Article

Effective real numbers in Mmxlib

Author:

Joris van der HoevenAuthors Info & Claims

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

Pages 138 - 145

https://doi.org/10.1145/1145768.1145795

Published: 09 July 2006 Publication History

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.

References

[1]

G. Alefeld and J. Herzberger. Introduction to interval analysis. Academic Press, 1983.

[2]

J. Blanck. General purpose exact real arithmetic. Technical Report CSR 21-200, Luleå University of Technology, Sweden, 2002. http://www.sm.luth.se/~jens/.

[3]

J. Blanck, V. Brattka, and P. Hertling, editors. Computability and complexity in analysis, volume 2064 of Lect. Notes in Comp. Sc. Springer, 2001.

[4]

A. Edalat and P. Sünderhauf. A domain-theoretic approach to real number computation. TCS, 210:73--98, 1998.

Digital Library

[5]

A. Gaganov. Computational complexity of the range of the polynomial in several variables. Cybernetics, pages 418--425, 1985.

[6]

T. Granlund et al. GMP, the GNU multiple precision arithmetic library. http://www.swox.com/gmp, 1991-2006.

[7]

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.

[8]

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.

[9]

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.

[10]

V. Kreinovich and S. Rump. Towards optimal use of multi-precision arithmetic: a remark. Technical Report UTEP-CS-06-01, UTEP-CS, 2006.

[11]

B. Lambov. The RealLib project. http://www.brics.dk/~barnie/RealLib, 2001-2006.

[12]

V. Ménissier-Morain. Arbitrary precision real arithmetic: design and algorithms. Unpublished.

[13]

N. Müller. iRRAM, exact arithmetic in C++. http://www.informatik.uni-trier.de/iRRAM/, 2000-2006.

[14]

R. O'Connor. A monadic, functional implementation of real numbers. Technical report, Institute for Computing and Information Science, Radboud University Nijmegen, 2005.

[15]

N. Revol. MPFI, a multiple precision interval arithmetic library. http://perso.ens-lyon.fr/nathalie.revol/software.html, 2001-2006.

[16]

S. Rump. Fast and parallel inteval arithmetic. BIT, 39(3):534--554, 1999.

[17]

A. Turing. On computable numbers, with an application to the Entscheidungsproblem. Proc. London Maths. Soc., 2(42):230--265, 1936.

[18]

J. van der Hoeven. GMPX, a C-extension library for gmp. http://www.math.u-psud.fr/~vdhoeven/, 1999. No longer maintained.

[19]

J. van der Hoeven. Relax, but don't be too lazy. JSC, 34:479--542, 2002.

Digital Library

[20]

J. van der Hoeven. Computations with effective real numbers. TCS, 351:52--60, 2006.

Digital Library

[21]

J. van der Hoeven et al. Mmxlib: the standard library for Mathemagix, 2002-2006. http://www.mathemagix.org/mml.html.

[22]

K. Weihrauch. Computable analysis. Springer-Verlag, Berlin/Heidelberg, 2000.

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Cited By

Johansson FChyzak FLabahn G(2021)CalciumProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465513(225-232)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465513
Chevillard S(2011)Automatic Generation of Code for the Evaluation of Constant Expressions at Any Precision with a Guaranteed Error BoundProceedings of the 2011 IEEE 20th Symposium on Computer Arithmetic10.1109/ARITH.2011.38(225-232)Online publication date: 25-Jul-2011
https://dl.acm.org/doi/10.1109/ARITH.2011.38
Yu JYap CDu ZPion SBrönnimann H(2010)The design of core 2Proceedings of the Third international congress conference on Mathematical software10.5555/1888390.1888421(121-141)Online publication date: 13-Sep-2010
https://dl.acm.org/doi/10.5555/1888390.1888421
Show More Cited By

Index Terms

Effective real numbers in Mmxlib
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

July 2006

374 pages

ISBN:1595932763

DOI:10.1145/1145768

General Chair:
Barry Trager
IBM T.J. Watson Research Center, USA

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 09 July 2006

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ISSAC06: International Symposium on Symbolic and Algebraic Computation

July 9 - 12, 2006

Genoa, Italy

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Johansson FChyzak FLabahn G(2021)CalciumProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465513(225-232)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465513
Chevillard S(2011)Automatic Generation of Code for the Evaluation of Constant Expressions at Any Precision with a Guaranteed Error BoundProceedings of the 2011 IEEE 20th Symposium on Computer Arithmetic10.1109/ARITH.2011.38(225-232)Online publication date: 25-Jul-2011
https://dl.acm.org/doi/10.1109/ARITH.2011.38
Yu JYap CDu ZPion SBrönnimann H(2010)The design of core 2Proceedings of the Third international congress conference on Mathematical software10.5555/1888390.1888421(121-141)Online publication date: 13-Sep-2010
https://dl.acm.org/doi/10.5555/1888390.1888421
Yu JYap CDu ZPion SBrönnimann H(2010)The Design of Core 2: A Library for Exact Numeric Computation in Geometry and AlgebraMathematical Software – ICMS 201010.1007/978-3-642-15582-6_24(121-141)Online publication date: 2010
https://doi.org/10.1007/978-3-642-15582-6_24

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