The Appearance of Big Integers in Exact Real Arithmetic based on Linear Fractional Transformations (1997)
| Venue: | In Proc. Foundations of Software Science and Computation Structures (FoSSaCS '98), volume 1378 of LNCS |
| Citations: | 7 - 4 self |
BibTeX
@INPROCEEDINGS{Heckmann97theappearance,
author = {Reinhold Heckmann},
title = {The Appearance of Big Integers in Exact Real Arithmetic based on Linear Fractional Transformations},
booktitle = {In Proc. Foundations of Software Science and Computation Structures (FoSSaCS '98), volume 1378 of LNCS},
year = {1997},
pages = {172--188},
publisher = {Springer-Verlag}
}
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Abstract
. 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...
