triangle_exactness

triangle_exactness, a C code which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

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. For a triangle, the degree of a monomial term is the sum of the exponents of x and y. Thus, for instance, the DEGREE of

x2y5
is 2+5=7.

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
When running the program, the user only enters the common prefix part of the file names, which is enough information for the program to find all three files.

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

• prefix_exact.txt

Usage:

triangle_exactness prefix degree_max
where
• 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. A value of 5 or 10 might be reasonable, but a value of 50 or 100 is probably never a good input!

Languages:

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

Related Data and Programs:

CUBE_EXACTNESS, a C code which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

HYPERCUBE_EXACTNESS, a C code which tests the exactness of integration rules over the interior of the hypercube in M dimensions.

PYRAMID_EXACTNESS, a C code which investigates the polynomial exactness of a quadrature rule over the interior of the pyramid in 3D.

SPHERE_EXACTNESS, a C code which tests the monomial exactness of a quadrature rule on the surface of the unit sphere in 3D.

SQUARE_EXACTNESS, a C code which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a rectangle in 2D.

TETRAHEDRON_EXACTNESS, a C code which investigates the polynomial exactness of a quadrature rule over the interior of the tetrahedron in 3D.

TRIANGLE_FEKETE_RULE, a C code which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a C code which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_MONTE_CARLO, a C code which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

TRIANGLE_SYMQ_RULE, a C code which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2D, by Hong Xiao and Zydrunas Gimbutas.

WEDGE_EXACTNESS, a C code which investigates the monomial exactness of a quadrature rule over the interior of the unit wedge in 3D.

Reference:

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

Source Code:

Last revised on 21 August 2019.