions are the closest rational numbers to the real numbers. However, ....

Vuillemin, J. (1990). Exact real computer arithmetic with continued fractions. IEEE Transactions on computers, 39(8):1087--1105.

....clearly that the second one is much faster than the first one. We studied this second representation and now we propose a complete and entirely proved set of algorithms for all elementary functions. This work leads to an implementation in the Caml implementation of the ML language. Finally, in (Vuillemin, 1987; Vuillemin, 1988; Vuillemin, 1990) Vuillemin interprets Bill Gosper s work on the continued fractions arithmetic (essentially rational operations) Gosper, 1972) and represents real numbers by continued fractions, with the underlying idea that continued fractions are the closest rational ....

Vuillemin, J. (1987). Exact real computer arithmetic with continued fractions. Research report 760, INRIA.

....accuracy from the root down to the leaves. Then the computations are performed upwards using a bigfloat library. Another way would be to use an on line arithmetic, which represent numbers by a (potentially infinite) stream of digits and in a redundant way (i.e. digits can take negative value) Vui90, BCRO86, Wie80, MM94, Sch89] This has the advantage that computations can be stopped as soon as the sign is known. However, on line arithmetic performances have been disappointing so far (conclusion in [MM94] this field still being in its infancy. Dub e and Yap have experimented their gap ....

J.E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Trans Computers, 39(8):1087--1105, 1990.

....[16] D. Scott [26] 26] Weihrauch [34] 1.1 Motivations for this work and results obtained. Although the theory of computable Analysis can be considered a well developed subject, there have been so far very few attempts of implementing computable Analysis on digital computers [6] 7] 10] [33]. Such implementations should lead to the realization of exact real number computation . In ordinary practice the computation on real numbers is performed by approximating real numbers by a subset of the rational numbers and by approximating the arithmetic on real numbers by a limited precision ....

....representation c) is used instead of representations a) or b) The representation d) is similar to the standard digit representation. The main difference consists in introducing negative digits. This representation has been studied in [2] 6] 36] The representation e) has been developed in [33] and is similar to the continuous fractions representation. The only difference is that in the standard continuous fraction notation only natural numbers are used. In this case it is necessary to use also negative integers. The representations described above do not make explicit use of intervals. ....

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J. Vuillemin, "Exact Real Computer Arithmetic with Continued Fraction." Proc. A.C.M. conference on Lisp and functional Programming (1988) 1427.

....functions, as quickly and easily as seen from outside as any rational operation. It is worth mentioning that, by principle, they can only compute continuous functions. Of course, they all use some kind of laziness, since they compute with (potentially) in nite objects. To quote a few, see [23, 2, 22, 15, 10] and the references therein. As is well known, algorithms in Computational Geometry do not withstand the inaccuracy of oatingpoint arithmetics, which results in topological inconsistencies or in nite loops at run time. To prevent those, computational geometers typically round initial numerical ....

J.E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Trans Computers, 39(8):1087-1105, 1990.

.... is that they are not injective [Wei00] More precisely, uncountably many real numbers have in nitely many names with respect to representations equivalent to the signed digit representation [BH00] This kind of redundancy is considered essential in many approaches to exact real arithmetic [BCRO86,EP97,Gia96,Gia97,Vui90]. Thus, computability of a real function is de ned in two steps: rst the computability of functions over in nite sequences is de ned using Type 2 machines, and then it is connected with the computability of real functions by representations. The redundancy of representations means that we cannot ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087-1105, 1990. 20

....of admissible representations is that they are not injective. That is, 1 some numbers have more than one representations. This redundancy is considered as essential in many approaches to exact real arithmetics [Boehm et al. 1986, Edalat and Potts 1997, Gianantonio 1996a, Gianantonio 1996b, Vuillemin 1990] In this way, the computability of real functions is de ned in two steps: rst the computability of in nite sequences is de ned using Type 2 machines, and then it is connected with the computability of real functions using representations. This redundancy of admissible representations means that ....

J. Vuillemin. (1990) Exact real computer arithmetic with continued fractions. In IEEE Transactions on Computers, 39(8) 1087-1105.

....where both digits and continuants intervene. Second, the continued fraction expansion of a rational number naturally provides an encoding for integer pairs that uses the digits of the continued fraction expansion. In computer systems that directly deal with continued fraction expansions [4] [36], it is important to analyse the average length of this continued fraction encoding. In this paper, we provide new analyses of the precise expected values of the main parameters in the discrete framework. We then obtain new results about the average bit complexity of classical Euclidean ....

Vuillemin, J. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers 39, 8 (Aug. 1990), 1087--1105.

....sequence of approximations with the Euler s and Runge Kutta methods as the number of nodes tends to in nity. We actually use the exact real arithmetic framework based on linear fractional transformations (LFTs) as developed by Edalat and Potts [8, 6, 20] based on the work of Gosper [12] Vuillemin [22] and Kornerup and Nielsen [19] on the one hand and Di Gianantonio [5] and Escard o [11] on the other. Using the complexity results of Heckmann [13] for computing elementary functions in the LFT framework, we obtain, as an example, the complexity of integrating the exponential function in the ....

.... Edalat and Potts, 8, 20] Recently, Heckmann [13] has considered the base interval [ 1; 1] In this section we summarize the representation of real numbers and functions using LFTs, based on the special base interval [ 1; 1] We work in R = R [ f1g, the one point compacti cation of R (as in [22]) identi ed with the unit circle in the plane as in Fig.1. s(0) 1 1 0 S 1 s( 8 ) s(x) x Fig. 1. The stereographic projection Let us denote the set of vectors, matrices and tensors with integer coecients respectively by: V = a b j a; b 2 Z ; M = a c b d j a; b; c; d 2 ....

Vuillemin, J. E.: Exact Real Computer Arithmetic with Continued Fractions. IEEE Transactions on Computers, 39(8):1087-1105 (1990).

....making error estimation very dicult. Another disadvantage of this method is that the two numbers (especially the denominator) grow very quickly. Expanding the value of a fractional number into continued fraction digits gives us an alternative approach to exact arithmetic which is proposed in [45]. An alternative approach to improve the accumulation of roundo error is to build compound arithmetic units. Compound arithmetic units join arithmetic operations together evaluating entire expressions. At each iteration the state of the arithmetic unit encodes the remaining error. Exact ....

....consumes an input quotient, or produces an output quotient at each iteration. However, ensuring that quotients are consumed and produced optimally requires the computation of a large error term which increases the overall computation time of the algebraic algorithm by an order of magnitude [45]. In order to avoid computing the error term in each iteration to determine if the algorithm should consume or produce a digit, Vuillemin[45] shows that in most common cases we can consume one input and produces one output at each iteration the positional algebraic algorithm. Consuming and ....

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J.E. Vuillemin, Exact Real Computer Arithmetic with Continued Fractions, IEEE Trans. on Computers, Vol. 39, No. 8, Aug. 1990.

....analysis should not only investigate the use of different implementation methods. Also the influence of generic programming on the performance has to be considered. As an alternative such an implementation could be contrasted with a realization of computable complex numbers in the style of [Vui87]. Again both approaches have to be analyzed. In form of libraries both are highly desirable for performing exact scientific computations. ....

J. Vuillemin. Exact Real Computer Arithmetic with Continued Fractions. Technical report, INRIA, Rocquencourt, France, 1987.

....the Metafont system: see [28] The HAKMEM algorithm has surfaced in a diversity of contexts of which we now discuss a few. First, there are general purpose computer arithmetics systems that are entirely based on continued fractions, a notable case being the one developed by Vuillemin around 1987 [49]; comparison of numbers in this context is likely to involve a version of the HAKMEM algorithm. Second, issues of correctness and robustness are central in the design of computational geometry systems: there, the comparison problem is identical to the problem of deciding the sign of 2 Theta 2 ....

Jean Vuillemin, Exact real computer arithmetic with continued fractions, IEEE Transactions on Computers 39 (1990), no. 8, 1087--1105.

....ln 2 = 0:0100110: 2 1 Definition of the Operators 1.1 Number Representation A generic real number cannot be represented in software with a finite word. It is usually written as the infinite sequence of numbers of its fi radix representation [3, 8] or as its continued fraction decomposition [15]. In our library, we define a real number X as a double infinite sequence of numbers (a n ; fi n ) 2 Z Theta R such that ja n j fi n Gamma1 fi n Gamma 1 and P 1 n 1 fi i fi n . The nth embedded interval X n is defined from its midpoint x n defined in (1) as X n = x n Gamma fi n ; x n ....

Jean Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990. 11

....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]. One dimensional LFT s x 7 ax c bx d are used in the representation of real numbers and to implement basic unary functions, while two dimensional 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 ....

....some tensor expressions proposed by Edalat s group. In certain cases, it is possible to modify these tensor expressions in order to achieve better convergence. 2 Exact Real Arithmetic by Linear Fractional Transformations In this section, we present the framework of exact real arithmetic by LFT s [8, 17, 11]. After a general introduction, we specialize to the version used by the group of Edalat and Potts at Imperial College [14, 12, 13, 16, 6] 2.1 From Digit Streams to Linear Fractional Transformations There are many ways to represent real numbers as infinite objects [3, 2, 4, 5] Here, we are ....

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J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990.

....this setting combining advantages of both approaches. Keywords: Exact real arithmetic, incrementality, signed digit expansions, real numbers as functions. 1 Introduction There exist several prototype implementations of exact real arithmetic packages, based on approaches via continued fractions [12, 8], Cauchy sequences [4, 9] signed digit expansions [13, 4, 6] and linear fractional transformations [10, 11, 6] Clearly, it is of primary interest to develop efficient algorithms for exact real arithmetic as an alternative to floating point computation. The two most recent and most promising ....

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990.

.... The main objective is to compute f(x) ax b cx d with [o 0 ; o 1 ; o 2 : f ( x 0 ; x 1 ; x 2 ; where x i s are the partial quotients of the simple input CF, and o i = f(x i ; state) are the partial quotients of the output cf. The positional algebraic algorithm introduced in [15] requires one state register for each coefficient of f(x) to consume an input quotient x i apply T 0 (x) T (x i 1 x ) to produce an output quotient o i apply T 00 (x) 1 T (x) o i . From transformations T 0 and T 00 we obtain the following iteration equations: a i 1 ....

J.E. Vuillemin, Exact Real Computer Arithmetic with Continued Fractions, IEEE Trans. on Computers, Vol. 39, No. 8, Aug. 1990.

....and the type that includes plus and minus infinity by the symbol I 1 . In order to treat 1 as any other number, we use the stereographic 6 Gamma1 1 Gamma1 1 r ffi(r) ffi(1) i ffi(0) Gammai Im Figure 1: Stereographic representation of IR. representation of the real line as defined in [Vuillemin 90 ] see Fig. 1] With this representation minus and plus infinity are treated as one point, and the interval j x j 1 is the upper part of the circle, and the reciprocal of this interval (e.g. j x j 1) is the lower part. We will adopt an alternative definition of the width of an interval [x; y] ....

Vuillemin, Jean: "Exact Real Computer Arithmetic with Continued Fractions"; IEEE Transactions on Computers, C-39, 8 (1990), 1087--1105.

....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]. One dimensional LFT s x 7 ax c bx d are used as digits and to implement basic functions, while two dimensional 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 ....

....in the case of matrices. Tensors are handled in Section 7. Finally, we discuss these results and their impact on the complexity of real number computation. 2 Exact Real Arithmetic by Linear Fractional Transformations In this section, we present the framework of exact real arithmetic by LFT s [8, 16, 11]. After a general introduction, we specialise to the version used by the group of Edalat and Potts at Imperial College [14, 12, 13, 15, 6] 2.1 From Digit Streams to Linear Fractional Transformations There are many ways to represent real numbers as infinite objects [3, 2, 4, 5] Here, we are only ....

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J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990.

.... the internal complexity in [4,5] we look for lower and upper bounds for the number of input digits in the paper at hand (input complexity) As in the previous papers, we work in the framework of Linear Fractional Transformations (LFT s) that provide an elegant approach to real number arithmetic [3,13,7,10,8,2]. One dimensional LFT s are used as digits and to implement basic unary functions, while two dimensional LFT s provide binary operations such as addition and multiplication, and can be combined to obtain infinite expression trees denoting transcendental functions. In Section 2, we present the LFT ....

....operations of exponential and square root, providing an analysis of the input complexity of p x for all x in [0; 1] including the two end points 0 and 1. 2 Exact Real Arithmetic by Linear Fractional Transformations In this section, we recall the framework of exact real arithmetic via LFT s [3,13,7]. In contrast to previous work of the author [4,6,5] we do not follow exactly the version used by the group of Edalat and Potts at Imperial College [10,8,9,12,2] but change the base interval from [0; 1] to [ Gamma1; 1] The reasons for this change and its pros and cons are discussed in Section ....

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J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990. 21

....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]. One dimensional LFT s x 7 ax c bx d are used as digits and to implement basic unary functions, while two dimensional 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 ....

....to combine many digits in a small basis to one digit in a large basis. By this method, the complexity is reduced to that of multiplying two n bit integers. 2 Exact Real Arithmetic by Linear Fractional Transformations In this section, we present the framework of exact real arithmetic via LFT s [5,14,9], specialised to the version used by the group of Edalat and Potts at Imperial College [12,10,11,13,4] 2.1 LFT s and Matrices General Linear Fractional Transformations (LFT s) are functions x 7 ax c bx d from reals to reals, parameterised by real numbers a, b, c, and d. In this paper, we ....

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J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990. 20

....to estimate [2] This problem, This work has been supported by EPSRC. with all its implications, has remained the main unresolved issue in computer arithmetic. Two main approaches have been proposed for exact real arithmetic as an alternative to floating point: The first, introduced by Vuillemin [19], uses redundant continued fractions to represent real numbers and has been implemented by Lester [10] The other approach, by Boehm and Cartwright, uses redundant sequences of Badic numbers to represent reals and has been implemented by Menissier Morain [11] There has been no attempt to provide ....

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990. 12

.... a 0 1 j j a1 b1 1 j j b1 b2 a 2 1 j j b2 b1 b3 a 3 1 j j b1 b3 b2 b4 a 4 Converting simple continued fractions with positive and negative integer quotients to regular continued fractions is achieved by applying the following equivalence. Equivalence 2 (see [3] 6][17]) For any fragment of a simple continued fraction with positive and negative partial quotients, a i ; 1; a i 2 ; a i 1; a i 2 1; a i 3 ; a i 4 ; a i ; 0; a i 2 ; a i a i 2 ; The theorem of Lochs gives information relating decimal numbers to ....

....as MapleV[21] compute minimax coe cients a ij automatically. Many elementary functions have straightforward simple continued fraction expansions the rational equivalent to Taylor series expansion around a point x 0 . For example: e x ; log(1 x) 1 x) n ; and tan 1 (x) are shown in [17]. The following theorem on the convergence of regular continued fractions ensures the convergence of the rational approximation algorithms with regular CF input. Theorem 6 Every regular continued fraction converges to a real number. For simple continued fractions it is much harder to guarantee ....

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J.E. Vuillemin, Exact Real Computer Arithmetic with Continued Fractions, IEEE Trans. on Computers, Vol. 39, No. 8, Aug. 1990.

....et al. [27] Boehm an Cartwright [2] Nielsen et al. [20] Menissier Morain [18] Di Gianantonio [7] and Escard o [4] The last two authors studied extensions of PCF with a real number data type. ii) Continued fraction expansions proposed by Gosper [8] developed by Peyton Jones [12] and Vuillemin [25] and advanced more recently by Kornerup et al. [17, 15, 14, 16] iii) Infinite composition of linear fractional transformations (also known as homographies or Mobius transformations) generalises the other two frameworks as demonstrated by Vuillemin [25] Nielsen et al. [20] showed that this ....

.... developed by Peyton Jones [12] and Vuillemin [25] and advanced more recently by Kornerup et al. [17, 15, 14, 16] iii) Infinite composition of linear fractional transformations (also known as homographies or Mobius transformations) generalises the other two frameworks as demonstrated by Vuillemin [25]. Nielsen et al. [20] showed that this framework can be used to represent quasi normalised floating point [27] Potts et al. [24] have developed a framework for exact real arithmetic in which the extended real numbers are represented by the composition of linear fractional transformations with ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on computers, 39(8):1087--1105, August 1990.

....a form of exact computation is feasible. In this case the results of the computations can be obtained with arbitrary precision. The exact computation on real numbers has been studied by several authors. Here we just mention Boehm and Cartwright ( 2, 3] Ko ( 9] Martin Lof ( 12] Vuillemin ([16]) and Weihrauch ( 17] In [2] and [6] an approach to real number computation via lazy functional languages is considered. In this approach a real number is represented by an infinite lazy sequence of digits. A real function is implemented by a lazy program that receives as input a lazy stream of ....

J. Vuillemin. Exact real computer arithmetic with continued fraction. In Proc. A.C.M. conference on Lisp and functional Programming, pages 14--27, 1988.

....of computation. In Lisp, the logical existing mechanism to use is is analogous to catch and throw, implemented through the error system. An example of such a technique is given in Appendix I. 5 Interval Arithmetic A useful application of extended rational numbers is in interval arithmetic [3] [11], 8] 7] 10] The model for interval arithmetic we use comprises both projective and affine real models: the projective model is implemented for intervals themselves, although an affine model is used for the endpoints. Let r, s, p, and q denote any extended affine rational numbers. If r s, ....

Jean Vuillemin, Exact Real Computer Arithmetic with Continued Fractions. IEEE Trans. on Cmptrs. 39 1990, 1087--1105.

.... of linear maps proposed by Avizienis [1] and appeared in the work of Watanuki et al. [23] Boehm an Cartwright [2] Di Gianantonio [5] Escardo [4] Nielsen et al. [18] and Menissier Morain [16] ii) Continued fraction expansions proposed by Gosper [7] developed by Peyton Jones [10] and Vuillemin [21] and advanced more recently by Kornerup et al. [15, 13, 12, 14] iii) Infinite composition of linear fractional transformations (also known as homographies or Mobius transformations) generalises the other two frameworks as demonstrated by Vuillemin [21] Nielsen et al. [18] showed that this ....

.... developed by Peyton Jones [10] and Vuillemin [21] and advanced more recently by Kornerup et al. [15, 13, 12, 14] iii) Infinite composition of linear fractional transformations (also known as homographies or Mobius transformations) generalises the other two frameworks as demonstrated by Vuillemin [21]. Nielsen et al. [18] showed that this framework can be used to represent quasi normalised floating point [23] We introduce here a new, feasible and incremental representation of the extended real numbers based on the composition of linear fractional transformations with either all non negative or ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on computers, 39(8):1087--1105, August 1990.

....never published, it has remained available throughout the intervening years, it was widely read, and sometimes even cited. But while the algorithms derived in it have been acknowledged as the Pratt source of inspiration for some of the most interesting modern approaches to exact real arithmetic [5,10,20], the underlying coalgebraic idea seems to have gone unnoticed. While hoping to point to this conceptual link, we must add that the constructions on the following pages should not be taken as a rational reconstruction of the Hakmem view of reals. In fact, they were obtained while we were trying to ....

J. Vuillemin, Exact real computer arithmetic with continued fractions. IEEE Trans. Comp. 39(1990) 1087--1105

.... several extensions of the programming language PCF with a real number data type [6,16,28] and a framework and an implementation of a package for exact real number computation [27,13] This latter work is based on the one hand on continued fractions and linear fractional transformations as in [38,24] and on the other hand on the domain of intervals. These promising results suggest that a marriage of domain theory and computable analysis will indeed be fruitful for both subjects. The recent survey paper [8] gives an overview of these applications of continuous domains. In this paper, we start ....

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990.

....constructive analysis and computable analysis. Although the theory of computable analysis can be considered a well developed subject, there have been so far very few attempts of implementing computable analysis on digital computers, see Boehm and Cartwright, Grue, Vuillemin, 4] 5] 9] [25]. Such implementations should lead to the realization of exact real number computation . In ordinary practice the computation on real numbers is performed by approximating real numbers by a subset of the rational numbers and by approximating the arithmetic on real numbers by a limited precision ....

....of representations a) or b) The representation d) is similar to the standard digited representation. The main difference consists in introducing negative digits. This representation has been studied in Avizienis [2] Boehm [4] and Wiedmer [28] The representation e) is developed in Vuillemin [25] and is similar to the standard continued fraction representation. The only difference is that in the standard continued fraction notation only natural numbers are used. In this case, however, negative integers are also used. The representations described above do not make explicit use of ....

J. Vuillemin, "Exact Real Computer Arithmetic with Continued Fraction." Proc. A.C.M. conference on Lisp and functional Programming (1988) 1427.

....clearly that the second one is much faster than the first one. We studied this second representation and now we propose a complete and entirely proved set of algorithms for all elementary functions. This work leads to an implementation in the Caml implementation of the ML language. Finaly, in [23, 24, 25], Vuillemin represents real numbers by contined fractions, with the underlying idea that continued fractions are the closest rational numbers to the real numbers. However, apart from the fact that these algorithms are principaly not proved, this representation is rather inadequate to current ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on computers 39, 8 (August 1990), pp. 1087--1105.

....clearly that the second one is much faster than the first one. We studied this second representation and now we propose a complete and entirely proved set of algorithms for all elementary functions. This work leads to an implementation in the Caml implementation of the ML language. Finaly, in [23, 24, 25], Vuillemin represents real numbers by contined fractions, with the underlying idea that continued fractions are the closest rational numbers to the real numbers. However, apart from the fact that these algorithms are principaly not proved, this representation is rather inadequate to current ....

J. Vuillemin. Exact real computer arithmetic with continued fractions. Research report 760, INRIA, 1987.

....to the value of the function at a given computable real number up to any threshold of accuracy. However, the computation is not incremental in the sense that to obtain any more accurate approximation one has to compute from scratch. Edalat and Potts Kornerup and Matula [9] and Vuillemin [16], proposed a representation of computable real numbers by redundant continued fractions and presented various incremental algorithms for basic arithmetic operations using the earlier work of Gosper [7] and for some transcendental functions. Any continued fraction expansion of a real number can be ....

....n ) with r n = a n 1 b n 1 a n 2 b n 2 a n 3 . and OE i (x) a i b i x for 0 i n. One can therefore identify the original continued fraction for r with the infinite composition OE 0 OE 1 OE 2 Delta Delta Delta. Such a representation of real numbers was already present in [16]. Nielson and Kornerup [12] later developed a general framework for exact arithmetic by representing real numbers by redundant infinite composition of linear fractional transformations (lft) Escardo s extension of PCF [6] is based on the redundant representation of a real number in [0; 1] as an ....

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, 1990.

....a mistake which we could not have committed. 4 Continued fractions and Pad e approximants The power series are not the only infinite data structures which can be processed by lazy algorithms, although here the co recursion is particularly simple. But already in 1972 Gosper [19] see also [13, 20]) has shown that the arithmetic of continued fractions can be very elegantly realized through incremental stream processing. We could give here a particularly simple realization of such arithmetic package, but for algebraic manipulation it might be more interesting to work with series than with ....

Jean Vuillemin, Exact Real Computer Arithmetic with Continued Fractions, IEEE Transactions on Computers 39(8), (1990), pp. 1087 -- 1105.

....is a degenerate interval representing a point. An ordinary interval will be designated by the symbol I normal , and the type that includes plus and minus infinity by the symbol I 1 . In order to treat 1 as any other number, we use the stereographic representation of the real line as defined in [96] (see Fig. 8.1) With this representation minus and plus infinity are treated as one point, and the interval j x j 1 is the upper part of the circle, and the reciprocal of this interval (e.g. j x j 1) is the lower part. We will adopt an alternative definition of the width of an interval [x; y] ....

Vuillemin, Jean: "Exact Real Computer Arithmetic with Continued Fractions "; IEEE Transactions on Computers, C-39, 8 (1990), 1087--1105. 248

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J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087--1105, Aug. 1990.

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Jean Vuillemin, Exact real computer arithmetic with continued fractions, IEEE Transactions on Computers 39 (1990), no. 8, 1087-1105.

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Jean Vuillemin, Exact real computer arithmetic with continued fractions, IEEE Transactions on Computers 39 (1990), no. 8, 1087-1105.

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Jean E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087-1105, August 1990.

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Jean E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087-1105, aug 1990.

No context found.

J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087-1105, 1990.

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