SIMPLEX_GM_RULE
GrundmannMoeller Quadrature Rules for the Simplex in M dimensions
SIMPLEX_GM_RULE
is a Python library which
defines GrundmannMoeller 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 Mdimensional
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 Mdimensional 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:

Paul Bratley, Bennett Fox, Linus Schrage,
A Guide to Simulation,
Second Edition,
Springer, 1987,
ISBN: 0387964673,
LC: QA76.9.C65.B73.

Bennett Fox,
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,
ACM Transactions on Mathematical Software,
Volume 12, Number 4, December 1986, pages 362376.

Axel Grundmann, Michael Moeller,
Invariant Integration Formulas for the NSimplex
by Combinatorial Methods,
SIAM Journal on Numerical Analysis,
Volume 15, Number 2, April 1978, pages 282290.

Pierre LEcuyer,
Random Number Generation,
in Handbook of Simulation,
edited by Jerry Banks,
Wiley, 1998,
ISBN: 0471134031,
LC: T57.62.H37.

Peter Lewis, Allen Goodman, James Miller,
A PseudoRandom Number Generator for the System/360,
IBM Systems Journal,
Volume 8, 1969, pages 136143.

Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
Academic Press, 1978,
ISBN: 0125192606,
LC: QA164.N54.

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:
Examples and Tests:
You can go up one level to
the Python source codes.
Last revised on 03 March 2017.