Joris van der Hoeven
ABSTRACT
Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. Based on earlier, theoretical work, the author has started to develop a systematic C++ library Mmxlib for mathematically correct computations with more analytic objects, like complex numbers and analytic functions. While implementing the library, we found that several of our theoretical ideas had to be further improved or adapted. In this paper, we report on the current implementation, we present several new results and suggest directions for future improvements.
- G. Alefeld and J. Herzberger. Introduction to interval analysis. Academic Press, 1983.Google Scholar
- J. Blanck. General purpose exact real arithmetic. Technical Report CSR 21-200, Luleå University of Technology, Sweden, 2002. http://www.sm.luth.se/~jens/.Google Scholar
- J. Blanck, V. Brattka, and P. Hertling, editors. Computability and complexity in analysis, volume 2064 of Lect. Notes in Comp. Sc. Springer, 2001.Google Scholar
- A. Edalat and P. Sünderhauf. A domain-theoretic approach to real number computation. TCS, 210:73--98, 1998. Google ScholarDigital Library
- A. Gaganov. Computational complexity of the range of the polynomial in several variables. Cybernetics, pages 418--425, 1985.Google Scholar
- T. Granlund et al. GMP, the GNU multiple precision arithmetic library. http://www.swox.com/gmp, 1991-2006.Google Scholar
- M. Grimmer, K. Petras, and N. Revol. Multiple precision interval packages: Comparing different approaches. Technical Report RR 2003-32, LIP, École Normale Supérieure de Lyon, 2003.Google Scholar
- G. Hanrot, V. Lefèvre, K. Ryde, and P. Zimmermann. MPFR, a C library for multiple-precision floating-point computations with exact rounding. http://www.mpfr.org, 2000-2006.Google Scholar
- V. Kreinovich. For interval computations, if absolute accuracy is NP-hard, then so is relative accuracy+optimization. Technical Report UTEP-CS-99-45, UTEP-CS, 1999.Google Scholar
- V. Kreinovich and S. Rump. Towards optimal use of multi-precision arithmetic: a remark. Technical Report UTEP-CS-06-01, UTEP-CS, 2006.Google ScholarCross Ref
- B. Lambov. The RealLib project. http://www.brics.dk/~barnie/RealLib, 2001-2006.Google Scholar
- V. Ménissier-Morain. Arbitrary precision real arithmetic: design and algorithms. Unpublished.Google Scholar
- N. Müller. iRRAM, exact arithmetic in C++. http://www.informatik.uni-trier.de/iRRAM/, 2000-2006.Google Scholar
- R. O'Connor. A monadic, functional implementation of real numbers. Technical report, Institute for Computing and Information Science, Radboud University Nijmegen, 2005.Google Scholar
- N. Revol. MPFI, a multiple precision interval arithmetic library. http://perso.ens-lyon.fr/nathalie.revol/software.html, 2001-2006.Google Scholar
- S. Rump. Fast and parallel inteval arithmetic. BIT, 39(3):534--554, 1999.Google ScholarCross Ref
- A. Turing. On computable numbers, with an application to the Entscheidungsproblem. Proc. London Maths. Soc., 2(42):230--265, 1936.Google Scholar
- J. van der Hoeven. GMPX, a C-extension library for gmp. http://www.math.u-psud.fr/~vdhoeven/, 1999. No longer maintained.Google Scholar
- J. van der Hoeven. Relax, but don't be too lazy. JSC, 34:479--542, 2002. Google ScholarDigital Library
- J. van der Hoeven. Computations with effective real numbers. TCS, 351:52--60, 2006. Google ScholarDigital Library
- J. van der Hoeven et al. Mmxlib: the standard library for Mathemagix, 2002-2006. http://www.mathemagix.org/mml.html.Google Scholar
- K. Weihrauch. Computable analysis. Springer-Verlag, Berlin/Heidelberg, 2000. Google ScholarDigital Library
Index Terms
- Effective real numbers in Mmxlib
Recommendations
Real algebraic numbers and polynomial systems of small degree
Based on precomputed Sturm-Habicht sequences, discriminants and invariants, we classify, isolate with rational points, and compare the real roots of polynomials of degree up to 4. In particular, we express all isolating points as rational functions of ...
Towards faster real algebraic numbers
Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)This paper presents a new encoding scheme for real algebraic number manipulations which enhances current Axiom's real closure. Algebraic manipulations are performed using different instantiations of sub-resultant-like algorithms instead of Euclidean-...
Effective Noether Irreducibility Forms and Applications
Selected papers of the 23rd annual ACM symposium on Theory of computingUsing recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a ...
Comments