TEST_INT
Quadrature Tests

TEST_INT is a library of FORTRAN90 routines, using double precision arithmetic, which evaluate test integrands.

The test integrands would normally be used to testing one dimensional quadrature software. It is possible to invoke a particular function by number, or to try out all available functions, as demonstrated in the sample calling program.

The library includes not just the integrand, but also the interval of integration, and the exact value of the integral. Thus, for each integrand function, three subroutines are supplied. For instance, for function #9, we have the routines:

So once you have the calling sequences for these routines, you can easily evaluate the function, or integrate it between the appropriate limits, or compare your estimate of the integral to the exact value.

Moreover, since the same interface is used for each function, if you wish to work with problem 16 instead, you simply change the "09" to "16" in your routine calls.

If you wish to call all of the functions, then you simply use the generic interface, which again has three subroutines, but which requires you to specify the problem number as an extra input argument:

Finally, some demonstration routines are built in for simple quadrature methods. These routines include

and can be used with any of the sample integrands.

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 FORTRAN90 library of routines for estimating integrals in one dimension.

QUADPACK is a FORTRAN90 library of routines for estimating integrals of functions in one dimension.

QUADRULE is a FORTRAN90 library of routines which define various quadrature rules.

TEST_INT_HERMITE is a FORTRAN90 library of routines which defines some test integration problems over infinite intervals.

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

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.

Reference:

  1. Kendall Atkinson,
    An Introduction to Numerical Analysis,
    Prentice Hall, 1989,
    ISBN: 0471624896,
    LC: QA297.A94.1989.
  2. Roger Broucke,
    Algorithm 446: Ten Subroutines for the Manipulation of Chebyshev Series,
    Communications of the ACM,
    Volume 16, 1973, pages 254-256.
  3. Charles Clenshaw, Alan Curtis,
    A Method for Numerical Integration on an Automatic Computer,
    Numerische Mathematik,
    Volume 2, Number 1, December 1960, pages 197-205.
  4. Richard Crandall,
    Projects in Scientific Computing,
    Springer, 2005,
    ISBN: 0387950095,
    LC: Q183.9.C733.
  5. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  6. Hermann Engels,
    Numerical Quadrature and Cubature,
    Academic Press, 1980,
    ISBN: 012238850X,
    LC: QA299.3E5.
  7. Leslie Fox, Ian Parker,
    Chebyshev Polynomials in Numerical Analysis,
    Oxford Press, 1968,
    LC: QA297.F65.
  8. John Halton,
    On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,
    Numerische Mathematik,
    Volume 2, Number 1, December 1960, pages 84-90.
  9. John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thacher, Christoph Witzgall,
    Computer Approximations,
    Wiley, 1968,
    LC: QA297.C64.
  10. David Kahaner,
    Comparison of Numerical Quadrature Formulas,
    in Mathematical Software,
    edited by John Rice,
    Academic Press, 1971,
    ISBN: 012587250X,
    LC: QA1.M766.
  11. Prem Kythe, Pratap Puri,
    Computational Methods for Linear Integral Equations,
    Birkhaeuser, 2002,
    ISBN: 0817641920,
    LC: QA431.K97.
  12. Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
    QUADPACK: A Subroutine Package for Automatic Integration,
    Springer, 1983,
    ISBN: 3540125531,
    LC: QA299.3.Q36.
  13. Herbert Salzer, Norman Levine,
    Table of a Weierstrass Continuous Nondifferentiable Function,
    Mathematics of Computation,
    Volume 15, Number 74, April 1961, pages 120-130.
  14. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.

Source Code:

Examples and Tests:

List of Routines:

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

Last revised on 30 August 2005.