SQUARE_ARBQ_RULE
Quadrature Rules for the Symmetric Square.

SQUARE_ARBQ_RULE is a FORTRAN90 library which returns quadrature rules, with exactness up to total degree 30, over the interior of the symmetric square in 2D, 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:

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

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.

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_HEX_GRID, a FORTRAN90 library which computes a hexagonal grid of points over the interior of a square in 2D.

SQUARE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

SQUARE_MINIMAL_RULE, a FORTRAN90 library which returns "almost minimal" quadrature rules, with exactness up to total degree 55, over the interior of the symmetric square in 2D, by Mattia Festa and Alvise Sommariva.

SQUARE_MONTE_CARLO, a FORTRAN90 library which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit square in 2D.

SQUARE_SYMQ_RULE, a FORTRAN90 library which returns efficient symmetric quadrature rules, with exactness up to total degree 15, over the interior of a 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.

TOMS886, a FORTRAN90 library which defines the Padua points for interpolation in a 2D region, including the rectangle, triangle, and ellipse, by Marco Caliari, Stefano de Marchi, Marco Vianello. This is a version of ACM TOMS algorithm 886.

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.

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:

• square_arbq_rule.f90, the source code.

Examples and Tests:

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

SQUARE08 is a degree 8 rule in the symmetric square.

• square08.txt, the node coordinates and weights.
• square08_commands.txt, gnuplot commands to create a plot.
• square08_nodes.txt, the node coordinates.
• square08_vertices.txt, the vertices.
• square08.png, the PNG image of the point locations.

List of Routines:

• GET_UNIT returns a free FORTRAN unit number.
• LEGE2EVA evaluates orthogonal polynomials on the symmetric square.
• 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.
• RULE16 returns the rule of degree 16.
• RULE17 returns the rule of degree 17.
• RULE18 returns the rule of degree 18.
• RULE19 returns the rule of degree 19.
• RULE20 returns the rule of degree 20.
• RULE21 returns the rule of degree 21.
• RULE22 returns the rule of degree 22.
• RULE23 returns the rule of degree 23.
• RULE24 returns the rule of degree 24.
• RULE25 returns the rule of degree 25.
• RULE26 returns the rule of degree 26.
• RULE27 returns the rule of degree 27.
• RULE28 returns the rule of degree 28.
• RULE29 returns the rule of degree 29.
• RULE30 returns the rule of degree 30.
• SQUARE_ARBQ returns a quadrature rule for the symmetric square.
• SQUARE_ARBQ_GNUPLOT: GNUPLOT plot a quadrature rule for the symmetric square.
• SQUARE_ARBQ_SIZE returns the size of quadrature rule for the symmetric square.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

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