Software Index

♦ Software Cross Index ♦ A Classification of Software ♦ Software Repositories ♦

§ Software Cross Index

	Open Source										With Book						Commercial
Package	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}Axiom\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}CEPHES\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}FN\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}GSL\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}MPFR\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}Maxima\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}PARI-GP\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}SAGE\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}SLATEC\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}mpmath\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}Baker\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}Lau\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}Num.~{}Recipes\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}Thompson\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}Watanabe\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}Zhang\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}IMSL\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}Maple\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}Mathematica\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col1}{\color{blue}\hbox to 60.0pt{~{}Matlab\hfil}}}}$	$\definecolor{col0}{rgb}{0.81,1.0,0.81}\definecolor{col1}{rgb}{0.94,1.0,0.94}\rotatebox{90.0}{{\pagecolor{col0}{\color{blue}\hbox to 60.0pt{~{}NAG\hfil}}}}$	See Also
Language	I	C	F	C	C	I	I C	I	F	Py	C	C J	C C++ F	C F MM	F	F	C F J	I	I	I	C F
4 Elementary Functions
§4.48(ii) Interval Arithmetic								✓		✓									✓			CoStLy
§4.48(iii) General Precision					✓	✓		✓		✓								✓	✓	a		REDUCE
§4.48(iv) Lambert $W$ -Function								a		✓								✓	✓	a
§4.48(v) Testing
5 Gamma Function
§5.24(ii) $\mathop{\Gamma\/}\nolimits\!\left(x\right)$ , $x\in\Real$	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	FDLIBM
§5.24(iii) $\mathop{\psi\/}\nolimits\!\left(x\right)$ , $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(x\right)$ , $x\in\Real$	✓	✓	✓	✓		✓	✓	a	✓	✓	✓			✓		✓	✓	✓	✓	✓	✓
§5.24(iv) $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ , $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(z\right)$ , $z\in\Complex$	✓		✓				✓	a	✓	✓	✓				✓	✓	✓	✓	✓	✓	✓
§5.24(v) $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ , $a,b\in\Real$	✓	✓	✓	✓		✓		a	✓	✓	✓			✓		✓	✓	✓	✓	✓
§5.24(vi) $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ , $a,b\in\Complex$	✓		✓					a	✓	✓							✓	✓	✓	a
6 Exponential, Logarithmic, Sine, and Cosine Integrals
§6.21(ii) $\mathop{E_{1}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{Si}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{Ci}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{Shi}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{Chi}\/}\nolimits\!\left(x\right)$ , $x\in\Real$	✓	✓	✓	✓	✓			✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
§6.21(iii) $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{Shi}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ , $z\in\Complex$	✓							✓		✓	✓							✓	✓	✓
7 Error Functions, Dawson’s and Fresnel Integrals
§7.25(ii) $\mathop{\mathrm{erf}\/}\nolimits x$ , $\mathop{\mathrm{erfc}\/}\nolimits x$ , $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(x\right)$ , $x\in\Real$		✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
§7.25(iii) $\mathop{\mathrm{erf}\/}\nolimits z$ , $\mathop{\mathrm{erfc}\/}\nolimits z$ , $z\in\Complex$								a		✓	✓							✓	✓	✓
§7.25(iv) $\mathop{C\/}\nolimits\!\left(x\right)$ , $\mathop{S\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{f}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{g}\/}\nolimits\!\left(x\right)$ , $x\in\Real$		✓						a		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
§7.25(v) $\mathop{C\/}\nolimits\!\left(z\right)$ , $\mathop{S\/}\nolimits\!\left(z\right)$ , $z\in\Complex$								a		✓								✓	✓	✓
§7.25(vi) $\mathop{\mathcal{F}\/}\nolimits\!\left(x\right)$ , $\mathop{G\/}\nolimits\!\left(x\right)$ , $\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right)$ , $\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right)$ , $x\in\Real$		✓	✓	✓					✓		✓	✓	✓	✓			✓	✓	✓	✓
§7.25(vii) $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ , $\mathop{G\/}\nolimits\!\left(z\right)$ , $z\in\Complex$																		✓	✓	✓
8 Incomplete Gamma and Related Functions
§8.28(ii) Incomplete Gamma Functions for Real Argument and Parameter		✓	✓	✓				✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
§8.28(iii) Incomplete Gamma Functions for Complex Argument and Parameter		✓	✓	✓				a	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓
§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters								a		✓									✓
§8.28(vi) Generalized Exponential Integral for Real Argument and Integer Parameter		✓		✓				a	✓	✓	✓	✓	✓			✓	✓	✓	✓	✓
§8.28(vii) Generalized Exponential Integral for Complex Argument and Parameter								a		✓								✓	✓	✓
9 Airy and Related Functions
§9.20(ii) $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $x\in\Real$	✓	✓	✓	✓		✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓
§9.20(iii) $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $z\in\Complex$	✓					✓		a	✓	✓	✓							✓	✓	✓	✓
§9.20(iv) Real and Complex Zeros				✓				a		✓									✓
§9.20(v) Integrals of $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , $x\in\Real$								a		✓	✓					✓			✓
§9.20(vi) Scorer Functions								a		✓
10 Bessel Functions
§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	FDLIBM, NMS
§10.77(iii) Bessel Functions–Real Order and Argument	✓	✓	✓	✓		✓		a	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	NMS
§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions	✓	✓						a		✓	✓			✓	✓	✓	✓	✓	✓	✓
§10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)	✓							a	✓	✓						✓	✓	✓	✓	✓
§10.77(vi) Bessel Functions–Imaginary Order and Real Argument	✓							a		✓								✓	✓	✓
§10.77(viii) Bessel Functions–Complex Order and Argument	✓							a		✓								✓	✓	✓
§10.77(ix) Integrals of Bessel Functions								a	✓	✓	✓					✓			✓
§10.77(x) Zeros of Bessel Functions				✓				a		✓	✓	✓				✓			✓
11 Struve and Related Functions
§11.16(ii) Struve Functions		✓						a		✓	✓			✓		✓		✓	✓	a
§11.16(iii) Integrals of Struve Functions											✓					✓		✓	✓	a
§11.16(iv) Lommel Functions								a		✓				✓				✓		a
§11.16(v) Anger and Weber Functions								a		✓				✓				✓	✓	a
§11.16(vi) Integrals of Anger and Weber Functions											✓								✓
12 Parabolic Cylinder Functions
§12.21(ii) Real Arguments and Parameters								a		✓				✓		✓		✓	✓	a
§12.21(iii) Complex Arguments and Parameters								a		✓								✓	✓	a
13 Confluent Hypergeometric Functions
§13.32(ii) Real Argument and Parameters		✓	✓	✓				✓		✓	✓			✓		✓		✓	✓	a
§13.32(iii) Complex Argument and/or Parameters								a		✓	✓					✓		✓	✓	a
14 Legendre and Related Functions
§14.34(ii) Legendre Functions: Real Argument and Parameters				✓				a	✓	✓	✓		✓	✓		✓		✓	✓	✓	✓
§14.34(iii) Legendre Functions: Complex Argument and/or Parameters								a		✓	✓							✓	✓	a
§14.34(iv) Conical (Mehler) and/or Toroidal Functions				✓							✓			✓
15 Hypergeometric Function
§15.20(ii) Real Parameters and Argument		✓		✓				a		✓	✓			✓		✓		✓	✓	a
§15.20(iii) Complex Parameters and Argument								a		✓			✓					✓	✓	a
16 Generalized Hypergeometric Functions and Meijer $G$ -Function
§16.27(ii) Real Argument and Parameters		✓		✓				a		✓								✓	✓	✓
§16.27(iii) Complex Argument and/or Parameters								a		✓								✓	✓	✓
18 Orthogonal Polynomials
§18.42 Software	✓							✓		✓								✓	✓	a		Koornwinder, Stembridge
19 Elliptic Integrals
§19.39(ii) Legendre’s and Bulirsch’s Complete Integrals		✓		✓		✓		✓		✓							✓	✓	✓	✓
§19.39(iii) Legendre’s and Bulirsch’s Incomplete Integrals		✓		✓		✓		✓		✓			✓					✓	✓	a	✓
§19.39(iv) Symmetric Integrals				✓				a	✓	✓			✓				✓				✓	Derive
20 Theta Functions
§20.16(ii) Real Argument and Parameter								a		✓	✓			✓				✓	✓	a	✓
§20.16(iii) Complex Argument and/or Parameter								a		✓	✓							✓	✓	a
21 Multidimensional Theta Functions
§21.11 Software																		✓	✓	a		JTEM
22 Jacobian Elliptic Functions
§22.22(ii) Real Argument		✓		✓		✓		✓		✓	✓		✓	✓		✓	✓	✓	✓	✓	✓
§22.22(iii) Complex Argument								a		✓	✓						✓	✓	✓	a	✓
23 Weierstrass Elliptic and Modular Functions
§23.24(ii) Real Argument																		✓	✓	a
§23.24(iii) Complex Argument											✓						✓	✓	✓	a
24 Bernoulli and Euler Polynomials
§24.21(ii) $\mathop{B_{{n}}\/}\nolimits$ , $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{E_{{n}}\/}\nolimits$ , and $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$	✓					✓	✓	a		✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
25 Zeta and Related Functions
§25.21(ii) Zeta Functions for Real Arguments		✓		✓	✓	✓		a		✓	✓			✓				✓	✓	✓
§25.21(iii) Zeta Functions for Complex Arguments								✓		✓	✓							✓	✓	✓
§25.21(iv) Hurwitz Zeta Function				✓				a		✓	✓			✓				✓	✓	a
§25.21(v) Dilogarithms, Polylogarithms		✓		✓	✓			a		✓	✓			✓				✓	✓	✓
§25.21(vi) Clausen’s Integral				✓				a		✓									✓			NetNUMPAC
§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals				✓							✓			✓					✓
§25.21(viii) Lerch’s Transcendent								a		✓								✓	✓	a
§25.21(ix) Dirichlet $L$ -series								a		✓									✓
26 Combinatorial Analysis
§26.22 Software																		✓	✓	a		Wolfram’s Mathworld, Zeilberger
27 Functions of Number Theory
§27.22 Software							✓	✓										✓	✓	a
28 Mathieu Functions and Hill’s Equation
§28.36(ii) Characteristic Exponents and Eigenvalues											✓					✓	✓	✓	✓	a
§28.36(iii) Mathieu and Modified Mathieu Functions											✓					✓	✓	✓	✓	a		Van Buren
30 Spheroidal Wave Functions
§30.18(ii) Eigenvalues $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$											✓			✓		✓			✓
§30.18(iii) Spheroidal Wave Functions											✓			✓		✓			✓			Van Buren
33 Coulomb Functions
§33.26(ii) Real Variable and Parameters				✓				a		✓	✓			✓
§33.26(iii) Complex Variable and/or Parameters								a		✓
34 $3j,6j,9j$ Symbols
§34.15 Software				✓				✓	✓		✓			✓				✓	✓	a
35 Functions of Matrix Argument
§35.12 Software																						Zeilberger

‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support.

§ A Classification of Software

In the list below we identify four main sources of software for computing special functions. Please see our Software Indexing Policy for rules that govern the indexing of software in the DLMF.

Research Software.: This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.
Open Source Collections and Systems.: These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.
Software Associated with Books.: An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.
Commercial Software.: Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

§ Software Repositories

The following are web-based software repositories with significant holdings in the area of special functions. Many research software packages are found here, as well as some open source software collections.

Collected Algorithms of the ACM: Software published by the journal ACM Transactions on Mathematical Software (TOMS).
Computer Physics Communications Program Library: Software associated with papers published in the journal Computer Physics Communications.
netlib: A collection of mathematical software, papers, and databases produced by the numerical analysis research community.
Guide to Available Mathematical Software: A cross index of mathematical software in use at NIST.