QUADRULE
Quadrature Rules


QUADRULE is a FORTRAN77 library which sets up a variety of quadrature rules, used to approximate the integral of a function over various domains.

QUADRULE returns the abscissas and weights for a variety of one dimensional quadrature rules for approximating the integral of a function. The best rule is generally Gauss-Legendre quadrature, but other rules offer special features, including the ability to handle certain weight functions, to approximate an integral on an infinite integration region, or to estimate the approximation error.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

QUADRULE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Programs:

CLENSHAW_CURTIS_RULE, a FORTRAN77 program which defines a Clenshaw Curtis quadrature rule.

INT_EXACTNESS, a FORTRAN90 program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_LAGUERRE, a FORTRAN90 program which checks the polynomial exactness of a 1-dimensional quadrature rule for a semi-infinite interval.

INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

KRONROD, a FORTRAN77 library which can compute a Gauss and Gauss-Kronrod pair of quadrature rules of arbitrary order, by Robert Piessens, Maria Branders.

LAGUERRE_RULE, a FORTRAN77 program which can compute and print a Gauss-Laguerre quadrature rule for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

LINE_FELIPPA_RULE, a FORTRAN77 library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

LINE_NCO_RULE, a FORTRAN77 library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

PATTERSON_RULE, a FORTRAN77 program which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

QUADMOM, a FORTRAN77 library which computes a Gaussian quadrature rule for a weight function rho(x) based on the Golub-Welsch procedure that only requires knowledge of the moments of rho(x).

QUADPACK, a FORTRAN77 library which contains a variety of routines for numerical estimation of integrals in 1D.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_TEST, a FORTRAN90 program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

R16_HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule, using "quadruple precision real" arithmetic.

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

TANH_SINH_RULE, a FORTRAN90 program which computes and writes out a tanh-sinh quadrature rule of given order.

TEST_INT, a FORTRAN77 library which contains a number of functions that may be used as test integrands for quadrature rules in 1D.

TEST_INT_HERMITE, a FORTRAN77 library which defines some test integration problems over infinite intervals.

TEST_INT_LAGUERRE, a FORTRAN77 library which defines test integrands for the interval [a,+oo) and weight exp(-x);

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.

TOMS655, a FORTRAN77 library which computes the weights for interpolatory quadrature rule;
this library is commonly called IQPACK;
this is a FORTRAN90 version of ACM TOMS algorithm 655.

TOMS793, a FORTRAN77 library which carries out Gauss quadrature for rational functions, by Walter Gautschi; this is ACM TOMS algorithm 793.

Reference:

  1. Milton Abramowitz, Irene Stegun,
    Handbook of Mathematical Functions,
    National Bureau of Standards, 1964,
    ISBN: 0-486-61272-4,
    LC: QA47.A34.
  2. Donald Anderson,
    Gaussian Quadrature Formulae for the integral from 0 to 1 of -ln(X) f(X) dx,
    Mathematics of Computation,
    Volume 19, Number 91, July 1965, pages 477-481.
  3. Claudio Canuto, Yousuff Hussaini, Alfio Quarteroni, Thomas Zang,
    Spectral Methods in Fluid Dynamics,
    Springer, 1993,
    ISNB13: 978-3540522058,
    LC: QA377.S676.
  4. Charles Clenshaw, Alan Curtis,
    A Method for Numerical Integration on an Automatic Computer,
    Numerische Mathematik,
    Volume 2, Number 1, December 1960, pages 197-205.
  5. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  6. Sylvan Elhay, Jaroslav Kautsky,
    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
    ACM Transactions on Mathematical Software,
    Volume 13, Number 4, December 1987, pages 399-415.
  7. Hermann Engels,
    Numerical Quadrature and Cubature,
    Academic Press, 1980,
    ISBN: 012238850X,
    LC: QA299.3E5.
  8. Gwynne Evans,
    Practical Numerical Integration,
    Wiley, 1993,
    ISBN: 047193898X,
    LC: QA299.3E93.
  9. Simeon Fatunla,
    Numerical Methods for Initial Value Problems in Ordinary Differential Equations,
    Academic Press, 1988,
    ISBN: 0122499301,
    LC: QA372.F35.
  10. Walter Gautschi,
    Numerical Quadrature in the Presence of a Singularity,
    SIAM Journal on Numerical Analysis,
    Volume 4, Number 3, September 1967, pages 357-362.
  11. Alan Genz, Bradley Keister,
    Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight,
    Journal of Computational and Applied Mathematics,
    Volume 71, 1996, pages 299-309.
  12. Florian Heiss, Viktor Winschel,
    Likelihood approximation by numerical integration on sparse grids,
    Journal of Econometrics,
    Volume 144, 2008, pages 62-80.
  13. Francis Hildebrand,
    Introduction to Numerical Analysis,
    Dover, 1987,
    ISBN13: 978-0486653631,
    LC: QA300.H5.
  14. Zdenek Kopal,
    Numerical Analysis,
    John Wiley, 1955,
    LC: QA297.K6.
  15. Vladimir Krylov,
    Approximate Calculation of Integrals,
    Dover, 2006,
    ISBN: 0486445798,
    LC: QA311.K713.
  16. Prem Kythe, Michael Schaeferkotter,
    Handbook of Computational Methods for Integration,
    Chapman and Hall, 2004,
    ISBN: 1-58488-428-2,
    LC: QA299.3.K98.
  17. Leon Lapidus, John Seinfeld,
    Numerical Solution of Ordinary Differential Equations,
    Mathematics in Science and Engineering, Volume 74,
    Academic Press, 1971,
    ISBN: 0124366503,
    LC: QA3.M32.v74
  18. Federico Paris, Jose Canas,
    Boundary Element Method: Fundamentals and Applications,
    Oxford, 1997,
    ISBN: 0-19-856543-7
    LC: TA347.B69.P34.
  19. Thomas Patterson,
    The Optimal Addition of Points to Quadrature Formulae,
    Mathematics of Computation,
    Volume 22, Number 104, October 1968, pages 847-856.
  20. Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
    QUADPACK: A Subroutine Package for Automatic Integration,
    Springer, 1983,
    ISBN: 3540125531,
    LC: QA299.3.Q36.
  21. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.
  22. Lloyd Trefethen,
    Is Gauss Quadrature Better Than Clenshaw-Curtis?,
    SIAM Review,
    Volume 50, Number 1, March 2008, pages 67-87.
  23. Joerg Waldvogel,
    Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
    BIT Numerical Mathematics,
    Volume 43, Number 1, 2003, pages 1-18.
  24. Stephen Wolfram,
    The Mathematica Book,
    Fourth Edition,
    Cambridge University Press, 1999,
    ISBN: 0-521-64314-7,
    LC: QA76.95.W65.
  25. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Source Code:

Examples and Tests:

List of Routines:

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


Last revised on 15 September 2012.