NINT_EXACTNESS
Exactness of Multidimensional Quadrature

NINT_EXACTNESS is a MATLAB program which investigates the polynomial exactness of a multidimensional quadrature rule which is defined over a finite rectangular product region.

The polynomial exactness of a quadrature rule is defined as the highest total degree D such that the quadrature rule is guaranteed to integrate exactly all polynomials of total degree DEGREE_MAX or less, ignoring roundoff. The total degree of a polynomial is the maximum of the degrees of all its monomial terms. The degree of a monomial term is the sum of the exponents. Thus, for instance, the DEGREE of

x²y z⁵

To be thorough, the program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates every possible monomial term, applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree is specified by the user as well.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt

For information on the form of these files, see the QUADRATURE_RULES directory listed below.

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

Usage:

nint_exactness ( 'prefix', degree_max )

• 'prefix' is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum total monomial degree to check. This should be a relatively small nonnegative number, particularly if the spatial dimension is high. A value of 5 or 10 might be reasonable, but a value of 50 or 100 is probably never a good input!

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:

NINT_EXACTNESS is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

NINT_EXACTNESS_MIXED, a MATLAB program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

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

PYRAMID_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the pyramid.

PRODUCT_RULE, a MATLAB program which can create a multidimensional quadrature rule as a product of 1D quadrature rules.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_TEST, a MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

QUADRULE, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions.

SPHERE_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule for the unit sphere;

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

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

TETRAHEDRON_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.

Source Code:

• nint_exactness.m, is the source code.

Examples and Tests:

CC_D1_O2 is a Clenshaw-Curtis order 2 rule for 1D.

• cc_d1_o2_x.txt, the abscissas of the rule.
• cc_d1_o2_w.txt, the weights of the rule.
• cc_d1_o2_r.txt, defines the region for the rule.
• cc_d1_o2_exact.txt, the results of the exactness test, up to degree 5.

CC_D1_O3 is a Clenshaw-Curtis order 3 rule for 1D. If you are paying attention, you may be surprised to see that a Clenshaw Curtis rule of odd order has one more degree of accuracy than you'd expect!

• cc_d1_o3_x.txt, the abscissas of the rule.
• cc_d1_o3_w.txt, the weights of the rule.
• cc_d1_o3_r.txt, defines the region for the rule.
• cc_d1_o3_exact.txt, the results of the exactness test, up to degree 5.

CC_D2_O3x3 is a Clenshaw-Curtis 3x3 product rule for 2D.

• cc_d2_o3x3_x.txt, the abscissas of the rule.
• cc_d2_o3x3_w.txt, the weights of the rule.
• cc_d2_o3x3_r.txt, defines the region for the rule.
• cc_d2_o3x3_exact.txt, the results of the exactness test, up to degree 8.

CC_D3_O3x3x3 is a Clenshaw-Curtis 3x3x3 product rule for 3D.

• cc_d3_o3x3x3_x.txt, the abscissas of the rule.
• cc_d3_o3x3x3_w.txt, the weights of the rule.
• cc_d3_o3x3x3_r.txt, defines the region for the rule.
• cc_d3_o3x3x3_exact.txt, the results of the exactness test, up to degree 5.

CCGL_D2_O3x2 is a product rule for 2D whose factors are a Clenshaw-Curtis of order 3 and a Gauss-Legendre rule of order 2.

• ccgl_d2_o3x2_x.txt, the abscissas of the rule.
• ccgl_d2_o3x2_w.txt, the weights of the rule.
• ccgl_d2_o3x2_r.txt, defines the region for the rule.
• ccgl_d2_o3x2_exact.txt, the results of the exactness test, up to degree 5.

CC_D2_LEVEL0 is a Clenshaw Curtis sparse rule for 2D of level 0 and order 1.

• cc_d2_level0_x.txt, the abscissas of the rule.
• cc_d2_level0_w.txt, the weights of the rule.
• cc_d2_level0_r.txt, defines the region for the rule.
• cc_d2_level0_exact.txt, the results of the exactness test, up to degree 4.

CC_D2_LEVEL1 is a Clenshaw Curtis sparse rule for 2D of level 1 and order 5.

• cc_d2_level1_x.txt, the abscissas of the rule.
• cc_d2_level1_w.txt, the weights of the rule.
• cc_d2_level1_r.txt, defines the region for the rule.
• cc_d2_level1_exact.txt, the results of the exactness test, up to degree 4.

CC_D2_LEVEL2 is a Clenshaw Curtis sparse rule for 2D of level 2 and order 13.

• cc_d2_level2_x.txt, the abscissas of the rule.
• cc_d2_level2_w.txt, the weights of the rule.
• cc_d2_level2_r.txt, defines the region for the rule.
• cc_d2_level2_exact.txt, the results of the exactness test, up to degree 6.

CC_D2_LEVEL3 is a Clenshaw Curtis sparse rule for 2D of level 3 and order 25.

• cc_d2_level3_x.txt, the abscissas of the rule.
• cc_d2_level3_w.txt, the weights of the rule.
• cc_d3_level2_r.txt, defines the region for the rule.
• cc_d2_level2_exact.txt, the results of the exactness test, up to degree 9.

CC_D2_LEVEL4 is a Clenshaw Curtis sparse grid rule for 2D of level 4 and order 65.

• cc_d2_level4_x.txt, the abscissas of the rule.
• cc_d2_level4_w.txt, the weights of the rule.
• cc_d3_level4_r.txt, defines the region for the rule.
• cc_d2_level4_exact.txt, the results of the exactness test, up to degree 17.

CCS_D2_LEVEL4 is a Clenshaw Curtis "Slow-Exponential-Growth" sparse grid rule for 2D of level 4 and order 49.

• ccs_d2_level4_x.txt, the abscissas of the rule.
• ccs_d2_level4_w.txt, the weights of the rule.
• ccs_d3_level4_r.txt, defines the region for the rule.
• ccs_d2_level4_exact.txt, the results of the exactness test, up to degree 17.

GL_D1_O3 is a Gauss-Legendre order 3 rule for 1D.

• gl_d1_o3_x.txt, the abscissas of the rule.
• gl_d1_o3_w.txt, the weights of the rule.
• gl_d1_o3_r.txt, defines the region for the rule.
• gl_d1_o3_exact.txt, the results of the exactness test, up to degree 5.

GL_D2_O3x3 is a Gauss-Legendre 3x3 product rule for 2D.

• gl_d2_o3x3_x.txt, the abscissas of the rule.
• gl_d2_o3x3_w.txt, the weights of the rule.
• gl_d2_o3x3_r.txt, defines the region for the rule.
• gl_d2_o3x3_exact.txt, the results of the exactness test, up to degree 5.

GL_D3_O3x3x3 is a Gauss-Legendre 3x3x3 product rule for 3D.

• gl_d3_o3x3x3_x.txt, the abscissas of the rule.
• gl_d3_o3x3x3_w.txt, the weights of the rule.
• gl_d3_o3x3x3_r.txt, defines the region for the rule.
• gl_d3_o3x3x3_exact.txt, the results of the exactness test, up to degree 5.

NCC_D1_O5 is a Newton-Cotes Closed order 5 rule for 1D.

• ncc_d1_o5_x.txt, the abscissas of the rule.
• ncc_d1_o5_w.txt, the weights of the rule.
• ncc_d1_o5_r.txt, defines the region for the rule.
• ncc_d1_o5_exact.txt, the results of the exactness test, up to degree 7.

NCC_D2_O5x5 is a Newton-Cotes Closed 5x5 product rule for 2D.

• ncc_d2_o5x5_x.txt, the abscissas of the rule.
• ncc_d2_o5x5_w.txt, the weights of the rule.
• ncc_d2_o5x5_r.txt, defines the region for the rule.
• ncc_d2_o5x5_exact.txt, the results of the exactness test, up to degree 7.

NCC_D3_O125 is a Newton-Cotes Closed 5x5x5 product rule for 3D.

• ncc_d3_o5x5x5_x.txt, the abscissas of the rule.
• ncc_d3_o5x5x5_w.txt, the weights of the rule.
• ncc_d3_o5x5x5_r.txt, defines the region for the rule.
• ncc_d3_o5x5x5_exact.txt, the results of the exactness test, up to degree 7.

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