We consider precise rational-fractional calculations for distributed computing environments with an MPI interface for the algorithmic analysis of large-scale problems sensitive to rounding errors in their software implementation. We can achieve additional software efficacy through applying heterogeneous computer systems that execute, in parallel, local arithmetic operations with large numbers o...

Interval arithmetic was introduced by Ramon Moore [Moo66] in the 1960s as an approach to bound rounding errors in mathematical computation. The theory of interval analysis emerged considering the computation of both the exact solution and the error term as a single entity, i.e. the interval. Though a simple idea, it is a very powerful technique with numerous applications in mathematics, compute...

New parallel algorithms for parsing arithmetic expressions on mesh-connected, shuffle-connected, cube-connected, and cube-connected cycles models of parallel computation are presented. On the meshconnected computer, the algorithm requires n processors and O(&) time; on the other models, n processors and O(log2 n ) time are required. Index Terns-Arithmetic expressions, parallel processing, parsi...

Most of the benefit of capacity planning for computer systems comes from applying common sense principles to easily understood (if not easily measured) data. If you can estimate business growth you need only simple arithmetic to figure out how busy your computer will be. But sometimes the fact that the system is subtle does matter: it’s hard to predict response times. Then you must use mathemat...

A binary representation of the rationals derived from their continued fraction expansions is described and analysed. The concepts \adjacency", \mediant" and \convergent" from the literature on Farey fractions and continued fractions are suitably extended to provide a foundation for this new binary representation system. Worst case representation-induced precision loss for any real number by a x...

Mathematics requires precise inferences about abstract objects inaccessible to perception. How is this possible? One proposal is that mathematical reasoning, while concerned with entirely abstract objects, nevertheless relies on neural resources specialized for interacting with the world-in other words, mathematics may be grounded in spatial or sensorimotor systems. Mental arithmetic, for insta...

131 W. E. Deskins, Abstract Algebra. London: Collier-Macmillan, 1964. [4] I. Flores, The Logic of Computer Arithmetic. Englewood Cliffs, N.J.: Prentice-Hall, 1963. 151 H. L. Garner, "The residue number system," IRE Trans. Electron. Comput., vol. EC-8, pp. 140-147, June 1959. 161 N. S. Szabo and R. 1. Tanaka, Residue Arithmetic and its Applications to Computer Technology. New York: McGraw-Hill, ...

The natural arithmetic operand in a computer is the binary integer. However, the range of numbers that can be represented is limited by the computer’s word size. We cannot represent very large or very small numbers. For example, in a computer with a 32 bit word, the largest signed number is 2 – 1. The range is further diminished if some bits of the word are used for fractions. There are techniq...

Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches — symbolic-algebraic and interval-arithmetic — are used separately. Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the “size” of the end-points. In this paper we propose a methodology for “true” symbolic-algebraic manipulations ...

Adders and multipliers are two main units of the computer arithmetic processors and play an important role in reversible computations. The binary multiplier consists of two main parts, the partial products generation circuit (PPGC) and the reversible parallel adders (RPA). This paper introduces a novel reversible 4×4 multiplier circuit that is based on an advanced PPGC with Peres gates only. Ag...