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On-line Algorithms for Multiplication and Division in Real and Complex Numeration Systems by Frougny, Christiane; Pavelka, Marta; Pelantova, Edita; Svobodova, Milena is a scholarly article available to read on EtoBox.

What is On-line Algorithms for Multiplication and Division in Real and Complex Numeration Systems about?

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

Author

Frougny, Christiane; Pavelka, Marta; Pelantova, Edita; Svobodova, Milena

Published

2016

Language

EN