QUADRATURE_TEST_2D
Quadrature Rule Applied to 2D Test Integrals

QUADRATURE_TEST_2D is a FORTRAN90 program which reads three files that define a 2D 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);

The test integrals come from the TEST_INT_2D library.

Usage:

quadrature_test_2d 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_2D is available in a C++ version and a FORTRAN90 version and a MATLAB 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.

PRODUCT_RULE, a FORTRAN90 program which can create a multidimensional quadrature rule as a product of one dimensional rules.

PRODUCT_RULE_GL, a dataset directory which contains multidimensional quadrature rules formed as products of 1D Gauss-Legendre rules.

PRODUCT_RULE_TANH_SINH, a dataset directory which contains multidimensional quadrature rules formed as products of 1D tanh-sinh 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 FORTRAN90 program which reads files defining a quadrature rule, and applies them to all the test integrals defined by TEST_NINT.

QUADRATURE_TEST_GENZ, a FORTRAN90 program which reads files defining a quadrature rule, and applies it to the Genz test integrands.

SPARSE_GRID_TANH_SINH, a dataset directory which contains multidimensional Smolyak sparse grids based on a 1D tanh-sinh rule;

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

TEST_INT_2D, a FORTRAN90 library which defines test integrands for 2D quadrature rules.

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.

Source Code:

• quadrature_test_2d.f90, the source code.

Examples and Tests:

PRODUCT_RULE_GL is a family of Gauss_Legendre product rules which includes 2D rules of orders 1x1, 2x2, 4x4, 8x8, 16x16 and 32x32.

• product_rule_gl.html, describes the Gauss-Legendre product rules.
• product_rule_gl_output, the output from running the test on the 2D rules.

PRODUCT_RULE_TANH_SINH is a family of tanh-sinh product rules which includes 2D rules of orders 1x1, 3x3, 7x7, 15x15, 31x31, 63x63, and 127x127.

• product_rule_tanh_sinh.html, describes the tanh-sinh product rules.
• product_rule_tanh_sinh_output, the output from running the test on the 2D rules.

SPARSE_GRID_CC is a family of Clenshaw Curtis sparse grid rules which includes rules of order 1, 5, 13, 29, 65, 145, 321, 705. 1537, 3329 and 7169. These rules have problems with many of the test integrands, which have endpoint singularities.

• sparse_grid_cc.html, describes the Clenshaw Curtis sparse grid rules.
• sparse_grid_cc_output, the output from running the test on the 2D rules.

SPARSE_GRID_F2 is a family of Fejer Type 2 sparse grid rules which includes rules of order 1, 5, 17, 49, 129, 321, 769, and 1793.

• sparse_grid_f2.html, describes the Fejer Type 2 sparse grid rules.
• sparse_grid_f2_output, the output from running the test on the 2D rules.

SPARSE_GRID_NCO is a family of Newton Cotes Open sparse grid rules which includes rules of order 1, 5, 17, 49, 129, 321, and 769 points. These rules do a terrible job, which reflects the instability of the Newton Cotes rules.

• sparse_grid_nco.html, describes the Newton Cotes Open sparse grid rules.
• sparse_grid_nco_output, the output from running the test on the 2D rules.

SPARSE_GRID_TANH_SINH is a family of tanh-sinh sparse grid rules which includes rules of order 1, 5, 17, 49, 129, 321, 769, 1793 and 4097 points.

• sparse_grid_tanh_sinh.html, describes the tanh-sinh sparse grid rules.
• sparse_grid_tanh_sinh_output, the output from running the test on the 2D rules.

List of Routines:

• MAIN is the main program for QUADRATURE_TEST_2D.
• 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.
• DTABLE_DATA_READ reads data from a DTABLE file.
• DTABLE_HEADER_READ reads the header from a DTABLE file.
• 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.
• 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.