Abstract:
An algorithm for constructing the cubature formulas on a sphere that are invariant with respect to the icosahedral group of rotations is proposed. This algorithm is applied to construct the new cubature formulas that have the algebraic order of accuracy n=19,20,21,23,24,25. Parameters of these cubature formulas are given to 16 significant digits. The table, which contains the main characteristics of all the best today cubature formulas of the icosahedral group of rotations up to the 35th order of accuracy, is given. A real variant of F. Klein's formula that states the connection between the basic invariant forms of the icosahedral group of rotations is given.
Key words:
numerical integration, invariant cubature formulas, invariant polynomials, icosahedral group of rotations.
Citation:
A. S. Popov, “The cubature formulas on a sphere that are invariant with respect to the icosahedral group of rotations”, Sib. Zh. Vychisl. Mat., 11:4 (2008), 433–440; Num. Anal. Appl., 1:4 (2008), 355–361
\Bibitem{Pop08}
\by A.~S.~Popov
\paper The cubature formulas on a~sphere that are invariant with respect to the icosahedral group of rotations
\jour Sib. Zh. Vychisl. Mat.
\yr 2008
\vol 11
\issue 4
\pages 433--440
\mathnet{http://mi.mathnet.ru/sjvm61}
\transl
\jour Num. Anal. Appl.
\yr 2008
\vol 1
\issue 4
\pages 355--361
\crossref{https://doi.org/10.1134/S199542390804006X}
Linking options:
https://www.mathnet.ru/eng/sjvm61
https://www.mathnet.ru/eng/sjvm/v11/i4/p433
This publication is cited in the following 19 articles:
A. S. Popov, “Cubature Formulas on the Sphere that are Invariant under Dihedral Rotation Groups”, Numer. Analys. Appl., 18:1 (2025), 78
A. S. Popov, “Poisk nailuchshikh kubaturnykh formul na sfere, invariantnykh otnositelno gruppy vraschenii ikosaedra”, Sib. zhurn. vychisl. matem., 26:4 (2023), 415–430
A. S. Popov, “Search for the Best Cubature Formulas on the Sphere Invariant under the Icosahedral Rotation Group”, Numer. Analys. Appl., 16:4 (2023), 348
R. È. Bairamov, Yu. A. Blinkov, I. V. Levichev, M. D. Malykh, V. S. Melezhik, “Analytical study of cubature formulas on a sphere in computer algebra systems”, Comput. Math. Math. Phys., 63:1 (2023), 77–85
Sara Shadmehri, Vladimir S Melezhik, “A hydrogen atom in strong elliptically polarized laser fields within discrete variable representation”, Laser Phys., 33:2 (2023), 026001
A. S. Popov, “The cubature formulas on a sphere invariant to the icosahedral group of rotations with inversion”, Num. Anal. Appl., 10:4 (2017), 339–346
Popov A.S., “Cubature Formulas on a Sphere Invariant Under the Symmetry Groups of Regular Polyhedrons”, Sib. Electron. Math. Rep., 14 (2017), 190–198
Popov A.S., “Cubature Formulas on a Sphere Invariant Under the Dihedral Group D2H”, Sib. Electron. Math. Rep., 13 (2016), 252–259
A. V. Shilkov, “Even- and odd-parity kinetic equations of particle transport. 3: Finite analytic scheme on tetrahedra”, Math. Models Comput. Simul., 7:5 (2015), 409–429
A. S. Popov, “Kubaturnye formuly na sfere, invariantnye otnositelno gruppy vraschenii diedra s inversiei $\mathrm{D_{4h}}$”, Sib. elektron. matem. izv., 12 (2015), 457–464
Kerstin Hesse, Ian H. Sloan, Robert S. Womersley, Handbook of Geomathematics, 2015, 1
Kerstin Hesse, Ian H. Sloan, Robert S. Womersley, Handbook of Geomathematics, 2015, 2671
A. S. Popov, “Kubaturnye formuly na sfere, invariantnye otnositelno gruppy tetraedra s inversiei”, Sib. elektron. matem. izv., 11 (2014), 372–379
A. S. Popov, “The cubature formulas on a sphere invariant with respect to a dihedral group of rotations with inversion $D_{6h}$”, Num. Anal. Appl., 6:1 (2013), 49–53
Kerstin Hesse, Ian H. Sloan, Robert S. Womersley, Handbook of Geomathematics, 2013, 1
Kerstin Hesse, Ian H. Sloan, Robert S. Womersley, Handbook of Geomathematics, 2010, 1185
Gräf M., Potts D., “Sampling sets and quadrature formulae on the rotation group”, Numerical Functional Analysis and Optimization, 30:7-8 (2009), 665–688
Ahrens C., Beylkin G., “Rotationally invariant quadratures for the sphere”, Proc. R. Soc. A, 465:2110 (2009), 3103–3125
Citation in format AMSBIB
Contact us: