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High degree efficient symmetrical Gaussian quadrature rules for the triangle. (English) Zbl 0589.65021
This paper provides a list of the numerical values (to 20 decimal digits) of weights and abscissas of quadrature rules for the triangle. These rules are the \(D_ 6\)-rules of the reviewer and D. Jesperson [J. Inst. Math. Appl. 15, 19-32 (1975; Zbl 0297.65018)]. The list includes one rule for each degree d up to \(d=20\). Each listed rule uses fewer than \((1/6)(d+1)(d+2)+5\) abscissas. The rules of degree 9 or less and 11 have been given before but only to 16 decimal digits. The other rules are new. The rules of degrees 10, 12, 13, 14, 17 and 19 are all PI; that is the weights are all positive and all abscissas are inside the triangle. The other new rules violate this condition but only by a very small margin.
The work continues that of the reviewer and Jesperson to degrees greater than about 10, and so involves the solution of sets of nonlinear equations. The author discusses this and provides a Fortran program. He also provides a convenient list of the polar moments for the triangle.
Reviewer: J.N.Lyness

MSC:
65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A63 Multidimensional problems
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