TESTPACK
Testing Multidimensional Integration Routines

TESTPACK is a C++ program, using double precision arithmetic, 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";

Related Data and Programs:

INTEGRAL_TEST is a FORTRAN90 executable 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 is a FORTRAN90 library of routines for estimating the integral of a function over a one-dimensional interval.

NINTLIB is a C++ library of routines for estimating the integral of a function over a one-dimensional interval.

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

QUADRULE is a C++ library of routines which define a variety of (mostly 1-dimensional) quadrature rules.

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

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

STROUD is a FORTRAN90 library of routines which define a variety of quadrature rules over various "interesting" geometric shapes.

TEST_NINT is a C++ library of routines that can be used to test N-dimensional quadrature routines.

TESTPACK is also available in a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Reference:

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:

• testpack.f90, the source code.
• testpack.csh, commands to compile the source code.
• testpack.out, the output from a run of the program.

List of Routines:

• MAIN is the main program for TESTPACK.
• ADAPT carries out adaptive multidimensional quadrature.
• FUNCTN evaluates one of the test integrand functions.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_POWER returns the value of I^J.
• I4VEC_SUM sums the entries of an I4VEC.
• MULTST tests a multidimensional integration routine.
• PHI estimates the normal cumulative density function.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8VEC_DOT computes the dot product of a pair of R8VEC's.
• R8VEC_MEDIAN computes the median of an R8VEC.
• R8VEC_PRODUCT returns the product of the entries of an R8VEC.
• R8VEC_SUM returns the sum of an R8VEC.
• RANDOM_SCHRAGE is a portable random number generator
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TUPLE_NEXT computes the next element of a tuple space.
• VALINT computes the exact integrals of the test functions.

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