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

SIMPLEX_GM_RULE is a C++ 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 C++ library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

CUBE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

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

SQUARE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

TETRAHEDRON_EXACTNESS, a C++ program which investigates the monomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

TETRAHEDRON_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_KEAST_RULE, a C++ library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_MONTE_CARLO, a C++ program which uses the Monte Carlo method to estimate integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a C++ library which defines Dunavant quadrature rules over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a C++ library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a C++ library which returns Lyness-Jespersen quadrature rules for the triangle.

TRIANGLE_MONTE_CARLO, a C++ program which uses the Monte Carlo method to estimate integrals over a triangle.

TRIANGLE_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_WANDZURA_RULE, a C++ library which defines Wandzura rules for quadrature over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a C++ library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

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:

• simplex_gm_rule.cpp, the source code.
• simplex_gm_rule.hpp, the include file.

Examples and Tests:

• simplex_gm_rule_test.cpp, a sample calling program.
• simplex_gm_rule_test.txt, the output file.

List of Routines:

• COMP_NEXT computes the compositions of the integer N into K parts.
• GM_RULE_SET sets a Grundmann-Moeller rule.
• GM_RULE_SET_OLD sets a Grundmann-Moeller rule. (OBSOLETE VERSION)
• GM_RULE_SIZE determines the size of a Grundmann-Moeller rule.
• I4_CHOOSE computes the binomial coefficient C(N,K).
• I4_HUGE returns a "huge" I4.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_POWER returns the value of I^J.
• MONOMIAL_VALUE evaluates a monomial.
• R8_ABS returns the absolute value of an R8.
• R8_FACTORIAL computes the factorial of N.
• R8VEC_DOT computes the dot product of a pair of R8VEC's.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• SIMPLEX_UNIT_MONOMIAL_INT integrates a monomial over a simplex.
• SIMPLEX_UNIT_MONOMIAL_QUADRATURE: quadrature of monomials in a unit simplex.
• SIMPLEX_UNIT_SAMPLE returns uniformly random points from a general simplex.
• SIMPLEX_UNIT_TO_GENERAL maps the unit simplex to a general simplex.
• SIMPLEX_UNIT_VOLUME computes the volume of the unit simplex.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.