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 .

**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****3.**Henri Cohen,*A course in computational algebraic number theory*, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR**1228206****4.**Felix Klein,*Elementary mathematics from an advanced standpoint*:*Geometry*, reprint (New York: Dover, 1939); see §II*ff*.**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****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****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****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**

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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:
https://doi.org/10.1090/S0025-5718-03-01518-7

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