QUADRATURE_TEST
Quadrature Rule Applied to Test Integrals

QUADRATURE_TEST is a MATLAB program which reads three files that define a quadrature rule, applies the quadrature rule to a set of test integrals, and reports the results.

The quadrature rule is defined by three text files:

  1. the "X" file lists the abscissas (N rows, M columns);
  2. the "W" file lists the weights (N rows);
  3. the "R" file lists the integration region corners (2 rows, M columns);
For more on quadrature rules, see the QUADRATURE_RULES listing below.

The test integrals come from the TEST_NINT library.

The list of integrand functions includes:

  1. f(x) = ( sum ( x(1:m) ) )**2;
  2. f(x) = ( sum ( 2 * x(1:m) - 1 ) )**4;
  3. f(x) = ( sum ( x(1:m) ) )**5;
  4. f(x) = ( sum ( 2 * x(1:m) - 1 ) )**6;
  5. f(x) = 1 / ( 1 + sum ( 2 * x(1:m) ) );
  6. f(x) = product ( 2 * abs ( 2 * x(1:m) - 1 ) );
  7. f(x) = product ( pi / 2 ) * sin ( pi * x(1:m) );
  8. f(x) = ( sin ( (pi/4) * sum ( x(1:m) ) ) )**2;
  9. f(x) = exp ( sum ( c(1:m) * x(1:m) ) );
  10. f(x) = sum ( abs ( x(1:m) - 0.5 ) );
  11. f(x) = exp ( sum ( abs ( 2 * x(1:m) - 1 ) ) );
  12. f(x) = product ( 1 <= i <= m ) ( i * cos ( i * x(i) ) );
  13. f(x) = product ( 1 <= i <= m ) t(n(i))(x(i)), t(n(i)) is a Chebyshev polynomial;
  14. f(x) = sum ( 1 <= i <= m ) (-1)**i * product ( 1 <= j <= i ) x(j);
  15. f(x) = product ( 1 <= i <= order ) x(mod(i-1,m)+1);
  16. f(x) = sum ( abs ( x(1:m) - x0(1:m) ) );
  17. f(x) = sum ( ( x(1:m) - x0(1:m) )**2 );
  18. f(x) = 1 inside an m-dimensional sphere around x0(1:m), 0 outside;
  19. f(x) = product ( sqrt ( abs ( x(1:m) - x0(1:m) ) ) );
  20. f(x) = ( sum ( x(1:m) ) )**power;
  21. f(x) = c * product ( x(1:m)^e(1:m) ) on the surface of an m-dimensional unit sphere;
  22. f(x) = c * product ( x(1:m)^e(1:m) ) in an m-dimensional ball;
  23. f(x) = c * product ( x(1:m)^e(1:m) ) in the unit m-dimensional simplex;
  24. f(x) = product ( abs ( 4 * x(1:m) - 2 ) + c(1:m) ) / ( 1 + c(1:m) ) );
  25. f(x) = exp ( c * product ( x(1:m) ) );
  26. f(x) = product ( c(1:m) * exp ( - c(1:m) * x(1:m) ) );
  27. f(x) = cos ( 2 * pi * r + sum ( c(1:m) * x(1:m) ) ),
    Genz "Oscillatory";
  28. f(x) = 1 / product ( c(1:m)**2 + (x(1:m) - x0(1:m))**2),
    Genz "Product Peak";
  29. f(x) = 1 / ( 1 + sum ( c(1:m) * x(1:m) ) )**(m+r),
    Genz "Corner Peak";
  30. f(x) = exp(-sum(c(1:m)**2 * ( x(1:m) - x0(1:m))**2 ) ),
    Genz "Gaussian";
  31. f(x) = exp ( - sum ( c(1:m) * abs ( x(1:m) - x0(1:m) ) ) ), Genz "Continuous";
  32. f(x) = exp(sum(c(1:m)*x(1:m)) for x(1:m) <= x0(1:m), 0 otherwise,
    Genz "Discontinuous";

Usage:

quadrature_test ( 'prefix' )
where
prefix
the common prefix for the files containing the abscissa (X), weight (W) and region (R) information of the quadrature rule;

If the arguments are not supplied on the command line, the program will prompt for them.

Licensing:

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

Languages:

QUADRATURE_TEST is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

INTEGRAL_TEST, a FORTRAN90 program which uses test integrals to evaluate sets of quadrature points.

NINT_EXACTNESS, a MATLAB program which demonstrates how to measure the polynomial exactness of a multidimensional quadrature rule.

QUADRATURE_RULES, a dataset directory which contains a description and examples of quadrature rules defined by a set of "X", "W" and "R" files.

QUADRATURE_TEST_2D, a MATLAB program which reads files defining a 2D quadrature rule, and applies them to all the test integrals defined by TEST_INT_2D.

STROUD, a MATLAB library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_NINT, a MATLAB library which defines a set of integrand functions to be used for testing multidimensional quadrature rules and routines.

TESTPACK, a MATLAB library which defines a set of integrands used to test multidimensional quadrature.

Reference:

  1. JD Beasley, SG Springer,
    Algorithm AS 111: The Percentage Points of the Normal Distribution,
    Applied Statistics,
    Volume 26, 1977, pages 118-121.
  2. Paul Bratley, Bennett Fox, Harald Niederreiter,
    Implementation and Tests of Low-Discrepancy Sequences,
    ACM Transactions on Modeling and Computer Simulation,
    Volume 2, Number 3, July 1992, pages 195-213.
  3. Roger Broucke,
    Algorithm 446: Ten Subroutines for the Manipulation of Chebyshev Series,
    Communications of the ACM,
    Volume 16, 1973, pages 254-256.
  4. William Cody, Kenneth Hillstrom,
    Chebyshev Approximations for the Natural Logarithm of the Gamma Function, Mathematics of Computation,
    Volume 21, Number 98, April 1967, pages 198-203.
  5. Richard Crandall,
    Projects in Scientific Computing,
    Springer, 2005,
    ISBN: 0387950095,
    LC: Q183.9.C733.
  6. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  7. Gerald Folland,
    How to Integrate a Polynomial Over a Sphere,
    American Mathematical Monthly,
    Volume 108, Number 5, May 2001, pages 446-448.
  8. Leslie Fox, Ian Parker,
    Chebyshev Polynomials in Numerical Analysis,
    Oxford Press, 1968,
    LC: QA297.F65.
  9. Alan Genz,
    Testing Multidimensional Integration Routines,
    in Tools, Methods, and Languages for Scientific and Engineering Computation,
    edited by B Ford, JC Rault, F Thomasset,
    North-Holland, 1984, pages 81-94.
  10. Alan Genz,
    A Package for Testing Multiple Integration Subroutines,
    in Numerical Integration: Recent Developments, Software and Applications,
    edited by Patrick Keast, Graeme Fairweather,
    D Reidel, 1987, pages 337-340.
  11. Kenneth Hanson,
    Quasi-Monte Carlo: halftoning in high dimensions?
    in Computatinal Imaging,
    Edited by CA Bouman, RL Stevenson,
    Proceedings SPIE,
    Volume 5016, 2003, pages 161-172.
  12. John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thatcher, Christoph Witzgall,
    Computer Approximations,
    Wiley, 1968,
    LC: QA297.C64.
  13. Stephen Joe, Frances Kuo
    Remark on Algorithm 659: Implementing Sobol's Quasirandom Sequence Generator,
    ACM Transactions on Mathematical Software,
    Volume 29, Number 1, March 2003, pages 49-57.
  14. David Kahaner, Cleve Moler, Steven Nash,
    Numerical Methods and Software,
    Prentice Hall, 1989,
    ISBN: 0-13-627258-4,
    LC: TA345.K34.
  15. Bradley Keister,
    Multidimensional Quadrature Algorithms,
    Computers in Physics,
    Volume 10, Number 2, March/April, 1996, pages 119-122.
  16. Arnold Krommer, Christoph Ueberhuber,
    Numerical Integration on Advanced Compuer Systems,
    Springer, 1994,
    ISBN: 3540584102,
    LC: QA299.3.K76.
  17. Anargyros Papageorgiou, Joseph Traub,
    Faster Evaluation of Multidimensional Integrals,
    Computers in Physics,
    Volume 11, Number 6, November/December 1997, pages 574-578.
  18. Thomas Patterson,
    On the Construction of a Practical Ermakov-Zolotukhin Multiple Integrator,
    in Numerical Integration: Recent Developments, Software and Applications,
    edited by Patrick Keast and Graeme Fairweather,
    D. Reidel, 1987, pages 269-290.
  19. Arthur Stroud,
    Approximate Calculation of Multiple Integrals,
    Prentice Hall, 1971,
    ISBN: 0130438936,
    LC: QA311.S85.
  20. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.
  21. Xiaoqun Wang, Kai-Tai Fang,
    The Effective Dimension and quasi-Monte Carlo Integration,
    Journal of Complexity,
    Volume 19, pages 101-124, 2003.

Source Code:

Examples and Tests:

CC_D2_LEVEL4 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 2 of level 4, 65 points.

CC_D2_LEVEL5 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 2 of level 5, 145 points.

CC_D6_LEVEL0 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 0, 1 point.

CC_D6_LEVEL1 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 1, 13 points.

CC_D6_LEVEL2 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 2, 85 points.

CC_D6_LEVEL3 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 3, 389 points.

CC_D6_LEVEL4 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 4, 1457 points.

CC_D6_LEVEL5 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 5, 4865 points.

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

Last revised on 11 November 2009.