Geodesic
The cubature formulas on a sphere invariant with respect to a dihedral group of rotations with inversion $D_{6h}$
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 1, pp. 57-62 
A. S. Popov. The cubature formulas on a sphere invariant with respect to a dihedral group of rotations with inversion $D_{6h}$. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 1, pp. 57-62. http://geodesic.mathdoc.fr/item/SJVM_2013_16_1_a5/
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     title = {The cubature formulas on a~sphere invariant with respect to a~dihedral group of rotations with inversion~$D_{6h}$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant with respect to a dihedral group of rotations with inversion D6h has been veloped. This algorithm was applied to find parameters of all the best cubature formulas of this group of symmetry up to the $23$rd order of accuracy $n$. In the course of the study carried out, the exact values of parameters of the corresponding cubature formulas were found for $n\le11$, and the approximate ones were obtained by the numerical solution of systems of nonlinear algebraic equations by a Newton-type method for the other $n$. For the first time, the ways of obtaining the best cubature formulas for the sphere were systematically investigated for the case of the group which is not a subgroup of the groups of symmetry of the regular polyhedrons.

[1] Sobolev S. L., “O formulakh mekhanicheskikh kubatur na poverkhnosti sfery”, Sib. mat. zhurnal., 3:5 (1962), 769–796 | MR | Zbl

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

[3] Lebedev V. I., “Znacheniya uzlov i vesov kvadraturnykh formul tipa Gaussa–Markova dlya sfery ot 9-go do 17-go poryadka tochnosti, invariantnykh otnositelno gruppy oktaedra s inversiei”, Zhurn. vychisl. matem. i mat. fiziki, 15:1 (1975), 48–54 | MR | Zbl

[4] Lebedev V. I., “O kvadraturakh na sfere”, Zhurn. vychisl. matem. i mat. fiziki, 16:2 (1976), 293–306 | MR | Zbl

[5] Lebedev V. I., “Kvadraturnye formuly dlya sfery 25–29-go poryadka tochnosti”, Sib. mat. zhurnal, 18:1 (1977), 132–142 | MR | Zbl

[6] Lebedev V. I., Laikov D. N., “Kvadraturnaya formula dlya sfery 131-go algebraicheskogo poryadka tochnosti”, Dokl. RAH, 366:6 (1999), 741–745 | MR | Zbl

[7] Konyaev S. I., “Kvadratury tipa Gaussa dlya sfery, invariantnye otnositelno gruppy ikosaedra s inversiei”, Mat. zametki, 25:4 (1979), 629–634 | MR | Zbl

[8] Konyaev S. I., “Formuly chislennogo integrirovaniya na sfere”, Teoremy vlozheniya i ikh prilozheniya, Tr. seminara akad. S. L. Soboleva, 1, Novosibirsk, 1982, 75–82 | MR | Zbl

[9] Mysovskikh I. P., Interpolyatsionnye kubaturnye formuly, Nauka, M., 1981 | MR | Zbl

[10] Popov A. S., “Kubaturnye formuly dlya sfery, invariantnye otnositelno gruppy tetraedra”, Zhurn. vychisl. matem. i mat. fiziki, 35:3 (1995), 459–466 | MR | Zbl

[11] Popov A. S., “Kubaturnye formuly vysokikh poryadkov tochnosti dlya sfery, invariantnye otnositelno gruppy tetraedra”, Zhurn. vychisl. matem. i mat. fiziki, 36:4 (1996), 5–9 | MR | Zbl

[12] Popov A. S., “Kubaturnye formuly na sfere, invariantnye otnositelno gruppy vraschenii oktaedra”, Zhurn. vychisl. matem. i mat. fiziki, 38:1 (1998), 34–41 | MR | Zbl

[13] Popov A. S., “Poisk nailuchshikh kubaturnykh formul dlya sfery, invariantnykh otnositelno gruppy vraschenii oktaedra”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 5:4 (2002), 367–372

[14] Popov A. S., “Poisk nailuchshikh kubaturnykh formul dlya sfery, invariantnykh otnositelno gruppy vraschenii oktaedra s inversiei”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 8:2 (2005), 143–148 | Zbl

[15] Popov A. S., “Kubaturnye formuly na sfere, invariantnye otnositelno gruppy vraschenii ikosaedra”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 11:4 (2008), 433–440 | Zbl

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

[17] Kazakov A. N., Lebedev V. I., “Kvadraturnye formuly tipa Gaussa dlya sfery, invariantnye otnositelno gruppy diedra”, Tr. MIRAN, 203, Nauka, M., 1994, 100–112 | MR | Zbl

[18] Klein F., Lektsii ob ikosaedre i reshenii uravnenii pyatoi stepeni, Editorial URSS, M., 2004