TRIANGLE_LYNESS_RULE
Quadrature Rules for the Triangle

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:

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

Languages:

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

Related Data and Programs:

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

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:

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:

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

Examples and Tests:

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

List of Routines:

• 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.