KEAST
Quadrature Rules for a Tetrahedron

KEAST is a MATLAB library which defines a set of quadrature rules for the tetrahedron.

The ten rules have the following orders and precisions:

Rule	Order	Precision
1	1	0
2	4	1
3	5	2
4	10	3
5	11	4
6	14	4
7	15	5
8	24	6
9	31	7
10	45	8

Licensing:

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

Languages:

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

Related Data and Programs:

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

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

NCC_TETRAHEDRON, a MATLAB library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCO_TETRAHEDRON, a MATLAB library which defines Newton-Cotes open quadrature rules on a tetrahedron.

NINTLIB, a MATLAB library which contains a variety of routines for numerical estimation of integrals in multiple dimensions.

QUADRATURE_RULES_TET, a dataset directory which contains files defining various quadrature rules on tetrahedrons.

QUADRULE, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions.

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

TETRAHEDRON_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a tetrahedron.

TETRAHEDRONS, a dataset directory which contains examples of tetrahedrons;

Reference:

1. Patrick Keast,
   Moderate Degree Tetrahedral Quadrature Formulas,
   Computer Methods in Applied Mechanics and Engineering,
   Volume 55, Number 3, May 1986, pages 339-348.

Source Code:

• comp_next.m returns the next composition of an integer into a given number of parts.
• i4_modp.m returns the nonnegative remainder of integer division.
• i4_wrap.m forces an integer to lie between given limits by wrapping.
• keast_degree.m returns the degree of a Keast rule for the tetrahedron.
• keast_order_num.m returns the order of a Keast rule for the tetrahedron.
• keast_rule.m returns the points and weights of a Keast rule.
• keast_rule_num.m returns the number of Keast rules for the tetrahedron.
• keast_suborder.m returns the suborders for a Keast rule.
• keast_suborder_num.m returns the number of suborders for a Keast rule.
• keast_subrule.m returns a compressed Keast rule.
• monomial_value.m evaluates a monomial function.
• r8mat_det_4d.m computes the determinant of a 4 by 4 R8MAT.
• tetrahedron_reference_to_physical.m maps T4 reference points to physical points.
• tetrahedron_volume.m computes the volume of a tetrahedron in 3D.
• timestamp.m prints the current YMDHMS date as a time stamp.

Examples and Tests:

• keast_test.m, calls all the tests.
• keast_test01.m, tests KEAST_RULE_NUM, KEAST_DEGREE, KEAST_ORDER_NUM.
• keast_test02.m, tests KEAST_RULE.
• keast_test03.m, tests KEAST_RULE.
• keast_test04.m, tests TETRAHEDRON_REFERENCE_TO_PHYSICAL.
• keast_test05.m, applies all the rules to a set of monomial integrands.
• keast_test07.m, prints out a rule.
• keast_test_output.txt, the output file.

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