CUBE_ARBQ_RULE
Quadrature Rules for Cubes.

CUBE_ARBQ_RULE is a FORTRAN90 library which returns quadrature rules, with exactness up to total degree 15, over the interior of a cube 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:

CUBE_ARBQ_RULE 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 FORTRAN90 library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

CUBE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

CUBE_GRID, a FORTRAN90 library which computes a grid of points over the interior of a cube in 3D.

CUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

CUBE_MONTE_CARLO, a FORTRAN90 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit cube in 3D;

DISK_RULE, a FORTRAN90 library which computes quadrature rules over the interior of a disk in 2D.

GNUPLOT, FORTRAN90 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 FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a FORTRAN90 library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_ARBQ_RULE, a FORTRAN90 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 FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

SQUARE_SYMQ_RULE, a FORTRAN90 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 FORTRAN90 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_ARBQ_RULE, a FORTRAN90 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.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_KEAST_RULE, a FORTRAN90 library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a FORTRAN90 library which defines Dunavant rules for quadrature over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a FORTRAN90 library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a FORTRAN90 library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

TRIANGLE_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

TRIANGLE_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_SYMQ_RULE, a FORTRAN90 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.

TRIANGLE_WANDZURA_RULE, a FORTRAN90 library which defines Wandzura rules for quadrature over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a FORTRAN90 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:

• cube_arbq_rule.f90, the source code.

Examples and Tests:

• cube_arbq_rule_test.f90, a sample calling program.
• cube_arbq_rule_test.txt, the output file.

CUBE08 is a degree 8 rule in the symmetric cube.

• cube08.txt, the node coordinates and weights.
• cube08_commands.txt, gnuplot commands to create a plot.
• cube08_nodes.txt, the node coordinates.
• cube08_vertices.txt, the vertices.
• cube08.png, the PNG image of the point locations.

List of Routines:

• CUBE_ARBQ_SIZE returns the size of quadrature rule for a cube.
• CUBE_ARBQ returns a quadrature rule for a cube.
• CUBE_ARBQ_GNUPLOT: GNUPLOT plot of a quadrature rule in the symmetric cube.
• GET_UNIT returns a free FORTRAN unit number.
• LEGE3EVA evaluates orthonormal polynomials in the cube.
• LLEGEPOLS1 evaluates orthogonal polynomials on the symmetric interval.
• R8MAT_ROW_COPY copies a vector into a row of an R8MAT.
• R8VEC_COPY copies an R8VEC.
• R8VEC_SUM sums the entries of an R8VEC.
• 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.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN90 source codes.