GM_RULES
Grundmann-Moeller Quadrature Rules for N-Simplexes.

GM_RULES is a MATLAB library which defines Grundmann-Moeller quadratures rules for a simplex.

The user can choose the spatial dimension, thus defining the region to be a triangle (N = 2), tetrahedron (N = 3) or a general N-dimensional simplex.

The user chooses the index S of the rule. Rules are available with index S = 0 on up. A rule of index S will exactly integrate any polynomial of total degree 2*S+1 or less.

The rules are defined on the unit N-dimensional simplex. A simple linear transformation can be used to map the vertices and weights to an arbitrary simplex, while preserving the accuracy of the rule.

The formula for the weight involves the ratio of factorials and a high power. Direct evaluation of this formula results in disastrous inaccuracies, even for relatively low order rules. The original code has been revised to take a more cautious and accurate approach to this computation.

Licensing:

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

Related Data and Programs:

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

DUNAVANT is a MATLAB library of routines for defining Dunavant rules for quadrature on a triangle.

FEKETE is a MATLAB library of routines for defining Fekete rules for interpolation or quadrature on a triangle.

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

GM_RULES is also available in a C++ version and a FORTRAN90 version.

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

NCC_TETRAHEDRON is a MATLAB library defining Newton-Cotes Closed quadrature rules on a tetrahedron.

NCC_TRIANGLE is a MATLAB library defining Newton-Cotes Closed quadrature rules on a triangle.

NCO_TETRAHEDRON is a MATLAB library defining Newton-Cotes Open quadrature rules on a tetrahedron.

NCO_TRIANGLE is a MATLAB library defining Newton-Cotes Open quadrature rules on a triangle.

NINT_EXACTNESS_TET is an executable MATLAB program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

NINT_EXACTNESS_TRI is an executable MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.

QUADRATURE_RULES_TET is a dataset directory of triples of files defining various quadrature rules on tetrahedrons.

QUADRATURE_RULES_TRI is a dataset directory of triples of files defining various quadrature rules on triangles.

SIMPACK is a FORTRAN77 library of routines which approximate the integral of a function or vector of functions over a multidimensional simplex, or a region which is the sum of multidimensional simplexes.

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

TEST_TRI_INT is a MATLAB library of functions that can be used to test algorithms for quadrature over a triangle.

WANDZURA is a MATLAB library of routines which definine Wandzura rules for quadrature on a triangle.

Reference:

1. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Second Edition,
   Springer, 1987,
   ISBN: 0387964673,
   LC: QA76.9.C65.B73.
2. Bennett Fox,
   Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, December 1986, pages 362-376.
3. Axel Grundmann, Michael Moeller,
   Invariant Integration Formulas for the N-Simplex by Combinatorial Methods,
   SIAM Journal on Numerical Analysis,
   Volume 15, Number 2, April 1978, pages 282-290.
4. Pierre LEcuyer,
   Random Number Generation,
   in Handbook of Simulation,
   edited by Jerry Banks,
   Wiley, 1998,
   ISBN: 0471134031,
   LC: T57.62.H37.
5. Peter Lewis, Allen Goodman, James Miller,
   A Pseudo-Random Number Generator for the System/360,
   IBM Systems Journal,
   Volume 8, 1969, pages 136-143.
6. Albert Nijenhuis, Herbert Wilf,
   Combinatorial Algorithms for Computers and Calculators,
   Second Edition,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.
7. ML Wolfson, HV Wright,
   Algorithm 160: Combinatorial of M Things Taken N at a Time,
   Communications of the ACM,
   Volume 6, Number 4, April 1963, page 161.

Source Code:

• comp_next.m, computes the compositions of the integer N into K parts.
• gm_rule_set.m, sets a Grundmann-Moeller rule.
• gm_rule_set_old.m, sets a Grundmann-Moeller rule. (OBSOLETE VERSION)
• gm_rule_size.m, determines the size of a Grundmann-Moeller rule.
• i4_choose.m, computes the binomial coefficient C(N,K).
• i4_huge.m, returns a "huge" I4.
• monomial_value.m, evaluates a monomial.
• r8_factorial.m, computes the factorial.
• r8vec_uniform_01.m, returns a unit pseudorandom R8VEC.
• simplex_unit_monomial_int.m, integrates a monomial over a simplex.
• simplex_unit_monomial_quadrature.m, quadrature of monomials in a unit simplex.
• simplex_unit_sample.m, returns uniformly random points from a general simplex.
• simplex_unit_to_general.m, maps the unit simplex to a general simplex.
• simplex_unit_volume.m, computes the volume of the unit simplex.
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• gm_rules_test.m, calls all the problems.
• gm_rules_test01.m, tests SIMPLEX_UNIT_TO_GENERAL.
• gm_rules_test02.m, tests SIMPLEX_UNIT_TO_GENERAL.
• gm_rules_test03.m, tests GM_RULE_SIZE.
• gm_rules_test04.m, tests GM_RULE_SET.
• gm_rules_test05.m, tests GM_RULE_SET.
• gm_rules_test06.m, tests GM_RULE_SET.
• gm_rules_test07.m, tests GM_RULE_SET.
• gm_rules_test_output.txt, the output file.

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