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TRIANGLE_LYNESS_RULE\ Quadrature Rules for the Triangle {#triangle_lyness_rule-quadrature-rules-for-the-triangle align=”center”} =================================

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TRIANGLE_LYNESS_RULE is a C++ library which produces the Lyness-Jespersen family of quadrature rules over the interior of a triangle in 2D.

The rules have the following orders (number of points) and precisions (maximum degree of polynomials whose integrals they can compute exactly):

Rule Order Precision —— ——- ———– 0 1 1 1 3 2 2 4 2 3 4 3 4 7 3 5 6 4 6 10 4 7 9 4 8 7 5 9 10 5 10 12 6 11 16 6 12 13 6 13 13 7 14 16 7 15 16 8 16 21 8 17 16 8 18 19 9 19 22 9 20 27 11 21 28 11

Licensing: {#licensing align=”center”}

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

Languages: {#languages align=”center”}

TRIANGLE_LYNESS_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs: {#related-data-and-programs align=”center”}

CUBE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a C++ library which returns Felippa’s quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a C++ library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

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

STROUD, a C++ library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and M-dimensions.

TETRAHEDRON_FELIPPA_RULE, a C++ library which returns Felippa’s quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_ANALYZE, a C++ program which reads a triangle defined in a file, and uses the triangle_properties() library to compute angles, area, centroid, circumcircle, edge lengths, incircle, orientation, orthocenter, and quality.

TRIANGLE_DUNAVANT_RULE, a C++ library which sets up a Dunavant quadrature rule over the interior of a triangle in 2D.

TRIANGLE_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a C++ library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a C++ library which returns Felippa’s quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

TRIANGLE_MONTE_CARLO, a C++ program which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

TRIANGLE_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_SYMQ_RULE, a C++ 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.

TRIANGLE_WANDZURA_RULE, a C++ library which sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30 over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a C++ library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

Reference: {#reference align=”center”}

1. James Lyness, Dennis Jespersen,\ Moderate Degree Symmetric Quadrature Rules for the Triangle,\ Journal of the Institute of Mathematics and its Applications,\ Volume 15, Number 1, February 1975, pages 19-32.

Source Code: {#source-code align=”center”}

• triangle_lyness_rule.cpp, the source code.
• triangle_lyness_rule.hpp, the include file.

Examples and Tests: {#examples-and-tests align=”center”}

• triangle_lyness_rule_prb.cpp, a sample calling program.
• triangle_lyness_rule_prb_output.txt, the output file.

List of Routines: {#list-of-routines align=”center”}

• I4_MAX returns the maximum of two I4’s.
• I4_MIN returns the minimum of two I4’s.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• I4VEC_COPY_NEW copies an I4VEC to a “new” I4VEC.
• LYNESS_ORDER returns the order of a Lyness quadrature rule.
• LYNESS_PRECISION returns the precision of a Lyness quadrature rule.
• LYNESS_RULE returns the points and weights of a Lyness quadrature rule.
• LYNESS_RULE_NUM returns the number of Lyness quadrature rules.
• LYNESS_SUBORDER returns the suborders for a Lyness rule.
• LYNESS_SUBORDER_NUM returns the number of suborders for a Lyness rule.
• LYNESS_SUBRULE returns a compressed Lyness rule.
• R8_ABS returns the absolute value of an R8.
• R8MAT_COPY copies one R8MAT to another.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_COPY copies an R8VEC.
• R8VEC_SUM returns the sum of an R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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

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Last revised on 30 September 2010.