QUADPACK

QUADPACK
Original authors	Robert Piessens Elise deDoncker-Kapenga Christoph W. Überhuber David Kahaner
Initial release	May 1981 (1981-05)
Stable release	11 October 2021
Repository	{{URL|example.com|optional display text}}
Written in	FORTRAN 77
Engine
Type	Library
License	Public domain
Website	nines.cs.kuleuven.be/software/QUADPACK/

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

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

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

1st letter	2nd letter	3rd letter	4th letter, if present
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) and R (the integrate function).

QAGS

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

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 $x=(1-t)/t$: $$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(x)dx=\underset{0}{\overset{1}{\int }}\frac{dt}{t^2}(f\left(\frac{1-t}{t}\right)+f(-\frac{1-t}{t})).$$ 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.

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

See also

• List of numerical libraries