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.


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


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:



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    A Guide to Simulation,
    Second Edition,
    Springer, 1987,
    ISBN: 0387964673,
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    Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
    ACM Transactions on Mathematical Software,
    Volume 12, Number 4, December 1986, pages 362-376.
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    Invariant Integration Formulas for the N-Simplex by Combinatorial Methods,
    SIAM Journal on Numerical Analysis,
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    A Pseudo-Random Number Generator for the System/360,
    IBM Systems Journal,
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    Second Edition,
    Academic Press, 1978,
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    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:

Last revised on 13 October 2022.