Construction of fully symmetric cubature formulae of degree 4k-3 for fully symmetric planar regions
The paper first outlines the T-method attributed to \textit{C. R. Morrow} and \textit{T. N. L. Patterson} [SIAM J. Numer. Analysis 15, No.5, 953-976 (1978; Zbl 0402.65013)] and to \textit{H. J. Schmid} [Multivariate approximation theory, Proc. Conf. Math. Res. Inst., Oberwolfach 1979, ISNM Vol. 51, 359-377 (1979; Zbl 0456.41026)]. This uses the connection between knots of a cubature formula for product regions and ideal theory. From this the authors develop their S-method. Using the ALTRAN formula manipulation system, they obtain several specific cubature formulas for the square, circle and the entire plane \((E_ 2^{r^ 2},E^ r_ 2)\). The 36 positive weights and abscissas of some rules of degree 13 are listed. Only two rules in this sequence of degree 17 have been constructed, and each has some negative weights. These, together with rules of degrees 5 and 9, are listed in the authors' corresponding Technical report TW71 of K. U. Leuven (1985).
scientific article; zbMATH DE number 4018217