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Monomial cubature rules since “Stroud”: A compilation. II. (English) Zbl 0954.65021
[For part I see R. Cools and P. Rabinowitz, ibid. 48, No. 3, 309-329 (1993; Zbl 0799.65027).]
This second part of a compilation of rules for multiple integration continues in the same format as the original. New rules, new references and some corrections are reported mostly in tabular form to clearly correlate with the previous tables. An appropriate commentary accompanies the tables. The references also augment the earlier set with numbering running on consecutively.
The two parts together, and with A. H. Stroud’s book [Approximate calculation of multiple integrals (1971; Zbl 0379.65013)], then give a near comprehensive listing, at least of work available in English. As before, the regions covered are the \(n\)-cube, \(n\)-sphere and the entire space for selected low values of \(n\). Some results for general \(n\) are also included.

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A63 Multidimensional problems
Full Text: DOI
[1] Cohen, A.; Gismalla, D., Some integration formulae for symmetric functions of two variables, Int. J. comput. math., 19, 57-68, (1986) · Zbl 0653.65018
[2] Cools, R.; Haegemans, A., An imbedded family of cubature formulae for n-dimensional product regions, J. comput. appl. math., 51, 251-262, (1994) · Zbl 0821.65008
[3] Cools, R.; Rabinowitz, P., Monomial cubature rules Since ‘stroud’: A compilation, J. comput. appl. math., 48, 309-326, (1993) · Zbl 0799.65027
[4] A. Genz, R. Cools, An adaptive numerical cubature algorithm for simplices, Report TW 273, Department of Computer Science, K.U. Leuven, 1997. · Zbl 1072.65032
[5] Genz, A.; Keister, B., Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight, J. comput. appl. math., 71, 299-309, (1996) · Zbl 0856.65011
[6] G. Godzina, Dreidimensionale Kubaturformeln für zentralsymmetrische Integrale, Ph.D. Thesis, Universität Erlangen-Nürnberg, 1994. · Zbl 0941.65507
[7] Griener, B.; Schmid, H., An interactive tool to visualize common zeros of two-dimensional polynomials, J. comput. appl. math., 112, 57-68, (1999)
[8] T. Jankewitz, Zweidimensionale Kubaturformeln, Master’s Thesis, Universität Erlangen, 1998.
[9] Kim, K.; Song, M., Symmetric quadrature formulas over a unit disk, Korean J. comput. appl. math., 4, 179-192, (1997) · Zbl 0915.65021
[10] Kim, K.; Song, M., Invariant cubature formulas over a unit cube, Commun. Korean math. soc., 13, 913-931, (1998) · Zbl 0972.65019
[11] Maeztu, J.; de la Maza, E.S., An invariant quadrature rule of degree 11 for the tetrahedron, C. R. acad. sci. Paris, 321, 1263-1267, (1995) · Zbl 0836.65030
[12] J. Mysovskih, Interpolatorische Kubaturformeln, Bericht Nr. 74, Institut für Geometrie und Praktische Mathematik der RWTH Aachen, April 1992. Übertragung aus dem Russischen von Irina Dietrich und Hermann Engels.
[13] G. Rasputin, Construction of cubature formulas containing prespecified knots, Metody Vychisl. 13 (1983) 122-128 (in Russian).
[14] Stoyanova, S., Cubature formulae of the seventh degree of accuracy for the hypersphere, J. comput. appl. math., 84, 15-21, (1997) · Zbl 0880.41023
[15] P. Verlinden, Cubature formulas and asymptotic expansions, Ph.D. Thesis, Katholieke Universiteit Leuven, 1993.
[16] P. Verlinden, R. Cools, The algebraic construction of a minimal cubature formula of degree 11 for the square, in: M. Noskov (Eds.), Cubature Formulas and their Applications, (in Russian) Krasnoyarsk, 1994, pp. 13-23.
[17] S. Waldron, Symmetries of linear functionals, in: C. Chui, L. Schumaker (Eds.), Approximation Theory VIII, World Scientific, Singapore, 1995, pp. 541-550. · Zbl 1137.46309
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