Computational Aspects of Various Number Representations

Abstract

Modern mathematical modeling problems, which exploit the power of supercomputers to the utmost extent, require increasing digit capacity of real numbers. However, the traditional representation of the reals is not efficient enough in the case of large bit sizes due to carry propagation. To make the arithmetic operations digit-wise parallel, several alternative numeration systems has been proposed and intensively studied during the last decades, starting with the seminal result by L. E. J. Brouwer a century ago. This paper continues this line of research and reveals some computational aspects of various number representation systems: exponential, recurrent (or linear), aliquot, overlaying exponential. Several classic results are generalized to broader classes of numeration systems. In particular, the representability theorem by A. Rényi for exponential systems is generalized to include convergent and nonconvergent numeration systems, uniform and non-uniform bases, and arbitrary sets of digits. As a contribution for practical use, a symmetric optimal positional overlay system is proposed, and an addition algorithm is described.