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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 j = (j1 ,..., jd ), jk = δjk , j=0,...d, k=1,...d, where |Δd| = 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 The integral Id is calculated analytically and for d=2,...,6 is equal to: Note that the obtained barycentric coordinates (yj1,...,yjd+1) of nodes can be used for integration over the d-dimensional arbitrary simplex Δq where the volume of the simplex |Δq| must be calculated separately. Download the INQSIM program archive. References: 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).