QUADPACK
Numerical Integration

QUADPACK is a FORTRAN90 library which estimates integrals using numerical quadrature, by Piessens, deDoncker-Kapenga, Ueberhuber, and Kahaner

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.

Languages:

QUADPACK is available in a FORTRAN90 version.

Related Data and Programs:

INTLIB, a FORTRAN90 library which numerically estimates integrals.

KRONROD, a FORTRAN90 library which can compute a Gauss and Gauss-Kronrod pair of quadrature rules of arbitrary order, by Robert Piessens, Maria Branders.

NMS, a FORTRAN90 library which includes QUADPACK.

PRODUCT_RULE, a FORTRAN90 program which can create a multidimensional quadrature rule as a product of one dimensional rules.

QUADPACK_DOUBLE, a FORTRAN90 library which estimates integrals using numerical quadrature, using double precision arithmetic, by Piessens, deDoncker-Kapenga, Ueberhuber, and Kahaner.

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

SLATEC, a FORTRAN90 library which collects together a number of standard numerical libraries, including BLAS, DASSL, DEPAC, EISPACK, FFTPACK, FISHPACK, FNLIB, LINPACK, PCHIP, QUADPACK, SDRIV, SLAP, XERROR.

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

TANH_QUAD, a FORTRAN90 library which sets up the tanh quadrature rule;

TEST_INT, a FORTRAN90 library which defines some test integration problems.

TEST_INT_HERMITE, a FORTRAN90 library which defines some test integration problems over infinite intervals.

TEST_INT_LAGUERRE, a FORTRAN90 library which defines test integrands for integration over [-ALPHA,+Infinity).

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.

Author:

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.f90, the source code.

Examples and Tests:

• quadpack_test.f90, a sample problem.
• quadpack_test.txt, the output file.

List of Routines:

• AAAA is a dummy subroutine with QUADPACK documentation in its comments.
• 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 over the interval [ A, +Infinity ).
• 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 carries out a 15 point Gauss-Kronrod quadrature rule.
• QK15I applies a 15 point Gauss-Kronrod quadrature on an infinite interval.
• QK15W applies a 15 point Gauss-Kronrod rule for a weighted integrand.
• QK21 carries out a 21 point Gauss-Kronrod quadrature rule.
• QK31 carries out a 31 point Gauss-Kronrod quadrature rule.
• QK41 carries out a 41 point Gauss-Kronrod quadrature rule.
• QK51 carries out a 51 point Gauss-Kronrod quadrature rule.
• QK61 carries out a 61 point Gauss-Kronrod quadrature rule.
• 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 defines the weight function used by QC25C.
• QWGTO defines the weight functions used by QC25O.
• QWGTS defines the weight functions used by QC25S.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.