Translation of Taylor series into LFT expansions

Abstract

In Exact Real Arithmetic, real numbers are represented as potentially infinite streams of information units, called digits. In this paper, we work in the framework of Linear Fractional Transformations (LFT’s, also known as Möbius 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, 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 apower series using Homer’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.