triangle_symq_rule_original

triangle_symq_rule_original, an Octave code which returns 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.

This "original" code is a MATLAB translation based on the original Fortran77 code. A newer version of the code has been written which is based on a reference triangle instead of the equilateral triangle, and returns barycentric coordinates.

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 information on this web page is distributed under the MIT license.

Languages:

triangle_symq_rule_original is available in a MATLAB version and an Octave version.

Related Data and Programs:

triangle_symq_rule_original_test

simplex_gm_rule, an Octave code which defines Grundmann-Moeller quadrature rules over the interior of a simplex in m dimensions.

triangle_dunavant_rule, an Octave code which returns a Dunavant quadrature rule over the interior of a triangle in 2d.

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

triangle_lyness_rule, an Octave code which returns a Lyness-Jespersen quadrature rule over the interior of a triangle in 2d.

triangle_monte_carlo, an Octave code which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit triangle in 2d.

triangle_ncc_rule, an Octave code which defines Newton-Cotes closed quadrature rules on a triangle.

triangle_nco_rule, an Octave code which defines Newton-Cotes open quadrature rules 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.

triangle_symq_to_ref, an Octave code which creates quadrature rules defined on the half-unit square from symmetric quadrature rules defined on an equilateral triangle, defined by triangle_symq_rule(), by Hong Xiao and Zydrunas Gimbutas.

triangle_twb_rule, an Octave code which generates the points and weights of quadrature rules over the interior of a triangle in 2D, determined by Taylor, Wingate, and Bos.

triangle_wandzura_rule, an Octave code which returns a Wandzura quadrature rule of exactness 5, 10, 15, 20, 25 and 30 over the interior of the triangle in 2D.

triangle_witherden_rule, an Octave code which returns a symmetric Witherden quadrature rule for the triangle, with exactness up to total degree 20.

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:

• kjacopols2.m, evaluates Jacobi polynomials and derivatives.
• kjacopols.m, evaluates Jacobi polynomials.
• klegeypols3.m, evaluate scaled Legendre polynomials and derivatives.
• klegeypols.m, evaluates scaled Legendre polynomials.
• ortho2eva0.m, evaluates the orthonormal polynomials on the triangle.
• ortho2eva30.m, orthonormal polynomials and derivatives on triangle.
• ortho2eva3.m, orthogonal polynomial values and derivatives, reference triangle.
• ortho2eva.m, evaluates orthogonal polynomials on the reference triangle.
• quaecopy2.m, copies a quadrature rule into a user arrays X, Y, and W.
• quaecopy.m, copies a quadrature rule into user arrays Z and W.
• quaeinside.m, checks whether a point is inside a triangle.
• quaenodes2.m, expands nodes from 1/6 to 1/3 of the triangle.
• quaenodes.m, expands nodes to the reference triangle.
• quaequad0.m, returns the requested quadrature rule.
• quaequad.m, returns a symmetric quadrature formula for a reference triangle.
• quaerotate.m, applies a rotation.
• r8vec_uniform_01.m, returns a unit pseudorandom R8VEC.
• ref_to_koorn.m, maps points from the reference to Koornwinder's triangle.
• ref_to_triangle.m, maps points from the reference triangle to a triangle.
• rule_compressed_size.m, returns the compressed size of the requested quadrature rule.
• rule_full_size.m, returns the full size 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.
• 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 29.
• rule28.m, returns the rule of degree 28.
• rule29.m, returns the rule of degree 29.
• rule30.m, returns the rule of degree 30.
• rule31.m, returns the rule of degree 31.
• rule32.m, returns the rule of degree 32.
• rule33.m, returns the rule of degree 33.
• rule34.m, returns the rule of degree 34.
• rule35.m, returns the rule of degree 35.
• rule36.m, returns the rule of degree 36.
• rule37.m, returns the rule of degree 37.
• rule38.m, returns the rule of degree 38.
• rule39.m, returns the rule of degree 39.
• rule40.m, returns the rule of degree 40.
• rule41.m, returns the rule of degree 41.
• rule42.m, returns the rule of degree 42.
• rule43.m, returns the rule of degree 43.
• rule44.m, returns the rule of degree 44.
• rule45.m, returns the rule of degree 45.
• rule46.m, returns the rule of degree 46.
• rule47.m, returns the rule of degree 47.
• rule48.m, returns the rule of degree 48.
• rule49.m, returns the rule of degree 49.
• rule50.m, the rule of degree 50.
• simplex_to_triangle.m, maps points from the simplex to a triangle.
• triangle_area.m, returns the area of a triangle.
• triangle_to_ref.m, maps points from any triangle to the reference triangle.
• triangle_to_simplex.m, maps points from any triangle to the simplex.
• trianmap.m, maps rules from the reference triangle to the user triangle.
• triasimp.m, maps a point from the reference triangle to the simplex.
• triasymq.m, returns a symmetric quadrature formula for a user triangle.
• triasymq_plot.m, plots a triangle quadrature rule.