On-line algorithms for multiplication and division in real and complex numeration systems

TL;DR

This paper generalizes on-line multiplication and division algorithms for real and complex base numeration systems, establishing conditions under which these algorithms are feasible and demonstrating their linear time complexity with concrete examples.

Contribution

It introduces a generalized framework for on-line arithmetic in real and complex bases, defining the OL Property and providing new feasibility conditions for these algorithms.

Findings

The OL Property ensures on-line multiplication and division are feasible.

For real bases, OL Property holds if the digit set size exceeds the base magnitude.

For complex bases, OL Property depends on the base's norm and the digit set size.

Abstract

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\beta$ such that $|\beta|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit. In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $\beta$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(\beta, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $\beta$ and a digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > |\beta|$. For a complex base $\beta$ and symmetric digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > \beta\overline{\beta} + |\beta + \overline{\beta}|$. Provided that addition and subtraction are realizable in parallel in the system $(\beta, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $\beta=\frac{3+\sqrt{5}}{2}$ with digits $A=\{-1,0,1\}$; base $\beta=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base $\beta = -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + \omega$, where $\omega = \exp{\frac{2i\pi}{3}}$, with digits $A = \{0, \pm 1, \pm \omega, \pm \omega^2 \}$.