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INQSIM - a program for converting PI-type fully symmetric quadrature rules
on 2-,..., 6-simplexes from compact to expanded forms


Authors: G.Chuluunbaatar, O.Chuluunbaatar, A.A.Gusev, S.I.Vinitsky
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Languages: Maple, Fortran


The program is designed to construct d-dimensional p-ordered quadrature rules in expanded form for integration over the d-dimensional standard unit simplex Δd with the vertices index0xj = (index0xj1 ,..., index0xjd ), index0xjk = δjk , j=0,...d, k=1,...d,

index4x
where d| = 1/d! is the volume of the simplex. Here Ndp is the number of nodes, wj are the weights, and (xj1,...,xjd) are the nodes.

A detailed description of the method for constructing fully symmetric quadrature rules with positive weights, and with points lying inside the 2-,...,6- simplex (so-called PI-type) is published in [1].

The *.mw and *.f files contain the Maple and Fortran programs for converting quadrature formulas up to the 20-th order on a triangle and a tetrahedron, the 16-th order on a 4-simplex, the 10-th order on 5- and 6- simplexes in expanded form, and examples of their application:

•  on INPUT:
       ○ `ddxoy_z.dat' file,
•  on OUTPUT:
       ○ wg is an array of weights with a dimension of gnodes,
       ○ xg is an array of barycentric coordinates of nodes with a dimension of (dim+1)*gnodes.
The `ddxoy_z.dat' files contain the dimension of the simplex, the order of the quadrature rule, the number of nodes, the information about orbits, and PI-type fully symmetric quadrature rules in barycentric coordinates (y1,...,yd+1) in compact form, where
•  x=dim means the dimension of the simplex,
•  y=p means the order of the quadrature rule,
•  z=gnodes means the number of nodes.

As an example, we consider the integral

index5x
The integral Id is calculated analytically and for d=2,...,6 is equal to:
index6x
Note that the obtained barycentric coordinates (yj1,...,yjd+1) of nodes can be used for integration over the d-dimensional arbitrary simplex Δq
index7x
where the volume of the simplex q| must be calculated separately.

Download the INQSIM program archive.

inqsim.zip Download the INQSIM program archive.

References:

  1. G. Chuluunbaatar, O. Chuluunbaatar, A.A. Gusev, and S.I. Vinitsky. PI-type fully symmetric quadrature rules on the 3-,...,6-simplexes. Computers & Mathematics with Applications, 124, 89--97 (2022).



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