QUADRATURE_TEST_GENZ
Quadrature Rule Applied to Genz Integrals

QUADRATURE_TEST_GENZ is a FORTRAN90 program which reads three files that define a quadrature rule, applies the quadrature rule to the set of Genz 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);

The Genz integrand functions include:

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

Usage:

quadrature_test_genz prefix

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_GENZ is available in a FORTRAN90 version.

Related Data and Programs:

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

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

NINT_EXACTNESS, a FORTRAN90 program which demonstrates how to measure the polynomial exactness of a multidimensional quadrature rule.

NINTLIB, a FORTRAN90 library which numerically estimates integrals in multiple dimensions.

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 FORTRAN90 program which reads files defining a quadrature rule, and applies them to all the test integrals defined by TEST_NINT.

STROUD, a FORTRAN90 library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_NINT, a FORTRAN90 library which defines a set of integrand functions to be used for testing multidimensional quadrature rules and routines.

TESTPACK, a FORTRAN90 library which defines a set of integrands used to test multidimensional quadrature.

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.

Source Code:

• quadrature_test_genz.f90, the source code.

Examples and Tests:

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

• cce_d2_level4_x.txt, the "X" file.
• cce_d2_level4_w.txt, the "W" file.
• cce_d2_level4_r.txt, the "R" file.
• cce_d2_level4_genz.txt, the test results.

List of Routines:

• MAIN is the main program for QUADRATURE_TEST_GENZ.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.