research-article

Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations

Author:

Christopher KormanyosAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 37, Issue 4

Article No.: 45, Pages 1 - 27

https://doi.org/10.1145/1916461.1916469

Published: 01 February 2011 Publication History

Get Access

Abstract

This article presents a portable C++ system for multiple precision calculations of special functions called e_float. It has an extendable architecture with a uniform C++ layer which can be used with any suitably prepared MP type. The system implements many high-precision special functions and extends some of these to very large parameter ranges. It supports calculations with 30 ⋯ 300 decimal digits of precision. Interoperabilities with Microsoft’s® CLR, Python, and Mathematica® are supported. The e_float system and its usage are described in detail. Implementation notes, testing results, and performance measurements are provided.

Supplementary Material

ZIP File (910.zip)

Software for A Portable C++ Multiple-Precision System for Special-Function Calculations

Download
11.07 MB

References

[1]

Abrahams, D. 2008. boost.python Index. http://www.boost.org/doc/libs/1_42_0/libs/python/.

Google Scholar

[2]

Abramowitz, M. and Stegun, I. A. 1972. Handbook of Mathematical Functions 9th Ed. Dover Publications, New York, NY.

Google Scholar

[3]

Amos, D. E. 1986. A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Softw. 12, 3, 265--273.

Digital Library

Google Scholar

[4]

Andrews, G., Askey, R., and Roy, R. 2000. Special Functions. Cambridge University Press, Cambridge, UK.

Google Scholar

[5]

Arndt, J. and Haenel, C. 2000. π Unleashed. Springer, New York, NY.

Digital Library

Google Scholar

[6]

Bailey, D., Borwein, J. M., Calkin, N. J., Girgensohn, R., Luke, D., and Moll, V. H. 2007. Experimental Mathematics in Action. A.K. Peters Press, Natick, MA.

Google Scholar

[7]

Bailey, D. H. 1993. Multiprecision translation and execution of Fortran programs. ACM Trans. Math. Softw. 19, 3, 288--319.

Digital Library

Google Scholar

[8]

Bailey, D. H. 1995. A Fortran 90--based multiprecision system. ACM Trans. Math. Softw. 21, 4, 379--387.

Digital Library

Google Scholar

[9]

Bailey, D. H. and Broadhurst, D. J. 2000. Parallel integer relation detection: Techniques and applications. Math. Comp. 70, 1719--1736.

Digital Library

Google Scholar

[10]

Bailey, D. H., Hida, Y., Li, X. S., and Thompson, B. 2002. ARPREC: An arbitrary precision computation package. Tech. rep. Lawrence Berkeley National Laboratory, Berkeley, CA.

Crossref

Google Scholar

[11]

Becker, P. 2006. The C++ Standard Library Extensions: A Tutorial and Reference. Addison Wesley, Reading, MA.

Digital Library

Google Scholar

[12]

Borwein, P. 1995. An efficient algorithm for the Riemann zeta function. Can. Math. Soc. Conf. Proc. 27, 29--34.

Google Scholar

[13]

Brent, R. P. 1978. A Fortran multiple-precision arithmetic package. ACM Trans. Math. Softw. 4, 1, 57--70.

Digital Library

Google Scholar

[14]

Cody, W. J. 1993. SPECFUN---A portable Fortran package of special function routines and test drivers. ACM Trans. Math. Softw. 19, 1, 22--30.

Digital Library

Google Scholar

[15]

Cody, W. J. and Waite, W. 1980. Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs, New Jersey.

Digital Library

Google Scholar

[16]

Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. 1981. Higher Transcendental Functions. Vols. 1--2. Krieger, New York, NY.

Google Scholar

[17]

Espinosa, O. and Moll, V. H. 2004. A generalized polygamma function. Integral Trans. Special Func. 15, 101--115.

Crossref

Google Scholar

[18]

Finch, S. R. 2003. Mathematical Constants. Cambridge University Press, Cambridge, UK.

Google Scholar

[19]

Foord, M. J. and Muirhead, C. 2009. IronPython in Action. Manning Publications, Greenwich, CT.

Digital Library

Google Scholar

[20]

Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., and Zimmermann, P. 2007. MPFR: A multiple--precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33, 2, 1--15.

Digital Library

Google Scholar

[21]

Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., Booth, M., and Rossi, F. 2009. GNU Scientific Library Reference Manual, for Version 1.12 3rd Ed. Network Theory Ltd., Bristol, UK.

Digital Library

Google Scholar

[22]

GCC. 2009. The GNU Compiler Collection Version 4.4.2. Free Software Foundation. http://gcc.gnu.org/.

Google Scholar

[23]

Gil, A., Segura, J., and Temme, N. M. 2006. Computing the real parabolic cylinder functions U(a, x), V(a, x). ACM Trans. Math. Softw. 32, 1, 70--101.

Digital Library

Google Scholar

[24]

Gil, A., Segura, J., and Temme, N. M. 2007. Numerical Methods for Special Functions. SIAM, Philadelphia, PA.

Digital Library

Google Scholar

[25]

Gladman, B. 2008. Building GMP and MPFR with Microsoft® Visual Studio 2008 and YASM. http://gladman.plushost.co.uk/oldsite/computing/gmp4win.php.

Google Scholar

[26]

GMP. 2008. GNU Multiple Precision Arithmetic Library Version 4.2.4. Free Software Foundation. http://gmplib.org/.

Google Scholar

[27]

GNU. 2006. GNUmake Version 3.81. Free Software Foundation. http://www.gnu.org/software/make/.

Google Scholar

[28]

Gourdon, X. and Sebah, P. 2008. Numbers, Constants and Computation. http://numbers.computation.free.fr/.

Google Scholar

[29]

Intel. 2008. Intel® C++ Compiler Professional Edition. http://software.intel.com/en-us/intel-compilers/.

Google Scholar

[30]

ISO. 2003. ISO/IEC 14882:2003: Information Technology---Programming Languages---C++. International Organization for Standardization, Geneva, Switzerland.

Google Scholar

[31]

ISO. 2006a. ISO/IEC 23270:2006: Information Technology Programming Languages---C#. International Organization for Standardization, Geneva, Switzerland.

Google Scholar

[32]

ISO. 2006b. ISO/IEC 23271:2006: Information Technology Programming Languages---Common Language Infrastructure (CLI) Partitions I to VI. International Organization for Standardization, Geneva, Switzerland.

Google Scholar

[33]

ISO. 2007. ISO/IEC 19768:2007: Information Technology---Programming Languages---Technical Report on C++ Library Extensions. International Organization for Standardization, Geneva, Switzerland.

Google Scholar

[34]

Jahnke, E. and Emden, F. 1945. Tables of Functions with Formulae and Curves 4th Ed. Dover, New York, NY.

Google Scholar

[35]

Josuttis, N. M. 1999. The C++ Standard Library: A Tutorial and Reference. Addison Wesley, Reading, MA.

Digital Library

Google Scholar

[36]

King, L. V. 1924. On the Direct Numerical Calculation of Elliptic Functions and Integrals. Cambridge University Press, Cambridge, UK.

Google Scholar

[37]

Knuth, D. E. 1998. The Art of Computer Programming Vols. 1--3 2nd Ed. Addison Wesley, Reading, MA.

Digital Library

Google Scholar

[38]

Luke, Y. L. 1977. Algorithms for the Computation of Mathematical Functions. Academic Press, New York, NY.

Google Scholar

[39]

Microsoft. 2008. Microsoft® Visual Studio Professional 2008. http://www.microsoft.com/visualstudio/.

Google Scholar

[40]

Microsoft. 2009a. IronPython. http://www.ironpython.net/.

Google Scholar

[41]

Microsoft. 2009b. NMAKE Reference. http://msdn.microsoft.com/en-us/library/dd9y37ha(VS.71).aspx.

Google Scholar

[42]

Microsoft. 2010. Common Language Runtime Overview. http://msdn.microsoft.com/en-us/library/ddk909ch.aspx.

Google Scholar

[43]

MISRA. 2004. MISRA--C 2004: Guidelines for the Use of the C Language in Critical Systems. http://www.misra.org.uk/.

Google Scholar

[44]

MISRA. 2008. MISRA--C++ 2008: Guidelines for the Use of the C++ Language in Critical Systems. http://www.misra-cpp.org/.

Google Scholar

[45]

MPMATH. 2009. Python Library for Arbitrary-Precision Floating-Point Arithmetic. http://code.google.com/p/mpmath/.

Google Scholar

[46]

Oldham, K., Myland, J., and Spanier, J. 2009. An Atlas of Functions 2nd Ed. Springer, New York.

Google Scholar

[47]

Olver, F. W. J. 1997. Asymptotics and Special Functions. A.K. Peters Press, Natick MA.

Google Scholar

[48]

Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. 2002. Numerical Recipes in C++ 2nd Ed. Cambridge University Press, Cambridge, UK.

Google Scholar

[49]

PSF. 2009. Python Programming Language---Official Website. http://www.python.org/.

Google Scholar

[50]

Richter, J. 2006. CLR Via C# 2nd Ed. Microsoft Press, Redmond, WA.

Google Scholar

[51]

Shoup, V. 2008. NTL: A Library for Doing Number Theory. http://www.shoup.net/ntl/.

Google Scholar

[52]

Sloane, N. J. A. 2009. Online Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/sequences/.

Google Scholar

[53]

Smith, D. 1991. A Fortran package for floating-point multiple-precision arithmetic. ACM Trans. Math. Softw. 17, 2, 273--283.

Digital Library

Google Scholar

[54]

Smith, D. 1998. Multiple-precision complex arithmetic and functions. ACM Trans. Math. Softw. 24, 4, 359--367.

Digital Library

Google Scholar

[55]

Temme, N. M. 1975. On the numerical evaluation of the modified Bessel function of the third kind. J. Comp. Phys. 19, 324.

Crossref

Google Scholar

[56]

Temme, N. M. 2007. Personal communications. 2007--2009.

Google Scholar

[57]

Vepštas, L. 2008. An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions. Numer. Algor. 47, 3, 211--252.

Crossref

Google Scholar

[58]

Watson, G. N. 1995. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, UK.

Google Scholar

[59]

Weisstein, E. W. 2010. MathWorld® --A Wolfram Web Resource. http://mathworld.wolfram.com/.

Google Scholar

[60]

Wikipedia. 2009. Wikipedia---The Free Encyclopedia. http://en.wikipedia.org/wiki/.

Google Scholar

[61]

Wimp, J. 1984. Computation with Recurrence Relations. Pitman Publishing, London, UK.

Google Scholar

[62]

Wolfram, S. 1999. The Mathematica® Book 4th Ed. Cambridge University Press, Cambridge, UK.

Digital Library

Google Scholar

[63]

Wolfram Research. 2010. The Wolfram Functions Site. http://functions.wolfram.com/.

Google Scholar

[64]

Zhang, S. and Jin, J. 1996. Computation of Special Functions. Wiley, New York, NY.

Google Scholar

Cited By

View all

Tang EZhang XMuller NChen ZLi X(2017)Software Numerical Instability Detection and Diagnosis by Combining Stochastic and Infinite-Precision TestingIEEE Transactions on Software Engineering10.1109/TSE.2016.264295643:10(975-994)Online publication date: 1-Oct-2017
https://doi.org/10.1109/TSE.2016.2642956
Ventura GBenvenuti E(2014)Equivalent polynomials for quadrature in Heaviside function enriched elementsInternational Journal for Numerical Methods in Engineering10.1002/nme.4679102:3-4(688-710)Online publication date: 13-May-2014
https://doi.org/10.1002/nme.4679

Index Terms

Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations

Recommendations

Generalization of matching extensions in graphs-combinatorial interpretation of orthogonal and q-orthogonal polynomials

In this paper, we present generalization of matching extensions in graphs and we derive combinatorial interpretation of wide classes of orthogonal and q-orthogonal polynomials. Specifically, we assign general weights to complete graphs, cycles and ...
Algebra of Orthogonal Series
Abstract
An algebra of orthogonal series has been developed for the operations of multiplication and division, differentiation and integration of signals represented by series of classical orthogonal polynomials and functions. It is shown that the ...
A combined symbolic and numerical algorithm for the computation of zeros of orthogonal polynomials and special functions
Special issue: Algebra and computer analysis

A Maple algorithm for the computation of the zeros of orthogonal polynomials (OPs) and special functions (SFs) in a given interval [ x ₁ , x ₂ ] is presented. The program combines symbolic and numerical calculations and it is based on fixed point ...

Reviews

Reviewer: Todor Todorov

Kormanyos presents a portable C++ system for multiple precision calculations. The system supports calculations with 30 to 300 decimal digits of precision, and interoperability with Microsoft's Common Language Runtime (CLR), Python, and Mathematica. The author presents the system architecture in great detail, as well as some software realizations of the calculation algorithms. A detailed discussion of interoperability follows, including test results and performance analysis, with three examples: Python, Microsoft CLR Export, and computer algebra systems. Finally, for future work, the author mentions increasing the extension to 1,000-digits, which will be quite interesting and useful. I recommend this paper to anybody interested in high-precision calculations and their software implementation. Online Computing Reviews Service

Access critical reviews of Computing literature here

Become a reviewer for Computing Reviews.

Comments

Information & Contributors

Information

Published In

ACM Transactions on Mathematical Software Volume 37, Issue 4

February 2011

212 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/1916461

Issue’s Table of Contents

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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 February 2011

Accepted: 01 October 2010

Revised: 01 July 2010

Received: 01 July 2009

Published in TOMS Volume 37, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
751
Total Downloads

Downloads (Last 12 months)13
Downloads (Last 6 weeks)3

Reflects downloads up to 07 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Tang EZhang XMuller NChen ZLi X(2017)Software Numerical Instability Detection and Diagnosis by Combining Stochastic and Infinite-Precision TestingIEEE Transactions on Software Engineering10.1109/TSE.2016.264295643:10(975-994)Online publication date: 1-Oct-2017
https://doi.org/10.1109/TSE.2016.2642956
Ventura GBenvenuti E(2014)Equivalent polynomials for quadrature in Heaviside function enriched elementsInternational Journal for Numerical Methods in Engineering10.1002/nme.4679102:3-4(688-710)Online publication date: 13-May-2014
https://doi.org/10.1002/nme.4679

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Index Terms

Recommendations

Generalization of matching extensions in graphs-combinatorial interpretation of orthogonal and q-orthogonal polynomials

Algebra of Orthogonal Series

A combined symbolic and numerical algorithm for the computation of zeros of orthogonal polynomials and special functions

Reviews

Access critical reviews of Computing literature here