Testing Multidimensional Integration Routines

TESTPACK is a C++ program which demonstrates the testing of a routine for multidimensional integration.

In this case, the code being tested is a subroutine known as ADAPT, written by Genz. The code is tested on six test integrand functions, also defined by Genz. The test is done with a variety of spatial dimensions, parameter values, and difficulty factors.

With M denoting the spatial dimension, R a parameter, C a scaling vector, and X0 a displacement vector, the test functions can be summarized as:

  1. f(x) = cos ( 2 * pi * r + sum ( c(1:m) * x(1:m) ) ),
    Genz "Oscillatory";
  2. f(x) = 1 / product ( c(1:m)^2 + (x(1:m) - x0(1:m))^2),
    Genz "Product Peak";
  3. f(x) = 1 / ( 1 + sum ( c(1:m) * x(1:m) ) )^(m+r),
    Genz "Corner Peak";
  4. f(x) = exp(-sum(c(1:m)^2 * ( x(1:m) - x0(1:m))^2 ) ),
    Genz "Gaussian";
  5. f(x) = exp ( - sum ( c(1:m) * abs ( x(1:m) - x0(1:m) ) ) ),
    Genz "Continuous";
  6. f(x) = exp(sum(c(1:m)*x(1:m)) for x(1:m) <= x0(1:m), 0 otherwise,
    Genz "Discontinuous";


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


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

Related Data and Programs:

INTEGRAL_TEST, a FORTRAN90 program which tests the suitability of a set of N points for use in an equal-weight quadrature rule over the M-dimensional unit hypercube.

INTLIB, a FORTRAN90 library which estimates the integral of a function over a one-dimensional interval.

NINTLIB, a C++ library which estimates the integral of a function over a one-dimensional interval.

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

QUADRULE, a C++ library which defines a variety of (mostly 1-dimensional) quadrature rules.

SMOLPACK, a C library which uses the Genz test functions to demonstrate quadrature in multiple dimensions using a sparse grid.

SPINTERP, a MATLAB library which uses the Genz test functions to demonstrate multilinear interpolation on a sparse grid.

STROUD, a C++ library which defines a variety of quadrature rules over various "interesting" geometric shapes.

TEST_NINT, a C++ library which can be used to test N-dimensional quadrature routines.


  1. 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.
  2. 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
  3. John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thatcher, Christoph Witzgall,
    Computer Approximations,
    Wiley, 1968,
    LC: QA297.C64.
  4. Linus Schrage,
    A More Portable Fortran Random Number Generator,
    ACM Transactions on Mathematical Software,
    Volume 5, Number 2, June 1979, pages 132-138.

Source Code:

List of Routines:

You can go up one level to the C++ source codes.

Last revised on 22 March 2007.