quadpack

quadpack, a FORTRAN77 code which estimates integrals using numerical quadrature, by Piessens, deDoncker-Kapenga, Ueberhuber, and Kahaner

The code estimates the integral of a function F(X). There are routines for nonadaptive or adaptive integration, finite, semi-infinite or fully infinite integration regions, integrands with singularities, and integrands that include a factor of SIN(X) or COS(X).

Routines for a finite region:

How to decide what routine to use, if your integration region is finite:

• If you can factor the integrand as F(X)=W(X)*G(X), where G is smooth on [A,B] and W(X)=COS(OMEGA*X) or SIN(OMEGA*X) then use QAWO.
• Otherwise, if you can factor F(X)=W(X)*G(X) where G is smooth and W(X)=(X-A)**ALFA * (B-X)**BETA * (LOG(X-A))**L * (LOG(B-X))**K with K, L = 0 or 1, and ALFA, BETA greater than -1, then use QAWS.
• Otherwise, if you can factor F(X)=W(X)*G(X) where G is smooth and W(X)=1/(X-C) for some constant C, use QAWC.
• Otherwise, if you do not care too much about possible inefficient use of computer time, and do not want to further analyze the problem, use QAGS.
• Otherwise, if the integrand is smooth, use QNG or QAG.
• Otherwise, if there are discontinuities or singularities of the integrand or of its derivative, and you know where they are, split the integration range at these points and analyze each subinterval. You can also use QAGP, which is to be provided with the x-locations of the singularities or discontinuities.
• Otherwise, if the integrand has end point singularities, use QAGS.
• Otherwise, if the integrand has an oscillatory behavior of nonspecific type, and no singularities, use QAG with KEY=6.
• Otherwise, use QAGS.

Routines for an infinite region:

• If the integrand decays rapidly to zero, truncate the interval and use the finite interval decision tree.
• Otherwise, if the integrand oscillates over the entire infinite range, and the integral is a Fourier transform, use QAWF.
• Or, if the integrand oscillates over the entire infinite range, but is not a Fourier transform, then sum the successive positive and negative contributions by integrating between the zeroes of the integrand. Apply convergence acceleration with QELG.
• Otherwise, if you are not constrained by computer time, and do not wish to analyze the problem further, use QAGI.
• Otherwise, if the integrand has a non-smooth behavior in the range, and you know where it occurs, split off these regions and use the appropriate finite range routines to integrate over them. Then begin this tree again to handle the remainder of the region.
• Otherwise, truncation of the interval, or application of a suitable transformation for reducing the problem to a finite range may be possible. And you may also call QAGI.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

quadpack is available in a FORTRAN77 version and a FORTRAN90 version.

Related Data and Programs:

quadpack_test

intlib, a FORTRAN90 library which numerically estimate sintegrals.

KRONROD, a FORTRAN77 library which can compute a Gauss and Gauss-Kronrod pair of quadrature rules of arbitrary order, by Robert Piessens, Maria Branders.

MACHINE, a FORTRAN77 library which defines machine arithmetic constants.

NMS, a FORTRAN77 library which includes QUADPACK.

QUADRULE, a FORTRAN90 library which defines quadrature rules for various intervals and weight functions.

SLATEC, a FORTRAN90 library which includes QUADPACK.

STROUD, a FORTRAN90 library which defines quadrature rules for various geometric shapes.

TEST_INT, a FORTRAN90 library which defines test integration problems.

TEST_INT_LAGUERRE, a FORTRAN77 library which defines test integrands for the interval [a,+oo) and weight exp(-x);

TOMS351, a FORTRAN77 library which estimates an integral using Romberg integration.

TOMS379, a FORTRAN77 library which estimates an integral.

TOMS418, a FORTRAN77 library which estimates the integral of a function with a sine or cosine factor.

TOMS424, a FORTRAN77 library which estimates the integral of a function using Clenshaw-Curtis quadrature.

TOMS468, a FORTRAN77 library which carries out the "automatic" integration of a function.

XERROR, a FORTRAN77 library which handles run-time errors.

Author:

Original Fortran77 version by Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner.

Reference:

1. Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
   QUADPACK: A Subroutine Package for Automatic Integration,
   Springer, 1983,
   ISBN: 3540125531,
   LC: QA299.3.Q36.

Source Code:

• quadpack.f, the source code.
• quadpack.sh, commands to compile the source code.