square_symq_rule

square_symq_rule, a MATLAB code which returns symmetric quadrature rules, with exactness up to total degree 20, over the interior of the unit square, 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 https://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 MIT license

Languages:

square_symq_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and a Octave version and a Python version.

Related Data and Programs:

square_symq_rule_test

alpert_rule, a MATLAB code which sets up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

annulus_rule, a MATLAB code which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2d.

cube_felippa_rule, a MATLAB code which returns a Felippa quadrature rule over the interior of a cube in 3d.

hexagon_stroud_rule, a MATLAB code which computes one of four Stroud quadrature rules over the interior of the unit hexagon.

pyramid_felippa_rule, a MATLAB code which returns a Felippa quadrature rule for approximating integrals over the interior of a pyramid in 3d.

square_arbq_rule, a MATLAB code which returns quadrature rules, with exactness up to total degree 20, over the interior of the square in 2d, by Hong Xiao and Zydrunas Gimbutas.

square_felippa_rule, a MATLAB code which returns a Felippa quadrature rule over the interior of a square in 2d.

square_grid, a MATLAB code which computes a grid of points over the interior of a square in 2d.

square_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit square in 2d.

square_minimal_rule, a MATLAB code which returns "almost minimal" quadrature rules, with exactness up to total degree 55, over the interior of the square in 2d, by Mattia Festa and Alvise Sommariva.

square_monte_carlo, a MATLAB code which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit square in 2d.

stroud, a MATLAB code 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 MATLAB code 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 MATLAB code which returns a Felippa quadrature rule over the interior of a tetrahedron.

triangle_fekete_rule, a MATLAB code which defines a Fekete rule for quadrature or interpolation over the interior of a triangle in 2d.

triangle_felippa_rule, a MATLAB code which returns a Felippa quadrature rule for approximating integrals over the interior of a triangle in 2d.

triangle_symq_rule, a MATLAB code which returns symmetric quadrature rules, with exactness up to total degree 50, over the interior of a triangle in 2d, by Hong Xiao and Zydrunas Gimbutas.

wedge_felippa_rule, a MATLAB code 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:

• comp_next.m, returns the next composition of an integer.
• lege2eva.m, evaluates orthogonal polynomials on the symmetric square.
• llegepols1.m, evaluates orthogonal polynomials on the symmetric interval.
• monomial_value.m, evaluates a multidimensional monomial.
• quadrilateral_unit_area.m, computes the area of a unit quadrilateral.
• quadrilateral_unit_monomial_integral.m, returns the exact integral of a given monomial over the unit quadrilateral.
• rule_order.m, returns the order of the requested quadrature rule.
• rule00.m, returns the rule of degree 0.
• rule01.m, returns the rule of degree 1.
• rule02.m, returns the rule of degree 2.
• rule03.m, returns the rule of degree 3.
• rule04.m, returns the rule of degree 4.
• rule05.m, returns the rule of degree 5.
• rule06.m, returns the rule of degree 6.
• rule07.m, returns the rule of degree 7.
• rule08.m, returns the rule of degree 8.
• rule09.m, returns the rule of degree 9.
• rule10.m, returns the rule of degree 10.
• rule11.m, returns the rule of degree 11.
• rule12.m, returns the rule of degree 12.
• rule13.m, returns the rule of degree 13.
• rule14.m, returns the rule of degree 14.
• rule15.m, returns the rule of degree 15.
• rule16.m, returns the rule of degree 16.
• rule17.m, returns the rule of degree 17.
• rule18.m, returns the rule of degree 18.
• rule19.m, returns the rule of degree 19.
• rule20.m, returns the rule of degree 20.
• square_symq.m, returns a symmetric quadrature rule for the square.
• square_symq_plot.m, plots the symmetric square quadrature rule.