Abstract
Closed approximate rational arithmetic systems are described and their number theoretic foundations are surveyed. The arithmetic is shown to implicitly contain an adaptive single-to-double precision natural rounding behavior that acts to recover true simple fractional results. The probability of such recovery is investigated and shown to be quite favorable.
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This research was supported in part by the National Science Foundation under Grant MCS77-21510.
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References
Bartley, D. H., The Utilization of Hybrid Techniques in Computer Systems for Symbolic Mathematics, Master's Thesis, Univ. Texas, Austin, 1972.
Cabay, S. and Lam, T.P.L., "Congruence Techniques for the Exact Solution of Integer Systems of Linear Equations," TOMS 3(1977), 386–397.
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1960.
Horn, B. K. P., "Rational Arithmetic for Minicomputers," Software-Prac. and Exper. 8(1978), 171–176.
Howell, J. and Gregory, R. T., "Solving Linear Equations Using Residue Arithmetic II," Nordisk Tidskr. Inf. 10(1970), 23–37.
Knuth, D. E., The Art of Computer Programming, Vol. II, Addison-Wesley, Reading, 1969.
Kornerup, P. and Matula, D. W., "A Feasibility Analysis of Fixed-Slash Rational Arithmetic," Proc. 4th Sym. on Comp. Arith., IEEE (1978), 39–47.
Kulisch, U., "An Axiomatic Approach to Rounded Computations," Numerische Mathematik 18(1971), 1–17.
Matula, D. W., "Fixed-Slash and Floating-Slash Rational Arithmetic," Proc. 3rd Sym. on Comp. Arith., IEEE (1975), 90–91.
Matula, D. W. and Kornerup, P., "A Feasibility Analysis of Binary Fixed-Slash and Floating-Slash Number Systems," Proc. 4th Sym. on Comp. Arith., IEEE (1978), 29–38.
Yun, D. and Gustavson, F., "Fast Computation of Rational Hermite Interpolants and Solving Toeplitz Systems of Equations via the Extended Euclidean Algorithm," this volume.
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© 1979 Springer-Verlag Berlin Heidelberg
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Matula, D.W., Kornerup, P. (1979). Approximate rational arithmetic systems: Analysis of recovery of simple fractions during expression evaluation. In: Ng, E.W. (eds) Symbolic and Algebraic Computation. EUROSAM 1979. Lecture Notes in Computer Science, vol 72. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09519-5_89
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DOI: https://doi.org/10.1007/3-540-09519-5_89
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