QUADPACK - Routines for Estimating Integrals

QUADPACK is a set of routines for estimating 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).

Many subroutines come in two versions, a "simple" interface and an "extended" interface. Compare, for example, the routines QAWF and QAWFE. The first one simply makes sensible choices for many parameter values. The second one gives the user full control over all the parameters.

Reference:: R Piessens, E de Doncker-Kapenger, C Ueberhuber, D Kahaner,
QUADPACK, A Subroutine Package for Automatic Integration,
Springer Verlag, 1983.

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.

Infinite Region Decision Tree

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.

Files you may copy include:

quadpack.f90, the source code.
quadpack_prb.f90, a sample problem.
quadpack_prb.out, sample problem output.

The list of routines includes:

PI returns the value of pi.
QAG approximates an integral over a finite interval.
QAGE estimates a definite integral.
QAGI estimates an integral over a semi-infinite or infinite interval.
QAGP computes a definite integral.
QAGS estimates the integral of a function.
QAWC computes a Cauchy principal value.
QAWCE computes a Cauchy principal value.
QAWF computes Fourier integrals.
QAWFE computes Fourier integrals.
QAWO computes the integrals of oscillatory integrands.
QAWS estimates integrals with algebraico-logarithmic endpoint singularities.
QAWSE estimates integrals with algebraico-logarithmic endpoint singularities.
QC25C returns integration rules for Cauchy Principal Value integrals.
QC25O returns integration rules for integrands with a COS or SIN factor.
QC25S returns rules for algebraico-logarithmic end point singularities.
QCHEB computes the Chebyshev series expansion.
QEXTR carries out the Epsilon extrapolation algorithm.
QFOUR estimates the integrals of oscillatory functions.
QK15 returns integration rules of order 15.
QK15I returns integration rules of order 15 for an infinite interval.
QK15W returns integration rules for a weighted integrand.
QK21 returns integration rules of order 21.
QK31 returns integration rules of order 31.
QK41 returns integration rules of order 41.
QK51 returns integration rules of order 51.
QK61 returns integration rules of order 61.
QMACO returns machine constants.
QMOMO computes modified Chebyshev moments.
QNG estimates an integral, using non-adaptive integration.
QSORT maintains the order of a list of local error estimates.
QWGTC ???
QWGTO ???
QWGTS ???
R_SWAP swaps two real values.
TIMESTAMP prints the current YMDHMS date as a time stamp.

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Last revised on 10 December 2001.