KEAST
Quadrature Rules for a Tetrahedron

KEAST is a FORTRAN90 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:

CUBPACK, a FORTRAN90 library which estimates the integral of a function (or vector of functions) over a collection of N-dimensional hyperrectangles and simplices.

FELIPPA, a FORTRAN90 library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GM_RULE, a FORTRAN90 library which defines Grundmann-Moeller rules for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

NCC_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes Closed quadrature rules on a tetrahedron.

NCO_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes Open quadrature rules on a tetrahedron.

NINTLIB, a FORTRAN90 library which contains a variety of routines for numerical estimation of integrals in multiple dimensions.

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

QUADRULE, a FORTRAN90 library which includes a library of routines for defining quadrature rules on a variety of intervals with different weight functions.

STROUD, a FORTRAN90 library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TETRAHEDRON_MONTE_CARLO, a FORTRAN90 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:

• keast.f90, the source code.
• keast.sh, commands to compile the source code.

Examples and Tests:

• keast_prb.f90, a sample calling program.
• keast_prb.sh, commands to compile and run the sample program.
• keast_prb_output.txt, the output file.

List of Routines:

• COMP_NEXT computes the compositions of the integer N into K parts.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• KEAST_DEGREE returns the degree of a Keast rule for the tetrahedron.
• KEAST_ORDER_NUM returns the order of a Keast rule for the tetrahedron.
• KEAST_RULE returns the points and weights of a Keast rule.
• KEAST_RULE_NUM returns the number of Keast rules for the tetrahedron.
• KEAST_SUBORDER returns the suborders for a Keast rule.
• KEAST_SUBORDER_NUM returns the number of suborders for a Keast rule.
• KEAST_SUBRULE returns a compressed Keast rule.
• MONOMIAL_VALUE evaluates a monomial.
• R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
• TETRAHEDRON_REFERENCE_TO_PHYSICAL maps T4 reference points to physical points.
• TETRAHEDRON_VOLUME computes the volume of a tetrahedron in 3D.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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