# square_arbq_rule

square_arbq_rule, an Octave code 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 https://www.cims.nyu.edu/cmcl/quadratures/quadratures.html ,

### Languages:

square_arbq_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

### Related Data and Programs:

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

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

cube_arbq_rule, an Octave code 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, an Octave code which returns a Felippa quadrature rule over the interior of a cube in 3d.

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

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

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

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

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

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

square_minimal_rule, an Octave code 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, an Octave code 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, an Octave code 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, an Octave 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, an Octave 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, an Octave code which returns a Felippa quadrature rule for approximating integrals over the interior of a tetrahedron in 3d.

toms886, an Octave code 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_fekete_rule, an Octave code which defines a Fekete rule for quadrature or interpolation over the interior of a triangle in 2d.

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

triangle_symq_rule, an Octave code 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, an Octave 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:

>
• lege2eva.m, evaluates orthogonal polynomials on the symmetric square.
• llegepols1.m, evaluates orthogonal polynomials on the symmetric interval.
• 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.
• rule21.m, returns the rule of degree 21.
• rule22.m, returns the rule of degree 22.
• rule23.m, returns the rule of degree 23.
• rule24.m, returns the rule of degree 24.
• rule25.m, returns the rule of degree 25.
• rule26.m, returns the rule of degree 26.
• rule27.m, returns the rule of degree 27.
• rule28.m, returns the rule of degree 28.
• rule29.m, returns the rule of degree 29.
• rule30.m, returns the rule of degree 30.
• square_arbq.m, returns a quadrature rule for the symmetric square.
• square_arbq_gnuplot.m, GNUPLOT plot of a quadrature rule for the symmetric square.
• square_arbq_size.m, returns the size of a quadrature rule for the symmetric square.

Last revised on 09 May 2023.