Algorithm 34 An algorithm for the automatic integration over a triangle

Abstract

--ZusammenfassungAlgorithm 34. An Algorithm for the Automatic Integration Over a Triangle. Numerical integration over an arbitrary triangular domain is considered. An adaptive algorithm with round-off error guard is described. A FORTRAN IV implementation is given and numerical results are reported. Algorithmus 34. Aigorithmus far das automatisehe Integrieren Uber Dreiecke. Es wird ein adaptiverAlgorithmus und ein FORTRAN IV Programm zur automatischen Berechnung yon Integralen fiber einem beliebigen Dreiecksbereich vorgestellt, und an numerischen Beispielen demonstriert.

Algorithm 584: CUBTRI: Automatic Cubature over a Triangle

“…This integral seems to be impossible to evaluate analytically; the numerical "exact" value is given in [1].…”

“…This effect is countered by employing a device given in [3] for sharpening the error estimate. It is, of course, possible to use two completely independent cubature rules (e.g., Haegemans [1] uses conic product Gauss formulas (see Stroud [8], Section 2.5) with 36 and 49 points, respectively), but it seems sensible to extend Kronrod's [2] idea, that Q2 should reuse the points of Q1, to the two-dimensional case.…”

“…Haegemans [1] has improved on this by keeping a separate list of vertices; each triangle can then be described by 3 integers. The length of the list is only n + 2, because each subdivision creates as many new vertices (three) as the net increase in the number of triangles.…”

“…Add these contributions to A and 7. The general strategy falls within the framework outlined by Rice [7], apart from the test for roundoff, which is due to Piessens [5] and also appears in Haegemans' routine TRIADA [1].…”

An Algorithm for Automatic Integration Over a Triangle Using Nonlinear Extrapolation

“…More recently, Haegemans [8], Kahaner and Wells [10] and Laurie [11] have focused attention on the use of the n-simplex as basic cell. K a h a n e r and Wells [10] list some advantages and disadvantages of the use of both types of regions.…”