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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A Note on NUCOMP


Author: Alfred J. van der Poorten
Journal: Math. Comp. 72 (2003), 1935-1946
MSC (2000): Primary 11Y40, 11E16, 11R11
Published electronically: April 29, 2003
MathSciNet review: 1986813
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Abstract | References | Similar Articles | Additional Information

Abstract: This note is a detailed explanation of Shanks-Atkin NUCOMP--composition and reduction carried out ``simultaneously''--for all quadratic fields, that is, including real quadratic fields. That explanation incidentally deals with various ``exercises'' left for confirmation by the reader in standard texts. Extensive testing in both the numerical and function field cases by Michael J Jacobson, Jr, reported elsewhere, confirms that NUCOMP as here described is in fact efficient for composition both of indefinite and of definite forms once the parameters are large enough to compensate for NUCOMP's extra overhead. In the numerical indefinite case that efficiency is a near doubling in speed already exhibited for discriminants as small as $10^7$.


References [Enhancements On Off] (What's this?)

  • 1. A. O. L. Atkin, Letter to Dan Shanks on the programs NUDUPL and NUCOMP, 12 December 1988; from the Nachlaß of D. Shanks and made available to me by Hugh C. Williams.
  • 2. Duncan A. Buell, Binary quadratic forms, Springer-Verlag, New York, 1989. Classical theory and modern computations. MR 1012948 (92b:11021)
  • 3. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206 (94i:11105)
  • 4. Felix Klein, Elementary mathematics from an advanced standpoint: Geometry, reprint (New York: Dover, 1939); see §IIff.
  • 5. Hermann Grassmann, A new branch of mathematics, Open Court Publishing Co., Chicago, IL, 1995. The Ausdehnungslehre of 1844 and other works; Translated from the German and with a note by Lloyd C. Kannenberg; With a foreword by Albert C. Lewis. MR 1637704 (99e:01015)
  • 6. W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. I, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Book I: Algebraic preliminaries; Book II: Projective space; Reprint of the 1947 original. MR 1288305 (95d:14002a)
  • 7. Michael J Jacobson Jr and Alfred J van der Poorten, ``Computational aspects of NUCOMP'', to appear in Claus Fieker and David Kohel eds, Algorithmic Number Theory (Proc. Fifth International Symposium, ANTS-V, Sydney, NSW, Australia July 2002), Springer Lecture Notes in Computer Science 2369 (2002), 120-133.
  • 8. H. W. Lenstra Jr., On the calculation of regulators and class numbers of quadratic fields, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, 1982, pp. 123–150. MR 697260 (86g:11080)
  • 9. Daniel Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR 0316385 (47 #4932)
    Mary Zimmerman, Matrix multiplication as an application of the principle of combinatorial analysis, Pi Mu Epsilon J. 6 (1975), no. 3, 166–175. MR 0389933 (52 #10762)
  • 10. Daniel Shanks, On Gauss and composition. I, II, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 163–178, 179–204. MR 1123074 (92e:11150)

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Additional Information

Alfred J. van der Poorten
Affiliation: ceNTRe for Number Theory Research, 1 Bimbil Pl. Killara, New South Wales 2071, Australia
Email: alf@math.mq.edu.au

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01518-7
PII: S 0025-5718(03)01518-7
Keywords: Binary quadratic form, composition
Received by editor(s): January 10, 2002
Published electronically: April 29, 2003
Additional Notes: The author was supported in part by a grant from the Australian Research Council
Article copyright: © Copyright 2003 American Mathematical Society