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:
-
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 362-376.
-
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.
-
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 Pseudo-Random Number Generator for the System/360,
IBM Systems Journal,
Volume 8, 1969, pages 136-143.
-
Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
Academic Press, 1978,
ISBN: 0-12-519260-6,
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.