Calcium: Computing in Exact Real and Complex Fields

ABSTRACT

Calcium is a C library for real and complex numbers in a form suitable for exact algebraic and symbolic computation. Numbers are represented as elements of fields Q(a₁,...,a_n) where the extension numbers a_k may be algebraic or transcendental. The system combines efficient field operations with automatic discovery and certification of algebraic relations, resulting in a practical computational model of R and C in which equality is rigorously decidable for a large class of numbers.

References

1. David H. Bailey, Jonathan M. Borwein, and Alexander D. Kaiser. 2014. Automated simplification of large symbolic expressions. Journal of Symbolic Computation, Vol. 60 (Jan. 2014), 120--136. https://doi.org/10.1016/j.jsc.2013.09.001
2. Hans-J Boehm. 2020. Towards an API for the real numbers. In Proceedings of the 41st ACM SIGPLAN Conference on Programming Language Design and Implementation. 562--576. https://doi.org/10.1145/3395658
3. Wieb Bosma, John Cannon, and Catherine Playoust. 1997. The Magma Algebra System I: The User Language. Journal of Symbolic Computation, Vol. 24, 3-4 (Sept. 1997), 235--265. https://doi.org/10.1006/jsco.1996.0125
4. Alin Bostan, Philippe Flajolet, Bruno Salvy, and É ric Schost. 2006. Fast computation of special resultants. Journal of Symbolic Computation, Vol. 41, 1 (Jan. 2006), 1--29. https://doi.org/10.1016/j.jsc.2005.07.001
5. Jacques Carette. 2004. Understanding expression simplification. In Proceedings of the 2004 international symposium on Symbolic and algebraic computation - ISSAC '04. ACM Press. https://doi.org/10.1145/1005285.1005298
6. Jacques Carette. 2020. Zero equivalence in computer algebra systems. Mathematics Stack Exchange, https://math.stackexchange.com/q/3607862 .
7. Timothy Y. Chow. 1999. What Is a Closed-Form Number? The American Mathematical Monthly, Vol. 106, 5 (May 1999), 440--448. https://doi.org/10.1080/00029890.1999.12005066
8. Henri Cohen. 1996. A Course in Computational Algebraic Number Theory. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-02945-9
9. Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann. 2019. Singular 4-1-2 -- A computer algebra system for polynomial computations. http://www.singular.uni-kl.de .
10. Jean Della Dora, Claire Dicrescenzo, and Dominique Duval. 1985. About a new method for computing in algebraic number fields. In EUROCAL textquotesingle85. Springer Berlin Heidelberg, 289--290. https://doi.org/10.1007/3-540-15984-3_279
11. Claus Fieker, William Hart, Tommy Hofmann, and Fredrik Johansson. 2017. Nemo/Hecke: computer algebra and number theory packages for the Julia programming language. In Proc. of the 42nd Intl. Symposium on Symbolic and Algebraic Computation (ISSAC '17). ACM, 157--164. https://doi.org/10.1145/3087604.3087611
12. S Fischler and T. Rivoal. 2019. Effective algebraic independence of values of E-functions. arxiv: 1906.05589 [math.NT]
13. William B. Hart. 2010. Fast Library for Number Theory: An Introduction. In Mathematical Software textendash ICMS 2010. Springer Berlin Heidelberg, 88--91. https://doi.org/10.1007/978-3-642-15582-6_18
14. William B. Hart. 2015. ANTIC: Algebraic number theory in C. Computeralgebra-Rundbrief: Vol. 56 (2015).
15. Fredrik Johansson. 2017. Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic. IEEE Trans. Comput., Vol. 66, 8 (Aug. 2017), 1281--1292. https://doi.org/10.1109/tc.2017.2690633
16. Manuel Kauers. 2005. Algorithms for Nonlinear Higher Order Difference Equations. Ph.D. Dissertation. RISC, Johannes Kepler University, Linz. http://www.algebra.uni-linz.ac.at/people/mkauers/publications/kauers05c.pdf.
17. Aaron Meurer et al. 2017. SymPy: symbolic computing in Python. PeerJ Computer Science, Vol. 3 (Jan. 2017), e103. https://doi.org/10.7717/peerj-cs.103
18. Michael Monagan and Roman Pearce. 2006. Rational simplification modulo a polynomial ideal. In Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06. ACM Press. https://doi.org/10.1145/1145768.1145809
19. Joel Moses. 1971. Algebraic simplification: a guide for the perplexed. Commun. ACM, Vol. 14, 8 (Aug. 1971), 527--537. https://doi.org/10.1145/362637.362648
20. Norbert Müller. 2001. The iRRAM: Exact Arithmetic in C++. In Computability and Complexity in Analysis. Springer Berlin Heidelberg, 222--252. https://doi.org/10.1007/3-540-45335-0_14
21. Daniel Richardson. 1992. The elementary constant problem. In Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92. ACM Press. https://doi.org/10.1145/143242.143284
22. Daniel Richardson. 1995. A simplified method of recognizing zero among elementary constants. In Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95. ACM Press. https://doi.org/10.1145/220346.220360
23. Daniel Richardson. 1997. How to Recognize Zero. Journal of Symbolic Computation, Vol. 24, 6 (Dec. 1997), 627--645. https://doi.org/10.1006/jsco.1997.0157
24. Daniel Richardson. 2007. Zero Tests for Constants in Simple Scientific Computation. Mathematics in Computer Science, Vol. 1, 1 (Oct. 2007), 21--37. https://doi.org/10.1007/s11786-007-0002-x
25. Daniel Richardson. 2009. Recognising zero among implicitly defined elementary numbers. Unpublished preprint.
26. Daniel Richardson and John Fitch. 1994. The identity problem for elementary functions and constants. In Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94. ACM Press. https://doi.org/10.1145/190347.190429
27. Allan Steel. 2002. A New Scheme for Computing with Algebraically Closed Fields. In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 491--505. https://doi.org/10.1007/3-540-45455-1_38
28. Allan Steel. 2010. Computing with algebraically closed fields. Journal of Symbolic Computation, Vol. 45, 3 (March 2010), 342--372. https://doi.org/10.1016/j.jsc.2009.09.005
29. The Sage Developers. 2020. SageMath, the Sage Mathematics Software System (Version 9.0). https://www.sagemath.org.
30. The PARI Group 2019. PARI/GP version 2.11.2. The PARI Group, University of Bordeaux. http://pari.math.u-bordeaux.fr/.
31. Joris van der Hoeven. 1995. Automatic numerical expansions. In Proc. of the conference "Real numbers and computers", Saint-Étienne, France. 261--274.
32. Joris van der Hoeven. 1997. Automatic asymptotics. Ph.D. Dissertation. École polytechnique, Palaiseau, France.
33. Joris van der Hoeven. 2006 a. Computations with effective real numbers. Theoretical Computer Science, Vol. 351, 1 (2006), 52--60. https://doi.org/10.1016/j.tcs.2005.09.060
34. Joris van der Hoeven. 2006 b. Effective real numbers in Mmxlib. In Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06. ACM Press. https://doi.org/10.1145/1145768.1145795
35. Jihun Yu, Chee Yap, Zilin Du, Sylvain Pion, and Hervé Brönnimann. 2010. The Design of Core 2: A Library for Exact Numeric Computation in Geometry and Algebra. In Mathematical Software textendash ICMS 2010. Springer Berlin Heidelberg, 121--141. https://doi.org/10.1007/978-3-642-15582-6_24

Index Terms

• Computing methodologies

  • Symbolic and algebraic manipulation

    • Representation of mathematical objects

      • Representation of exact numbers

    • Symbolic and algebraic algorithms

      • Exact arithmetic algorithms

      • Hybrid symbolic-numeric methods