General Approach to the Description of Computer Number Systems

Abstract

General relationships, algorithms, and differences for three numerical systems used in the field of computing technology are analyzed: positional, polyadic and modular. Their power-law modifications are obtained. A single basic algorithm for the formation of components of vectors representing numerical values in these systems, which are residues in relatively simple modules from the set of bases of a numerical system, is analyzed with an estimate of complexity. Vector representation of numerical values is a mathematical and algorithmic basis for generating arithmetic computer formats focused on high-performance parallel processing. Generalization of the representation of numerical values in three computer number systems in the form of vectors with an indication of the limits of change of representation components is intended for the analysis and parallelization of complex computational processes. Analysis from a unified position of computer number systems makes it possible to find computational problems for which the use of individual of them is effective. For tasks in which it is necessary to simultaneously process data in various formats associated with number systems, a correct description from a general point of view of the corresponding data formats is used. The analysis of number systems is intended for the development of mathematical structures as a base of software and algorithmic tools and the generation of machine data formats associated with these systems, designed to increase the efficiency of complex computational processes and assess their convergence for calculations in large computer ranges, focused on serial computers and scalable multiprocessor systems SIMD — architectures.