# SIMPLEX_GM_RULE Grundmann-Moeller Quadrature Rules for the Simplex in M dimensions

SIMPLEX_GM_RULE is a Python library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

The user can choose the spatial dimension M, thus defining the region to be a triangle (M = 2), tetrahedron (M = 3) or a general M-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 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 GNU LGPL license.

### Languages:

SIMPLEX_GM_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

ANNULUS_RULE, a Python library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

SIMPLEX_GRID, a Python library which generates a regular grid of points over the interior of an arbitrary simplex in M dimensions.

### 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.

### Examples and Tests:

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

Last revised on 03 March 2017.