Abstract
In this paper we consider the problem of fast computation of n-ary products (and also sums of such products), for large n, over arbitrary precision integer or rational number domains. We analyze the complexity of different algorithms for such computations and point out computational framework for which such computations can be accelerated. A combination of binary splitting algorithm, accelerating algorithm based on the chains of recurrences technique and predicted cancellation is described.
Résumé
Dans cet article, nous considérons le probléme du calcul rapide de produits n-aires (et aussi de sommes de tels produits) d’entiers ou de rationnels en précision arbitraire, pour de grandes valeurs de n. Nous analysons la complexité de différents algorithmes effectuant ces calculs et nous indiquons pour quels types de problémes de tels calculs peuvent être accélérés. Une combinaison d’algorithme de dichotomie, d’algorithme d’accélération basé sur la technique des chaînes de récurrences et de prédiction d’annulation est dé;crite.
This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.
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References
Amdeberhan T., Zeilberger D. Hypergeometric Series Acceleration via the WZ Method, Electronic Journal of Combinatorics (Wilf Festschrift Volume) 4 (1997).
Bini D., Pan V. Polynomial and Matrix Computations Fundamental Algorithms, vol.1, Birkhauser, 1994.
Borwein J., Borwein P. Pi and the AGM. Wiley, 1987.
Char B.W., Geddes K.O. (and others) Maple-V. Language Reference Manual. Springer-Verlag, 1991.
Cohen H. A Course in Computational Algebraic Number Theory. Springer-Verlag, 1995.
Haible B., Papanikolaou T. Fast multiprecision evaluation of series of rational numbers. Technical report TI-97/7, University of Darmstadt, 1997.
Kislenkov V., Mitrofanov V., Zima E. How fast can we compute products? Proc. of ISSAC’99, Vancouver, Canada, ACM Press, pp. 75–82.
Knuth D.E. The art of computer programming. V. 2. Seminumerical Algorithms. Third edition. Addison-Wesley, 1997.
Shoup V. http://www.cs.visc.edu/'shoup/ntl/.
Wedeniwsld S. http://www.math.temple.edu/zeilberg/mamarim/mamarimhtml/Zeta3.txt.
Zima E.V. Automatic Construction of Systems of Recurrence Relations. USSR Comput. Maths. Math. Phys., Vol. 24, N 6, 1984, pp. 193–197.
Zima E.V. Simplification and Optimization Transformations of Chains of Recurrences. Proc. of ISSAC’95, Montreal, Canada, ACM Press, pp. 42–50.
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Zima, E.V. (2000). On Acceleration of Multiprecision Computation of Products and Sums of Products of Rational Numbers. In: Krob, D., Mikhalev, A.A., Mikhalev, A.V. (eds) Formal Power Series and Algebraic Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04166-6_76
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DOI: https://doi.org/10.1007/978-3-662-04166-6_76
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