Results

**1 - 6**of**6**### 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. ..."

Abstract
- Add to MetaCart

(Show Context)
which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy.

### Admissible Digit Sets and a Modified Stern-Brocot Representation

, 2004

"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "dig ..."

Abstract
- Add to MetaCart

We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "digit set" yields an admissible representation of [0; +1]. Furthermore we establish the productivity and correctness of the homographic algorithm for such "admissible" digit sets. In the second part of the paper we discuss representation of positive real numbers based on the Stern-Brocot tree. We show how we can modify the usual Stern-Brocot representation to yield a ternary admissible digit set.

### Translation of Taylor series into LFT expansions

, 1999

"... this paper, we work in the framework of Linear Fractional Transformations (LFT's, also known as Mobius transformations) that provide an elegant approach to real number arithmetic (Gosper 1972, Vuillemin 1990, Nielsen and Kornerup 1995, Potts and Edalat 1996, Edalat and Potts 1997, Potts 1998b). ..."

Abstract
- Add to MetaCart

(Show Context)
this paper, we work in the framework of Linear Fractional Transformations (LFT's, also known as Mobius transformations) that provide an elegant approach to real number arithmetic (Gosper 1972, Vuillemin 1990, Nielsen and Kornerup 1995, Potts and Edalat 1996, Edalat and Potts 1997, Potts 1998b). Onedimensional 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. Peter Potts (1998a, 1998b) derived these expression trees from continued fraction expansions of the transcendental functions. In contrast, we show how to derive LFT expression trees from power series expansions, which are available for a greater range of functions. In Section 2, we present the LFT approach in some detail. Section 3 contains the main results of the paper. We first derive an LFT expansion from a power series using Horner's scheme (Section 3.1). The results are not very satisfactory. Thus, we show how LFT expansions may be modified using algebraic transformations (Section 3.2). A particular such transformation, presented in Section 3.3, yields satisfactory results for standard functions, as shown in the final examples section 4.

### Computation with Real Numbers -- Exact Arithmetic, Computational Geometry and Solid Modelling

"... ..."

(Show Context)
### Admissible Digit Sets and a Modified Stern–Brocot Representation

"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “dig ..."

Abstract
- Add to MetaCart

(Show Context)
We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set ” yields an admissible representation of [0, +∞]. Furthermore we establish the productivity and correctness of the homographic algorithm for such “admissible” digit sets. In the second part of the paper we discuss representation of positive real numbers based on the Stern–Brocot tree. We show how we can modify the usual Stern–Brocot representation to yield a ternary admissible digit set.

### Arbitrary precision real arithmetic: design and algorithms Valerie Menissier-Morain

"... We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite B-adic numbers and for each classical function (rational, algebraic or transcendental), we describe ..."

Abstract
- Add to MetaCart

(Show Context)
We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite B-adic numbers and for each classical function (rational, algebraic or transcendental), we describe how to produce a sequence representing the result of the application of this function to its arguments, according to the sequences representing these arguments. For each algorithm we prove that the resulting sequence is a valid representation of the exact real result. This arithmetic is the rst abritrary precision real arithmetic with mathematically proved algorithms. Resume Nous proposons une representation des nombres reels calculables ainsi que des algorithmes pour les fonctions elementaires usuelles pour cette representation. Un nombre reel est represente par une suite de nombres B-adiques nis et pour chaque fonction classique (rationnelle, algebrique ou transcendante), nous montrons comment produire une suite representant le resultat a partir de suites representant les parametres. Pour chacun de ces algorithmes nous demontrons que la suite qui en resulte represente bien le resultat reel exact. Cette arithmetique est la premiere arithmetique reelle en precision arbitraire dotee d'un jeu complet d'algorithmes