Scalability of Algorithms for Arithmetic Operations in Radix Notation∗ Anatoly V. Panyukov y Department of Computational Mathematics and Informatics, South Ural State University, Chelyabinsk, Russia
[email protected] Abstract We consider precise rational-fractional calculations for distributed com- puting environments with an MPI interface for the algorithmic analysis of large-scale problems sensitive to rounding errors in their software imple- mentation. We can achieve additional software efficacy through applying heterogeneous computer systems that execute, in parallel, local arithmetic operations with large numbers on several threads. Also, we investigate scalability of the algorithms for basic arithmetic operations and methods for increasing their speed. We demonstrate that increased efficacy can be achieved of software for integer arithmetic operations by applying mass parallelism in het- erogeneous computational environments. We propose a redundant radix notation for the construction of well-scaled algorithms for executing ba- sic integer arithmetic operations. Scalability of the algorithms for integer arithmetic operations in the radix notation is easily extended to rational- fractional arithmetic. Keywords: integer computer arithmetic, heterogeneous computer system, radix notation, massive parallelism AMS subject classifications: 68W10 1 Introduction Verified computations have become indispensable tools for algorithmic analysis of large scale unstable problems (see e.g., [1, 4, 5, 7, 8, 9, 10]). Such computations require spe- cific software tools; in this connection, we mention that our library \Exact computa- tion" [11] provides appropriate instruments for implementation of such computations ∗Submitted: January 27, 2013; Revised: August 17, 2014; Accepted: November 7, 2014. yThe author was supported by the Federal special-purpose program "Scientific and scientific-pedagogical personnel of innovation Russia", project 14.B37.21.0395.