QUADPACK
Numerical Integration

QUADPACK is a library of FORTRAN77 routines, using single precision arithmetic, for estimating integrals.

The QUADPACK estimate 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.

Related Data and Programs:

CLENSHAW_CURTIS is a FORTRAN90 library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

INTLIB is a library of FORTRAN90 routines for numerically estimating integrals.

MACHINE is a FORTRAN77 library of machine arithmetic constants, which is called by QUADPACK.

NMS is a FORTRAN90 library which includes QUADPACK.

QUADPACK is also available in a FORTRAN77 version .

QUADRULE is a library of FORTRAN90 routines that define quadrature rules for various intervals and weight functions.

SLATEC is a FORTRAN90 library which includes QUADPACK.

STROUD is a library of FORTRAN90 routines that define quadrature rules for various geometric shapes.

TEST_INT includes a library of FORTRAN90 routines that define test integration problems.

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

TOMS379 is a FORTRAN77 library which estimates an integral.

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

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

TOMS468 is a FORTRAN77 library for the "automatic" integration of a function.

XERROR is a FORTRAN77 library of error handling routines, which is called by QUADPACK.

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.csh, commands to compile the source code.

Examples and Tests:

• quadpack_prb.f, a sample problem.
• quadpack_prb.csh, commands to compile, link and run the sample problem.
• quadpack_prb.out, sample problem output.

List of Routines:

• D1MACH returns double precision machine-dependent constants.
• DGTSL solves a general tridiagonal linear system.
• DQAGE estimates a definite integral.
• DQAG approximates an integral over a finite interval.
• DQAGIE estimates an integral over a semi-infinite or infinite interval.
• DQAGI estimates an integral over a semi-infinite or infinite interval.
• DQAGPE computes a definite integral.
• DQAGP computes a definite integral.
• DQAGSE estimates the integral of a function.
• DQAGS estimates the integral of a function.
• DQAWCE computes a Cauchy principal value.
• DQAWC computes a Cauchy principal value.
• DQAWFE computes Fourier integrals.
• DQAWF computes Fourier integrals over the interval [ A, +Infinity ).
• DQAWOE computes the integrals of oscillatory integrands.
• DQAWO computes the integrals of oscillatory integrands.
• DQAWSE estimates integrals with algebraico-logarithmic endpoint singularities.
• DQAWS estimates integrals with algebraico-logarithmic endpoint singularities.
• DQC25C returns integration rules for Cauchy Principal Value integrals.
• DQC25F returns integration rules for functions with a COS or SIN factor.
• DQC25S returns rules for algebraico-logarithmic end point singularities.
• DQCHEB computes the Chebyshev series expansion.
• DQELG carries out the Epsilon extrapolation algorithm.
• DQK15 carries out a 15 point Gauss-Kronrod quadrature rule.
• DQK15I applies a 15 point Gauss-Kronrod quadrature on an infinite interval.
• DQK15W applies a 15 point Gauss-Kronrod rule for a weighted integrand.
• DQK21 carries out a 21 point Gauss-Kronrod quadrature rule.
• DQK31 carries out a 31 point Gauss-Kronrod quadrature rule.
• DQK41 carries out a 41 point Gauss-Kronrod quadrature rule.
• DQK51 carries out a 51 point Gauss-Kronrod quadrature rule.
• DQK61 carries out a 61 point Gauss-Kronrod quadrature rule.
• DQMOMO computes modified Chebyshev moments.
• DQNG estimates an integral, using non-adaptive integration.
• DQPSRT maintains the order of a list of local error estimates.
• DQWGTC defines the weight function used by DQC25C.
• DQWGTF defines the weight functions used by DQC25F.
• DQWGTS defines the weight functions used by DQC25S.
• FDUMP produces a symbolic dump.
• J4SAVE saves and recalls global variables.
• S88FMT copies Hollerith versions of integers into a string.
• XERABT aborts program execution.
• XERCTL allows the user control over individual errors.
• XERPRT prints a message on each file receiving error messages.
• XERROR processes a diagnostic error message.
• XERRWV processes a diagnostic error message.
• XERSAV records that a particular error has occurred.
• XGETUA reports the unit numbers of files receiving error messages.

You can go up one level to the FORTRAN77 source codes.