QUADPACK

QUADPACK is a FORTRAN 77 library for numerical integration of one-dimensional functions.^[2] It was included in the SLATEC Common Mathematical Library and is therefore in the public domain.^[3] The individual subprograms are also available on netlib.^[4]

QUADPACK

Original author(s)	Robert Piessens Elise deDoncker-Kapenga Christoph W. Überhuber David Kahaner
Initial release	May 1981 (1981-05)
Final release	May 1987[1]
Written in	FORTRAN 77
Type	Library
License	Public domain
Website	nines.cs.kuleuven.be/software/QUADPACK/

The GNU Scientific Library reimplemented the QUADPACK routines in C. SciPy provides a Python interface to part of QUADPACK.^[5][6]

The pm_quadpack module of the ParaMonte library offers a 100% type-kind-generic multi-precision implementation of QUADPACK library in modern Fortran.

Routines

The main focus of QUADPACK is on automatic integration routines in which the user inputs the problem and an absolute or relative error tolerance and the routine attempts to perform the integration with an error no larger than that requested. There are nine such automatic routines in QUADPACK, in addition to a number of non-automatic routines. All but one of the automatic routines use adaptive quadrature.^[7]

Summary of naming scheme for automatic routines^[8]

1st letter	2nd letter	3rd letter	4th letter
Q Quadrature	N Non-adaptive A Adaptive	G General integrand W Weight function of specified form	Simple integrator S Singularities handled P Specified points of local difficulty (singularities, discontinuities …) I Infinite interval O Oscillatory weight function (cos or sin) over a finite interval F Fourier transform (cos or sin) C Cauchy principal value

Each of the adaptive routines also have versions suffixed by E that have an extended parameter list that provides more information and allows more control. Double precision versions of all routines were released with prefix D.

General-purpose routines

The two general-purpose routines most suitable for use without further analysis of the integrand are QAGS for integration over a finite interval and QAGI for integration over an infinite interval.^[7] These two routines are used in GNU Octave (the quad command)^[5] and R (the integrate function).^[9]

QAGS

uses global adaptive quadrature based on 21-point Gauss–Kronrod quadrature within each subinterval, with acceleration by Peter Wynn's epsilon algorithm.^[7][10]

QAGI

is the only general-purpose routine for infinite intervals, and maps the infinite interval onto the semi-open interval (0,1] using a transformation then uses the same approach as QAGS, except with 15-point rather than 21-point Gauss–Kronrod quadrature.^[2] For an integral over the whole real line, the transformation used is ${\displaystyle x=(1-t)/t}$:^[2] $${\displaystyle \int _{-\infty }^{+\infty }f(x)dx=\int _{0}^{1}{dt \over t^{2}}\left(f\left({\frac {1-t}{t}}\right)+f\left(-{\frac {1-t}{t}}\right)\right)\;.}$$ This is not the best approach for all integrands: another transformation may be appropriate, or one might prefer to break up the original interval and use QAGI only on the infinite part.^[7]

Brief overview of the other automatic routines

QNG

simple non-adaptive integrator

QAG

simple adaptive integrator

QAGP

similar to QAGS but allows user to specify locations of internal singularities, discontinuities etc.

QAWO

integral of cos(ωx) f(x) or sin(ωx) f(x) over a finite interval

QAWF

Fourier transform

QAWS

integral of w(x) f(x) from a to b, where f is smooth and w(x) = (x–a)^α (b–x)^β log^k(x–a) log^l(b–x), with k, l = 0 or 1 and α, β > –1

QAWC

Cauchy principal value of the integral of f(x)/(x–c) for user-specified c and f ^[2]

See also

• List of numerical libraries

References

1. "quadpack/changes". Netlib. Retrieved November 16, 2010.
2. Piessens, Robert; de Doncker-Kapenga, Elise; Überhuber, Christoph W.; Kahaner, David (1983). QUADPACK: A subroutine package for automatic integration. Springer-Verlag. ISBN 978-3-540-12553-2.
3. Fong, Kirby W.; Jefferson, Thomas H.; Suyehiro, Tokihiko; Walton, Lee (July 1993). "Guide to the SLATEC Common Mathematical Library". netlib.org. Retrieved November 13, 2010.
4. "quadpack". Netlib. Retrieved November 13, 2010.
5. "QUADPACK". Numerical Integration, Nonlinear Equations & Software (NINES) Group, Katholieke Universiteit Leuven. Retrieved November 13, 2010.
6. "scipy.integrate.quad -- SciPy v0.14.0 Reference Guide". Retrieved 1 July 2014.
7. Piessens, Robert; De Doncker, Elise; Kahaner, David (1984-04-17). "Subroutine QPDOC". QUADPACK. netlib. Retrieved 16 November 2010.
8. Zwillinger, Daniel (1992). Handbook of integration. A K Peters. p. 255. ISBN 978-0-86720-293-9.
9. R Development Core Team and contributors worldwide (October 2010). "integrate {stats}: Integration of One-Dimensional Functions". Documentation for package ‘stats’ version 2.13.0. Retrieved 16 November 2010. {{cite web}}: |author= has generic name (help)
10. "17.4 QAGS adaptive integration with singularities". GNU Scientific Library -- Reference. Free Software Foundation. Retrieved 16 November 2010.

Further reading

• Favati, P.; Lotti, G.; Romani, F. (1991). "Algorithm 691; Improving QUADPACK automatic integration routines". ACM Transactions on Mathematical Software. 17 (2): 218–232. doi:10.1145/108556.108580. S2CID 19675880.
• Cools, R.; Haegemans, A. (2003). "Algorithm 824: CUBPACK: a package for automatic cubature; framework description". ACM Transactions on Mathematical Software. 29 (3): 287–296. doi:10.1145/838250.838253. S2CID 6855610.