TETRAHEDRON_ARBQ_RULE
Quadrature Rules for Tetrahedrons.

TETRAHEDRON_ARBQ_RULE is a C++ library which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

The original source code, from which this library was developed, is available from the Courant Mathematics and Computing Laboratory, at http://www.cims.nyu.edu/cmcl/quadratures/quadratures.html ,

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

ANNULUS_RULE, a C++ library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

CUBE_ARBQ_RULE, a C++ library which returns quadrature rules, with exactness up to total degree 15, over the interior of the symmetric cube in 3D, by Hong Xiao and Zydrunas Gimbutas.

CUBE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

GNUPLOT, C++ programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

PYRAMID_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a C++ library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_ARBQ_RULE, a C++ library which returns quadrature rules, with exactness up to total degree 20, over the interior of the symmetric square in 2D, by Hong Xiao and Zydrunas Gimbutas.

SQUARE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

SQUARE_SYMQ_RULE, a C++ library which returns symmetric quadrature rules, with exactness up to total degree 20, over the interior of the symmetric square in 2D, by Hong Xiao and Zydrunas Gimbutas.

STROUD, a C++ library which defines quadrature rules for a variety of M-dimensional regions, including the interior of the square, cube and hypercube, the pyramid, cone and ellipse, the hexagon, the M-dimensional octahedron, the circle, sphere and hypersphere, the triangle, tetrahedron and simplex, and the surface of the circle, sphere and hypersphere.

TETRAHEDRON_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit tetrahedron in 3D.

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

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

TRIANGLE_SYMQ_RULE, a C++ library 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_FELIPPA_RULE, a C++ library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

Reference:

1. Hong Xiao, Zydrunas Gimbutas,
   A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions,
   Computers and Mathematics with Applications,
   Volume 59, 2010, pages 663-676.

Source Code:

• tetrahedron_arbq_rule.cpp, the source code.
• tetrahedron_arbq_rule.hpp, the source code.

Examples and Tests:

• tetrahedron_arbq_rule_test.cpp, a sample calling program.
• tetrahedron_arbq_rule_test.txt, the output file.

TETRAHEDRON08 is a degree 8 rule in the reference tetrahedron.

• tetrahedron08.txt, the node coordinates and weights.
• tetrahedron08_commands.txt, gnuplot commands to create a plot.
• tetrahedron08_nodes.txt, the node coordinates.
• tetrahedron08_vertices.txt, the vertices.
• tetrahedron08.png, the PNG image of the point locations.

List of Routines:

• KJACOYPOLS3 evaluates modified Jacobi polynomials.
• KLEGEYPOLS evaluates scaled Legendre polynomials.
• ORTHO3EVA evaluates polynomials orthogonal in the reference triangle.
• R8MAT_ROW_COPY copies a vector into a row of an R8MAT.
• R8VEC_COPY copies an R8VEC.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_SUM returns the sum of the entries of an R8VEC.
• REF_TO_KOORN maps points from the reference to Koornwinder's tetrahedron.
• RULE01 returns the rule of degree 1.
• RULE02 returns the rule of degree 2.
• RULE03 returns the rule of degree 3.
• RULE04 returns the rule of degree 4.
• RULE05 returns the rule of degree 5.
• RULE06 returns the rule of degree 6.
• RULE07 returns the rule of degree 7.
• RULE08 returns the rule of degree 8.
• RULE09 returns the rule of degree 9.
• RULE10 returns the rule of degree 10.
• RULE11 returns the rule of degree 11.
• RULE12 returns the rule of degree 12.
• RULE13 returns the rule of degree 13.
• RULE14 returns the rule of degree 14.
• RULE15 returns the rule of degree 15.
• TETRAHEDRON_ARBQ returns a quadrature rule for a tetrahedron.
• TETRAHEDRON_ARBQ_GNUPLOT: plot of a quadrature rule in the tetrahedron.
• TETRAHEDRON_ARBQ_SIZE returns the size of quadrature rule for a tetrahedron.
• TETRAHEDRON_REF returns the vertices of the reference tetrahedron.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.