Geodesic
Cubature formulas on a sphere that are invariant under the transformations of the dihedral group of rotations with inversion $D_{3d}$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1196-1204.

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An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant under the transformations of the dihedral group of rotations with inversion D3d is described. This algorithm is applied to find the parameters of all the best cubature formulas of this symmetry type up to the 35th order of accuracy.
Keywords: numerical integration, dihedral group of rotations.
Mots-clés : invariant cubature formulas, invariant polynomials 
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A. S. Popov. Cubature formulas on a sphere that are invariant under the transformations of the dihedral group of rotations with inversion $D_{3d}$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1196-1204. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a117/

[1] A.N. Kazakov, V.I. Lebedev, “Gauss type quadrature formulas for a sphere that are invariant with respect to the group of dihedron”, Tr. Mat. Inst. Steklova, 203, 1994, 100–112 (in Russian) | MR | Zbl

[2] A.S. Popov, “Cubature formulae for a sphere invariant under cyclic rotation groups”, Russ. J. Numer. Anal. Math. Modelling, 9:6 (1994), 535–546 | DOI | MR | Zbl

[3] A.S. Popov, “Cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D6h”, Numerical Analysis and Applications, 6:1 (2013), 49–53 | DOI | MR | Zbl

[4] A.S. Popov, “Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion D4h”, Siberian Electronic Mathematical Reports, 12 (2015), 457–464 (in Russian) | MR | Zbl

[5] A.S. Popov, “Cubature formulas on a sphere invariant under the dihedral group D2h”, Siberian Electronic Mathematical Reports, 13 (2016), 252–259 (in Russian) | MR | Zbl

[6] A.S. Popov, “Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion D5d”, Siberian Electronic Mathematical Reports, 15 (2018), 389–396 (in Russian) | MR

[7] F. Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Dover, New York, 1956 | MR | Zbl

[8] L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Nauka, M., 1989 (in Russian) | MR | Zbl

[9] A.S. Popov, “The search for the sphere of the best cubature formulae invariant under octahedral group of rotations”, Siberian J. Num. Math., 5:4 (2002), 367–372 (in Russian) | MR

[10] S.L. Sobolev, “On mechanical cubature formulas for the surface of a sphere”, Sibirskii Mat. Zh., 3:5 (1962), 769–796 (in Russian) | MR | Zbl

[11] A.S. Popov, “The search for the best cubature formulae invariant under the octahedral group of rotations with inversion for a sphere”, Siberian J. of Numer. Mathematics, 8:2 (2005), 143–148 (in Russian) | Zbl

[12] A.S. Popov, “Cubature formulas invariant under the icosahedral group of rotations with inversion on a sphere”, Numerical Analysis and Applications, 10:4 (2017), 339–346 | DOI | MR | Zbl

[13] V.A. Ditkin, “On some approximate formulas for calculating triple integrals”, Dokl. Akad. Nauk SSSR, 62:4 (1948), 445–447 (in Russian) | MR | Zbl

[14] V.A. Ditkin, L.A. Lyusternik, “On a method of practical harmonic analysis on a sphere”, Vychisl. Matematika i Vychisl. Tekhnika, 1 (1953), 3–13 (in Russian) | MR | Zbl

[15] A.D. McLaren, “Optimal numerical integration on a sphere”, Math. Comput., 17:83 (1963), 361–383 | DOI | MR | Zbl

[16] V.I. Lebedev, “Nodes and weights of Gauss-Markov type quadrature formulas from 9th to 17th accuracy orders for a sphere which are invariant under the octahedral group with inversion”, Zh. Vychisl. Mat. Mat. Fiz., 15:1 (1975), 48–54 (in Russian) | MR | Zbl