TEST_NINT
Multi-dimensional Integration
Test Functions.


TEST_NINT is a MATLAB library which defines a set of test problems for the approximate computation of integrals over multi-dimensional regions.

Routines are available to evaluate the integrand, return the exact value of the integral, report the name of the problem, report the integration limits, get, set or modify a base point.

The integrands is assigned an index. The integrands can be invoked by index. Most integrands may be defined for any value of the spatial dimension, which we denote here by m. Most integrands are defined on the unit m-dimensional hypercube. Some integrands include one or more parameters. These generally have default values, which can be altered by the user.

For each problem, a set of routines are available with a standard interface, for manipulating and evaluating the problem. For a problem with index "87", for instance, we might have the following set of routines. The most important is P87_F which evaluates the integrand. We probably also need P87_LIM to determine the limits of integration, and P87_EXACT to get the exact value of the integral (if known). A number of routines are available to set, get, or randomize parameters associated with the problem.

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";

An Important Quote:

"When good results are obtained in integrating a high-dimensional function, we should conclude first of all that an especially tractable integrand was tried and not that a generally successful method has been found. A secondary conclusion is that we might have made a very good choice in selecting an integration method to exploit whatever features of f made it tractable."
Art Owen,
Latin Supercube Sampling for Very High Dimensional Simulation,
ACM Transactions on Modeling and Computer Simulations,
Volume 8, Number 1, January 1998, pages 71-102.

Licensing:

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

Languages:

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

Related Data and Programs:

CUBPACK, a FORTRAN90 library which estimates the integral of a function over a collection of N-dimensional hyperrectangles and simplices.

GSL, a C++ library which includes routines for estimating multidimensional integrals.

INTEGRAL_TEST, a FORTRAN90 program which uses some of these 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.

NINTLIB, a MATLAB library which numerically estimates integrals in multiple dimensions.

PRODUCT_RULE, a MATLAB program which can create a multidimensional quadrature rule as a product of one dimensional rules.

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, a MATLAB 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.

SANDIA_SPARSE, a MATLAB library which can produce a multidimensional sparse grid, based on a variety of 1D quadrature rules; only isotropic grids are generated, that is, the same rule is used in each dimension, and the same maximum order is used in each dimension.

SIMPACK, a FORTRAN77 library which approximates the integral of a function over a multidimensional simplex.

SPARSE_GRID_CC, a MATLAB library which creates sparse grids based on Clenshaw-Curtis rules.

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

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,
    ISBN: 0444875700,
    LC: Q183.9.I53.
  10. Alan Genz,
    A Package for Testing Multiple Integration Subroutines,
    in Numerical Integration: Recent Developments, Software and Applications,
    edited by Patrick Keast, Graeme Fairweather,
    Reidel, 1987, pages 337-340,
    ISBN: 9027725144,
    LC: QA299.3.N38.
  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. Claude Itzykson, Jean-Michel Drouffe,
    Statistical Field Theory,
    Volume 1,
    Cambridge, 1991,
    ISBN: 0-521-40806-7,
    LC: QC174.8.I89.
  14. 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.
  15. David Kahaner, Cleve Moler, Steven Nash,
    Numerical Methods and Software,
    Prentice Hall, 1989,
    ISBN: 0-13-627258-4,
    LC: TA345.K34.
  16. Bradley Keister,
    Multidimensional Quadrature Algorithms,
    Computers in Physics,
    Volume 10, Number 2, March/April, 1996, pages 119-122.
  17. Arnold Krommer, Christoph Ueberhuber,
    Numerical Integration on Advanced Compuer Systems,
    Springer, 1994,
    ISBN: 3540584102,
    LC: QA299.3.K76.
  18. Anargyros Papageorgiou, Joseph Traub,
    Faster Evaluation of Multidimensional Integrals,
    Computers in Physics,
    Volume 11, Number 6, November/December 1997, pages 574-578.
  19. 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.
  20. Arthur Stroud,
    Approximate Calculation of Multiple Integrals,
    Prentice Hall, 1971,
    ISBN: 0130438936,
    LC: QA311.S85.
  21. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.
  22. 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:

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


Last revised on 06 June 2007.