QUADRATURE_TEST_2D
Quadrature Rules Applied to 2D Test Integrals

quadrature_2d_test, a MATLAB 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_2d_test ( '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.

Related Data and Programs:

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

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

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

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

TEST_NINT, a MATLAB library which defines integrand functions for testing multidimensional quadrature rules and routines.

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

Source Code:

• quadrature_2d_test.m, calls all the tests.
• quadrature_2d_test.sh, runs the tests.
• quadrature_2d_test.txt, the output file.

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.txt, 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 points.

• product_rule_tanh_sinh.html, describes the tanh-sinh product rules.
• product_rule_tanh_sinh.txt, 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 points. 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.txt, 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 points.

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