Algorithm 720: An algorithm for adaptive cubature over a collection of 3-dimensional simplices

An adaptive algorithm for computing an approximation to the integral of each element in a vector of functions over a 3-dimensional region covered by simplices is presented. The algorithm is encoded in FORTRAN 77. Locally, a cubature formula of degree 8 with 43 points is used to approximate an integral. The local error estimate is based on the same evaluation points. The error estimation procedure tries to decide whether the approximation for each function has asymptotic behavior, and different actions are taken depending on that decision. The simplex with the largest error is subdivided into 8 simplices. The local procedure is then applied to each new region. This procedure is repeated until convergence.

[1]  Terje O. Espelid,et al.  Algorithm 698: DCUHRE: an adaptive multidemensional integration routine for a vector of integrals , 1991, TOMS.

[2]  A. Haegemans,et al.  The construction of cubature formulae for the tetrahedron , 1990 .

[3]  James N. Lyness,et al.  A Technique for Comparing Automatic Quadrature Routines , 1977, Comput. J..

[4]  James N. Lyness,et al.  Comments on the Nature of Automatic Quadrature Routines , 1976, TOMS.

[5]  J. N. Lyness.  Symmetric Integration Rules for Hypercubes III. Construction of Integration Rules Using Null Rules , 1965 .

[6]  Terje O. Espelid,et al.  An Adaptive Multidimensional Integration Routine for a Vector of Integrals , 1991 .

[7]  Elise de Doncker,et al.  New euler-maclaurin expansions and their application to quadrature over the s-dimensional simplex , 1979 .

[8]  Terje O. Espelid,et al.  Algorithm 706: DCUTRI: an algorithm for adaptive cubature over a collection of triangles , 1992, TOMS.

[9]  H. M. Möller,et al.  Invariant Integration Formulas for the n-Simplex by Combinatorial Methods , 1978 .

[10]  Terje O. Espelid,et al.  Error estimation in automatic quadrature routines , 1991, TOMS.

[11]  Terje O. Espelid,et al.  An adaptive algorithm for the approximate calculation of multiple integrals , 1991, TOMS.

[12]  Alan Genz,et al.  An adaptive algorithm for numerical integration over an n-dimensional rectangular region , 1980 .

[13]  Paul Van Dooren,et al.  An adaptive algorithm for numerical integration over the n-cube , 1976 .