We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with non-negative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S¹, is expressed as the image of the base interval [0�1] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first so-called sign matrix determines an interval on which the real number lies. The subsequent so-called digit matrices have non-negative integer coe cients and successively re ne that interval. Based on the classi cation of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S¹ by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.

In this paper, we incorporate a representation of the non-negative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.

: We develop a formal account of digit serial number representations by describing them as strings from a language. A prefix of a string represents an interval approximating a number by enclosure. Standard on-line representations are shown to be a special case of the general digit serial representations. Matrices are introduced as representations of intervals and a finite-state transducer is used for mapping strings into intervals. Homographic and bi-homographic functions are used for representing basic arithmetic operations on digit serial numbers, and finally a digit serial representation of floating point numbers is introduced. Key Words: Computer Arithmetic, On-line Computation, Number Representations, Redundant Digit sets, Continued Fractions, Intervals. Category: B.2 1 Introduction A number is usually represented as a string of digits belonging to some digit set \Sigma . The number representation specifies a function that maps the string to its value. In the context of this pa...

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One possible approach to exact real arithmetic is to use linear fractional transformations (LFT's) to represent real numbers and computations on real numbers. Recursive expressions built from LFT's are only convergent (i.e., denote a well-defined real number) if the involved LFT's are sufficiently contractive. In this paper, we define a notion of contractivity for LFT's. 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 expression trees ...

. 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]. 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 functions. In Section 2, we present the details of the LFT approach. This provides the background for understanding the r...

Computers manipulate approximations of real numbers, called floating-point numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floating-point operations lose so much precision that they quickly become irrelevant. In this article, we review some of the problems one can encounter, focussing on the IEEE754-1985 norm. We give a (sketch of a) semantics of its basic operations then abstract them (in the sense of abstract interpretation) to extract information about the possible loss of precision. The expected application is abstract debugging of software ranging from simple on-board systems (which use more and more on-the-shelf micro-processors with floating-point units) to scientific codes. The abstract analysis is demonstrated on simple examples and compared with related work. 1

Abstract: A binary representation of the rationals derived from their continued fraction expansions is described and analysed. The concepts \adjacency", \mediant " and \convergent " from the literature on Farey fractions and continued fractions are suitably extended to provide a foundation for this new binary representation system. Worst case representation-induced precision loss for any real number by a xed length representable number of the system is shown to be at most 19 % of bit word length, with no precision loss whatsoever induced in the representation of any reasonably sized rational number. The representation is supported by a computer arithmetic system implementing exact rational and approximate real computations in an on-line fashion.

this paper. By introducing an extra termination symbol, which signals that an operand was merely terminated due to its length exceeding some bound, operands can be kept as intervals, representing an imprecise operand. Operands terminated in the ordinary way can be taken to represent exact numbers. The cube modeling a function of two variables, can be generalized to a hypercube modeling a poly-homographic function of n variables. For n = 3 the function is defined as:

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