Description

QUADPACK is a Fortran library for the numerical computation of definite one-dimensional integrals (numerical quadrature). Development of this library, which had ceased in the 1980s, has been restarted. The original code is being modernized, and new methods are being added. The goal is a comprehensive and modern Fortran library that includes both classic and state-of-the-art methods for numerical integration.

Overview

The original QUADPACK code (written in the early 1980s) has been extensively refactored:

• It has been converted from FORTRAN 77 fixed form to modern free form syntax. This includes elimination of all GOTOs and other obsolescent language features.
• It is now a single stand-alone module, and has no dependencies on any other code from SLATEC or LINPACK.
• The SLATEC docstrings have been converted to Ford style, which allows for auto-generation of the API docs.
• Some typos have been corrected in the comments.
• General code cleanup and formatting.
• Added automated unit testing in GitHub CI.
• The separate routines for single and double precision versions have been eliminated. The library now exports a single (real32), double (real64) and quadruple (real128) precision interface using the same code by employing a preprocessor scheme.
• New procedures not present in the original QUADPACK have been added.
• The coefficients have been regenerated with full quadruple precision. (Note: this has not yet been done for all the coefficients in DQNG)
• Some bugs have been fixed in the original code. Note that this version includes the recent (Oct 2021) updates (see here and here) reported by the Scipy project.

To do list

• Additional docstring cleanups.
• Add more unit tests.
• In the unit tests, the "truth" values for the cases without analytical solutions need to be regenerated with some more precision so we have the exact results for the quad precision test.

Compiling

A Fortran Package Manager manifest file is included, so that the library and test cases can be compiled with FPM. For example:

fpm build --profile release
fpm test --profile release

To use quadpack within your fpm project, add the following to your fpm.toml file:

[dependencies]
quadpack = { git="https://github.com/jacobwilliams/quadpack.git" }

Or, to use a specific version:

[dependencies]
quadpack = { git="https://github.com/jacobwilliams/quadpack.git", tag = "2.1.0" }

Example

A simple example is given here (see the test folder for more examples):

subroutine test_qag
  use quadpack, only: dqag
  use iso_fortran_env, only: wp => real64 ! double precision
  implicit none

  real(wp), parameter :: a = 0.0_wp
  real(wp), parameter :: b = 1.0_wp
  integer, parameter :: key = 6
  integer, parameter :: limit = 100
  integer, parameter :: lenw = limit*4
  real(wp), parameter :: answer = 2.0_wp/sqrt(3.0_wp)

  real(wp) :: abserr, result, work(lenw)
  integer :: ier, iwork(limit), last, neval

  call dqag(f, a, b, epsabs, epsrel, key, result, &
            abserr, neval, ier, limit, lenw, last, &
            iwork, work)

  write(*,'(1P,A,1X,*(E13.6,1X))') &
        'result, error = ', result, abs(result-answer)

contains

  real(wp) function f(x)
    implicit none
    real(wp), intent(in) :: x
    real(wp), parameter :: pi = acos(-1.0_wp)
    f = 2.0_wp/(2.0_wp + sin(10.0_wp*pi*x))
  end function f

end subroutine test_qag

Which outputs:

 result, error = 1.154701E+00 2.220446E-16

Survey of procedures

The following list gives an overview of the QUADPACK integrators. The routine names for the double precision versions are preceded by the letter D, and the quadruple precision versions are preceded by Q.

• QNG : Is a simple non-adaptive automatic integrator, based on a sequence of rules with increasing degree of algebraic precision (Patterson, 1968).

• QAG : Is a simple globally adaptive integrator using the strategy of Aind (Piessens, 1973). It is possible to choose between 6 pairs of Gauss-Kronrod quadrature formulae for the rule evaluation component. The pairs of high degree of precision are suitable for handling integration difficulties due to a strongly oscillating integrand.

• QAGS : Is an integrator based on globally adaptive interval subdivision in connection with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

• QAGP : Serves the same purposes as QAGS, but also allows for eventual user-supplied information, i.e. the abscissae of internal singularities, discontinuities and other difficulties of the integrand function. The algorithm is a modification of that in QAGS.

• QAGI : Handles integration over infinite intervals. The infinite range is mapped onto a finite interval and then the same strategy as in QAGS is applied.

• QAWO : Is a routine for the integration of COS(OMEGA*X)*F(X) or SIN(OMEGA*X)*F(X) over a finite interval (A,B). OMEGA is is specified by the user The rule evaluation component is based on the modified Clenshaw-Curtis technique. An adaptive subdivision scheme is used connected with an extrapolation procedure, which is a modification of that in QAGS and provides the possibility to deal even with singularities in F.

• QAWF : Calculates the Fourier cosine or Fourier sine transform of F(X), for user-supplied interval (A,INFINITY), OMEGA, and F. The procedure of QAWO is used on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.

• QAWS : Integrates W(X)*F(X) over (A,B) with A<B finite, and W(X) = ((X-A)**ALFA)*((B-X)**BETA)*V(X) where V(X) = 1 or LOG(X-A) or LOG(B-X) or LOG(X-A)*LOG(B-X) and ALFA>(-1), BETA>(-1). The user specifies A, B, ALFA, BETA and the type of the function V. A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain A or B.

• QAWC : Computes the Cauchy Principal Value of F(X)/(X-C) over a finite interval (A,B) and for user-determined C. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point X = C.

  Each of the routines above also has a "more detailed" version with a name ending in E, as QAGE. These provide more information and control than the easier versions.

  The preceding routines are all automatic. That is, the user inputs his problem and an error tolerance. The routine attempts to perform the integration to within the requested absolute or relative error. There are, in addition, a number of non-automatic integrators. These are most useful when the problem is such that the user knows that a fixed rule will provide the accuracy required. Typically they return an error estimate but make no attempt to satisfy any particular input error request.

  • QK15, QK21, QK31, QK41, QK51, QK61: Estimate the integral on [a,b] using 15, 21,..., 61 point rule and return an error estimate.
  • QK15I: 15 point rule for (semi)infinite interval.
  • QK15W: 15 point rule for special singular weight functions.
  • QC25C: 25 point rule for Cauchy Principal Values
  • QC25F: 25 point rule for sin/cos integrand.
  • QMOMO: Integrates k-th degree Chebyshev polynomial times function with various explicit singularities.

Other procedures

The following procedures were not in the original QUADPACK, but are included in the new library:

• QUAD : The result is obtained using a sequence of 1, 3, 7, 15, 31, 63, 127, and 255 point interlacing formulae. The formulae are based on the optimal extension of the 3-point gauss formula. See: Patterson, 1968. See also QNG. This code is based on QUAD from NSWC Mathematical Library, with the addition of full quadruple-precision coefficients.

• AVINT : Integrates a function tabulated at arbitrarily spaced abscissas using overlapping parabolas. This procedure was originally from SLATEC.

• QNC79 : Integrate a function over a finite interval using a 7-point adaptive Newton-Cotes quadrature rule. This procedure was originally from SLATEC.

• GAUSS8 : Integrate a function over a finite interval using an adaptive 8-point Legendre-Gauss algorithm. This procedure was originally from SLATEC.

• SIMPSON : Integrate a function over a finite interval using an adaptive Simpson rule. See: Gander & Gautschi, 2000.

• LOBATTO : Integrate a function over a finite interval using an adaptive Lobatto rule. See: Gander & Gautschi, 2000.

Guidelines for the use of QUADPACK

Here it is not our purpose to investigate the question when automatic quadrature should be used. We shall rather attempt to help the user who already made the decision to use QUADPACK, with selecting an appropriate routine or a combination of several routines for handling his problem.

For both quadrature over finite and over infinite intervals, one of the first questions to be answered by the user is related to the amount of computer time he wants to spend, versus his -own- time which would be needed, for example, for manual subdivision of the interval or other analytic manipulations.

1. The user may not care about computer time, or not be willing to do any analysis of the problem. especially when only one or a few integrals must be calculated, this attitude can be perfectly reasonable. In this case it is clear that either the most sophisticated of the routines for finite intervals, QAGS, must be used, or its analogue for infinite intervals, GAGI. These routines are able to cope with rather difficult, even with improper integrals. This way of proceeding may be expensive. But the integrator is supposed to give you an answer in return, with additional information in the case of a failure, through its error estimate and flag. Yet it must be stressed that the programs cannot be totally reliable.

2. The user may want to examine the integrand function. If bad local difficulties occur, such as a discontinuity, a singularity, derivative singularity or high peak at one or more points within the interval, the first advice is to split up the interval at these points. The integrand must then be examined over each of the subintervals separately, so that a suitable integrator can be selected for each of them. If this yields problems involving relative accuracies to be imposed on -finite- subintervals, one can make use of QAGP, which must be provided with the positions of the local difficulties. However, if strong singularities are present and a high accuracy is requested, application of QAGS on the subintervals may yield a better result.

For quadrature over finite intervals we thus dispose of QAGS and

• QNG for well-behaved integrands,
• QAG for functions with an oscillating behaviour of a non specific type,
• QAWO for functions, eventually singular, containing a factor COS(OMEGA*X) or SIN(OMEGA*X) where OMEGA is known,
• QAWS for integrands with Algebraico-Logarithmic end point singularities of known type,
• QAWC for Cauchy Principal Values.

Remark

On return, the work arrays in the argument lists of the adaptive integrators contain information about the interval subdivision process and hence about the integrand behaviour: the end points of the subintervals, the local integral contributions and error estimates, and eventually other characteristics. For this reason, and because of its simple globally adaptive nature, the routine QAG in particular is well-suited for integrand examination. Difficult spots can be located by investigating the error estimates on the subintervals.

For infinite intervals we provide only one general-purpose routine, QAGI. It is based on the QAGS algorithm applied after a transformation of the original interval into (0,1). Yet it may eventuate that another type of transformation is more appropriate, or one might prefer to break up the original interval and use QAGI only on the infinite part and so on. These kinds of actions suggest a combined use of different QUADPACK integrators. Note that, when the only difficulty is an integrand singularity at the finite integration limit, it will in general not be necessary to break up the interval, as QAGI deals with several types of singularity at the boundary point of the integration range. It also handles slowly convergent improper integrals, on the condition that the integrand does not oscillate over the entire infinite interval. If it does we would advise to sum succeeding positive and negative contributions to the integral -e.g. integrate between the zeros- with one or more of the finite-range integrators, and apply convergence acceleration eventually by means of QUADPACK subroutine QELG which implements the Epsilon algorithm. Such quadrature problems include the Fourier transform as a special case. Yet for the latter we have an automatic integrator available, QAWF.

Documentation

The API documentation for the current master branch can be found here. This is generated by processing the source files with FORD. Note that the procedures listed in the API documentation are the double precision version (DQNG, etc.)

License

The original Quadpack was a public domain work of the United States government. The modifications are released under a permissive (BSD-3) license.

References

• R. Piessens, E. deDoncker-Kapenga, C. Uberhuber, D. Kahaner Quadpack: a Subroutine Package for Automatic Integration Springer Verlag, 1983. Series in Computational Mathematics v.1 515.43/Q1S 100394Z
• Paola Favati, Grazia Lotti, Francesco Romani, Algorithm 691: Improving QUADPACK automatic integration routines, ACM Transactions on Mathematical Software, Volume 17, Issue 2, June 1991, pp 218-232.
• Original SLATEC code from Netlib. Last modified 11 Oct 2021.
• W. Gander and W. Gautschi, "Adaptive Quadrature - Revisited", BIT Vol. 40, No. 1, March 2000, pp. 84--101.

Other versions

There are other versions of Quadpack out there. There is at least one project to provide module interface to the unmodified Fortran 77 code (see nshaffer/modern_quadpack). The license for these are not specified. Another fixed to free conversion can be found at John Burkardt's site (this is not an aggressive modernization though and also has an LGPL license). Also note that the Quadpack code in SLATEC is slightly modified from the stand-alone one at Netlib. It is not known if these modifications were anything significant.

See also

• quadrature-fortran

Keywords

• survey of integrators, guidelines for selection, quadpack, automatic integrator, general-purpose, integrand examinator, globally adaptive, gauss-kronrod, infinite intervals, transformation, extrapolation, singularities at user specified points, (end-point) singularities, cauchy principal value, clenshaw-curtis method, special-purpose, fourier integral, integration between zeros, convergence acceleration, integrand with oscillatory cos or sin factor, (end point) singularities, 25-point clenshaw-curtis integration, smooth integrand, non-adaptive, gauss-kronrod (patterson), epsilon algorithm, algebraico-logarithmic end point singularities, chebyshev series expansion, fast fourier transform