SQUARE_MINIMAL_RULE
Quadrature Rules for the Symmetric Square.

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

Licensing:

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

Languages:

SQUARE_MINIMAL_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

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

SQUARE_ARBQ_RULE, a MATLAB 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_EXACTNESS, a MATLAB library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

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

SQUARE_GRID, a MATLAB library which computes a grid of points over the interior of a square in 2D.

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

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

square_minimal_rule_test

SQUARE_MONTE_CARLO, a MATLAB 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 MATLAB 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 MATLAB 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.

TOMS886, a MATLAB 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.

Reference:

1. Mattia Festa, Alvise Sommariva,
   Computing almost minimal formulas on the square,
   Journal of Computational and Applied Mathematics,
   Volume 17, Number 236, November 2012, pages 4296-4302.

Source Code:

• set_amr_square.m, the original code by Festa and Sommariva.
• square_minimal_rule.m, the original code by Festa and Sommariva.
• square_minimal_rule_degree_max.m, returns the maximum degree for which a rule is available.
• square_minimal_rule_error_max.m, returns the maximum error made by a rule over all the monomials it should integrate exactly.
• square_minimal_rule_order.m, returns the order (number of points used) for the rule of given degree.
• square_minimal_rule_print.m, prints a minimal rule for the symmetric unit square.
• squaresym_monomial_integral.m, returns the exact integral of a momomial over the symmetric unit square.
• timestamp.m, prints out the current YMDHMS date as a timestamp.