simplex_gm_rule

simplex_gm_rule, an Octave code which defines Grundmann-Moeller quadrature rules over the interior of a triangle in 2D, a tetrahedron in 3D, or over the interior of the simplex in M dimensions.

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 M-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.

Licensing:

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

Languages:

simplex_gm_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

simplex_gm_rule_test

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.
• monomial_value.m, evaluates a monomial.
• 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.