QUADPACK explained

QUADPACK
QUADPACK
Author:	Robert Piessens Elise deDoncker-Kapenga Christoph W. Überhuber David Kahaner
Discontinued:	yes
Latest Release Version:	May 1987^[1]
Programming Language:	FORTRAN 77
License:	Public domain

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

The GNU Scientific Library reimplemented the QUADPACK routines in C. SciPy provides a Python interface to part of QUADPACK.^[4]

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.

Summary of naming scheme for automatic routines^[5] ! 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. These two routines are used in GNU Octave (the quad command)^[6] and R (the integrate function).^[7]

QAGS : uses global adaptive quadrature based on 21-point Gauss–Kronrod quadrature within each subinterval, with acceleration by Peter Wynn's epsilon algorithm.^[8]

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. For an integral over the whole real line, the transformation used is

x = ( 1- t)/t :^[9] $$\backslash int\_^\; f(x)\; dx\; =\; \backslash int\_0^1\; \backslash left(f\backslash left(\backslash frac\backslash right)+\; f\backslash left(-\backslash frac\backslash right)\backslash right)\; \backslash ;.$$ 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.^[10]

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 or over a finite interval

QAWF : Fourier transform

QAWS : integral of from to, where is smooth and, with and

QAWC : Cauchy principal value of the integral of for user-specified and ^[9]

See also

• List of numerical libraries

Further reading

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

Notes and References

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