Languages: Maple, Fortran

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The program is designed to construct d-dimensional p-ordered quadrature rules in expanded form for integration over the d-dimensional standard unit simplex Δ_d with the vertices _j = (_j1 ,..., _jd ), _jk = δ_jk , j=0,...d, k=1,...d,

where |Δ_d| = 1/d! is the volume of the simplex. Here N_dp is the number of nodes, w_j are the weights, and (x_j1,...,x_jd) 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,...,y_d+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

The integral I_d is calculated analytically and for d=2,...,6 is equal to: Note that the obtained barycentric coordinates (y_j1,...,y_jd+1) of nodes can be used for integration over the d-dimensional arbitrary simplex Δ_q where the volume of the simplex |Δ_q| must be calculated separately.

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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