LYNESS_RULE
Quadrature Rules for the Triangle

LYNESS_RULE is a MATLAB library which produces the Lyness-Jespersen family of quadrature rules for the triangle.

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:

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

Related Data and Programs:

DUNAVANT, a MATLAB library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a MATLAB library which defines Fekete rules for interpolation or quadrature on a triangle.

FELIPPA, a MATLAB library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GM_RULE, a MATLAB library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

NCC_TRIANGLE, a MATLAB library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE, a MATLAB library which defines Newton-Cotes open quadrature rules on a triangle.

QUADRATURE_RULES_TRI, a dataset directory of triples of files which defines various quadrature rules on triangles.

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

TEST_TRI_INT, a MATLAB library which tests algorithms for quadrature over a triangle.

TRIANGLE_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.

TRIANGLE_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a triangle.

WANDZURA, a MATLAB library which definines Wandzura rules for quadrature on a triangle.

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:

• i4_modp.m returns the nonnegative remainder of I4 division.
• i4_wrap.m forces an I4 to lie between given limits by wrapping.
• lyness_order.m returns the order of a Lyness quadrature rule.
• lyness_precision.m returns the precision of a Lyness quadrature rule.
• lyness_rule.m returns the points and weights of a Lyness quadrature rule.
• lyness_rule_num.m returns the number of Lyness quadrature rules.
• lyness_suborder.m returns the suborders for a Lyness rule.
• lyness_suborder_num.m returns the number of suborders for a Lyness rule.
• lyness_subrule.m returns a compressed Lyness rule.
• timestamp.m prints the current YMDHMS date as a time stamp.

Examples and Tests:

• lyness_rule_test.m, a sample calling program.
• lyness_rule_test_output.txt, the output file.
• lyness_rule_test01.m, tests LYNESS_RULE_NUM, LYNESS_DEGREE, LYNESS_ORDER_NUM.
• lyness_rule_test02.m, performs the weight sum test on Lyness rules.
• lyness_rule_test03.m, performs the barycentric coordinate sum test on Lyness rules.
• lyness_rule_test04.m, prints a rule generated by LYNESS_RULE.
• lyness_rule_test05.m, writes a rule created by LYNESS_RULE to a file.
• lyness_rule_test06.m, tests the Lyness rules for exact integration of monomials.

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