Geodesic
Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion $\mathrm{D}_{5d}$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 389-396.

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 under the transformations of dihedral group of rotations with inversion D5d 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 
@article{SEMR_2018_15_a112,
     author = {A. S. Popov},
     title = {Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion $\mathrm{D}_{5d}$},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {389--396},
     publisher = {mathdoc},
     volume = {15},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2018_15_a112/}
}
TY  - JOUR
AU  - A. S. Popov
TI  - Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion $\mathrm{D}_{5d}$
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2018
SP  - 389
EP  - 396
VL  - 15
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2018_15_a112/
LA  - ru
ID  - SEMR_2018_15_a112
ER  - 
%0 Journal Article
%A A. S. Popov
%T Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion $\mathrm{D}_{5d}$
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2018
%P 389-396
%V 15
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2018_15_a112/
%G ru
%F SEMR_2018_15_a112
A. S. Popov. Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion $\mathrm{D}_{5d}$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 389-396. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a112/

[1] 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

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

[3] 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

[4] V.I. Lebedev, “On quadratures for a sphere”, Zh. Vychisl. Mat. Mat. Fiz., 16:2 (1976), 293–306 (in Russian) | MR | Zbl

[5] S.I. Konyaev, “Gauss type quadratures for a sphere invariant under icosahedral group with inversion”, Mat. Zametki, 25:4 (1979), 629–634 (in Russian) | MR | Zbl

[6] I.P. Mysovskikh, Interpolation Cubature Formulas, Nauka, M., 1981 (in Russian) | MR | Zbl

[7] 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

[8] A.S. Popov, “Cubature formulas on a sphere invariant under the symmetry groups of regular polyhedrons”, Siberian Electronic Mathematical Reports, 14 (2017), 190–198 (in Russian) | MR | Zbl

[9] 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

[10] 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

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

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

[13] A.S. Popov, “Cubature formulas on a sphere invariant under the dihedral group $D_{2h}$”, Siberian Electronic Mathematical Reports, 13 (2016), 252–259 (in Russian) | MR | Zbl

[14] 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

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

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

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

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