Claus Fieker
ABSTRACT
We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a low-level C implementation. For specialised algorithms, we use Julia's efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic.
- E. Bach, Improved approximations for Euler products, Number theory (Halifax, NS, 1994), vol. 15, CMS Conference Proceedings, pages 13--28. Amer. Math. Soc., Providence, RI, 1995.Google Scholar
- K. Belabas, Topics in computational algebraic number theory, J. Théor. Nombres Bordeaux, 16(1):19--63, 2004.Google ScholarCross Ref
- K. Belabas, E. Friedman. Computing the residue of the Dedekind zeta function, Math. Comp., 84(291):357--369, 2015.Google ScholarCross Ref
- J. Bezanson, A. Edelman, S. Karpinski, V. B. Shah, Julia: A fresh approach to numerical computing, https://arxiv.org/abs/1411.1607Google Scholar
- J.-F. Biasse, C. Fieker, Subexponential class group and unit group computation in large degree number fields, LMS J. Comput. Math., 17(suppl. A):385--403, 2014.Google Scholar
- W. Bosma, J. J. Cannon, C. Fieker, A. Steel (Eds.), Handbook of Magma Functions, Edition 2.22 (2016).Google Scholar
- W. Brown, Null ideals and spanning ranks of matrices, Comm. Algebra 26, (1998), pp. 2401--2417.Google ScholarCross Ref
- H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. Google ScholarDigital Library
- A. Danilevsky, The numerical solution of the secular equation, (Russian),Matem. Sbornik, 44(2), 1937. pp. 169--171.Google Scholar
- W. Decker, G. M. Greuel, G. Pfister and H. Schönemann, Singular -- A computer algebra system for polynomial computations, http://www.singular.uni-kl.deGoogle Scholar
- E. Dobrowolski, On the maximal modulus of conjugates of an algebraic integer, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 26(4):291--292, 1978.Google Scholar
- L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, P. Zimmermann, MPFR: A multiple-precision binary floating-point library with correct rounding, ACM Trans. Math. Soft., vol. 33, (2), 13, (2007), pp. 1--15. Google ScholarDigital Library
- J. von zur Gathen, J. Gerhard, Modern Computer Algebra, Cambridge University Press, 1999. Google ScholarDigital Library
- M. Giesbrecht, A. Storjohann, Computing rational forms of integer matrices. Journal of Symbolic Computation, 2002, v. 34(3), pp. 157--172. Google ScholarDigital Library
- T. Granlund and the GMP development team, The GNU Multi Precision Arithmetic Library, http://gmplib.org/ Google ScholarDigital Library
- W. Hart, ANTIC: Algebraic Number Theory in C, Computer algebra Rundbrief 56, Fachgruppe Computer algebra, 2015, http://www.fachgruppe-computeralgebra.de/data/CA-Rundbrief/car56.pdfGoogle Scholar
- W. Hart, F. Johansson, S. Pancratz, FLINT: Fast Library for Number Theory, http://www.flintlib.orgGoogle Scholar
- F. Johansson, Arb: A C library for ball arithmetic, http://arblib.orgGoogle Scholar
- M. Monagan, R. Pearce, Sparse polynomial division using a heap, Journal of Symbolic Computation, vol. 46, (7), July 2011, pp. 807--822. Google ScholarDigital Library
- M. Monagan, R. Pearce, Parallel sparse polynomial multiplication using heaps, ISSAC'09, ACM Press (2009), pp. 263--269. Google ScholarDigital Library
- M. Monagan, R. Pearce, Sparse polynomial powering using heaps, Gerdt V.P., Koepf W., Mayr E.W., Vorozhtsov E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg (2012), pp. 236--247. Google ScholarDigital Library
- PARI Group, PARI/GP version 2.9.0, Univ. Bordeaux, 2016, http://pari.math.u-bordeaux.fr/Google Scholar
- B. Parisse, R. De Graeve, Giac/Xcas version 1.2.2, 2016, http://www-fourier.ujf-grenoble.fr/ parisse/giac.htmlGoogle Scholar
- M. Pohst, H. Zassenhaus, Algorithmic algebraic number theory, Cambridge University Press, Cambridge, 1997. Google ScholarDigital Library
- Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.4), 2016, http://www.sagemath.orgGoogle Scholar
- R. J. Schoof, Quadratic fields and factorization, In Computational methods in number theory, Part II, volume 155 of Math. Centre Tracts, pages 235--286. Math. Centrum, Amsterdam, 1982.Google Scholar
- M. Soltys, Berkowitz's algorithm and clow sequences, The Electronic Journal of Linear Algebra, Int. Linear Algebra Society, vol 9, Apr. 2002, pp. 42--54.Google ScholarCross Ref
- A. Steel, A new algorithm for the computation of canonical forms of matrices over fields, J. Symbolic Computation (1997) 24, pp. 409--432. Google ScholarDigital Library
- E. Vinberg, A Course in Algebra, American Mathematical Society, 2003, pp. 229.Google Scholar
Index Terms
- Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language
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