INT_EXACTNESS
Exactness of One Dimensional Quadrature Rules

INT_EXACTNESS is a MATLAB program which investigates the polynomial exactness of a one dimensional quadrature rule defined on a finite interval.

Although this program was intended for use with 1D rules, it should also work for quadrature rules defined for multiple dimensions. The spatial dimension is inferred from the form of the "X" file that is one of the input quadrature files. See the last example for a demonstration of the use of this program for a 2D quadrature rule. See the companion program NINT_EXACTNESS for a program that was explicitly intended for use in multiple dimensions.

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

3*x⁵ - 7*x⁹ + 27

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 the corresponding 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:

int_exactness ( 'prefix', degree_max )

• 'prefix' is a quoted string, the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.

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:

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

Related Data and Programs:

INT_EXACTNESS_CHEBYSHEV1, a MATLAB program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a MATLAB program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

INT_EXACTNESS_GEGENBAUER, a MATLAB program which tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.

INT_EXACTNESS_GEN_HERMITE, a MATLAB program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.

INT_EXACTNESS_GEN_LAGUERRE, a MATLAB program which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

INT_EXACTNESS_HERMITE, a MATLAB program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

INT_EXACTNESS_JACOBI, a MATLAB program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

INT_EXACTNESS_LAGUERRE, a MATLAB program which tests the polynomial exactness of Gauss-Laguerre quadrature rules.

INT_EXACTNESS_LEGENDRE, a MATLAB program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

NINT_EXACTNESS, a MATLAB program which tests the polynomial exactness of multidimensional integration rules.

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

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

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

TRIANGLE_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.

Reference:

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

Source Code:

• int_exactness.m, 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 command:

  int_exactness ( 'cc_d1_o2', 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 command:

  int_exactness ( 'cc_d1_o3', 5 )

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 command:

  int_exactness ( 'gl_d1_o3', 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 command:

  int_exactness ( 'ncc_d1_o5', 7 )

NCC_D2_O5X5 is a Newton-Cotes Closed order 5 tensor 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 command:

  int_exactness ( 'ncc_d2_o5x5', 7 )

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