Abstract
A process of searching on the sphere for the best (in a sense) cubature formulas that are invariant under the transformations of the icosahedral rotation group is described. The parameters of the best cubature formulas of this symmetry type up to the 30th order of accuracy are given to 16 significant digits. A table which contains the main characteristics of all the best to date cubature formulas of the icosahedral rotation group up to the 79th order of accuracy is given.
Similar content being viewed by others
REFERENCES
Haxton, D.J., Lebedev Discrete Variable Representation, J. Phys. Ser. B, 2007, vol. 40, no. 23, pp. 4443–4451.
Shadmehri, S., Saeidian, S., and Melezhik, V.S., 2D Nondirect Product Discrete Variable Representation for Schrödinger Equation with Nonseparable Angular Variables, J. Phys. Ser. B, 2020, vol. 53, no. 8, pp. 085001–085011.
Becke, A.D., A Multicenter Numerical Integration Scheme for Polyatomic Molecules, J. Chem. Phys., 1988, vol. 88, no. 4, pp. 2547–2553.
Laikov, D.N., Fast Evaluation of dEnsity Functional Exchange-Correlation Terms Using the Expansion of the Electron Density in Auxiliary Basis Sets, Chem. Phys. Lett., 1997, vol. 281, no. 2, pp. 151–156.
Kazakov, A.N. and Lebedev, V.I., Gauss-Type Quadrature Formulas for the Sphere, Invariant with Respect to Dihedral Group, Trudy Steklov Mat. Inst, 1994, vol. 203, pp. 100–112.
Ahrens, C. and Beylkin, G., Rotationally Invariant Quadratures for the Sphere, Proc. Royal Soc. Ser. A, 2009, vol. 465, no. 2110, pp. 3103–3125.
Sobolev, S.L., Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations, Soviet Math. Dokl., 1962, vol. 3, pp. 1307–1310.
Sobolev, S.L., The Formulas of Mechanical Cubature on the Surface of a Sphere, Sib. Mat. Zh., 1962, vol. 3, no. 5, pp. 769–796.
McLaren, A.D., Optimal Numerical Integration on a Sphere, Math. Comput., 1963, vol. 17, no. 83, pp. 361–383.
Lebedev, V.I., Values of the Nodes and Weights of Ninth to Seventeenth Order Gauss–Markov Quadrature Formulae Invariant under the Octahedron Group with Inversion, USSR Comput. Math. Math. Phys., 1975, vol. 15, no. 1, pp. 44–51.
Lebedev, V.I., Quadratures on a Sphere, USSR Comput. Math. Math. Phys., 1976, vol. 16, no. 2, pp. 10–24.
Lebedev, V.I., Spherical Quadrature Formulas Exact to Orders 25–29, Sib. Math. J., 1977, vol. 18, no 1, pp. 99–107.
Lebedev, V.I. and Laikov, D.N., A Quadrature Formula for the Sphere of the 131st Algebraic Order of Accuracy, Dokl. Mat., 1999, vol. 59, no. 3, pp. 477–481.
Konyaev, S.I., Quadrature Formulas on a Sphere Invariant with Respect to the Icosahedron Group, Preprint of the Institute of Atomic Energy, USSR Acad. Sci., Moscow, 1975, IAE-2516.
Konyaev, S.I., Quadratures of Gaussian Type for a Sphere Invariant under the Icosahedral Group with Inversion, Math. Notes Academy Sci. USSR, 1979, vol. 25, pp. 326–329.
Konyaev, S.I., Formulas for Numerical Integration on the Sphere. Embedding Theorems and Their Applications, Trudy Seminara akad. S.L. Soboleva (Proc. Acad. S.L. Sobolev’s Seminar), Novosibirsk, 1982, no. 1, pp. 75–82.
Konyaev, S.I., Quadrature Formulas for the Sphere of the 23rd and 27th Order Invariant under the Icosahedral Group with Inversion, Preprint of the Institute of Atomic Energy, USSR Acad. Sci., Moscow, 1990, IAE-5072/16.
Mysovskikh, I.P., Interpolyatsionnye kubaturnye formuly (Interpolation Cubature Formulas), Moscow: Nauka, 1981.
Popov, A.S., Cubature Formulae for a Sphere which are Invariant with Respect to the Tetrahedral Group, Comput. Math. Math. Phys., 1995, vol. 35, no. 3, pp. 369–374.
Popov, A.S., Cubature Formulae of High Orders of Accuracy for a Sphere which are Invariant with Respect to the Tetrahedral Group, Comput. Math. Math. Phys., 1996, vol. 36, no. 4, pp. 417–421.
Popov, A.S., Cubature Formulas on a Sphere that are Invariant with Respect to Octahedron Rotation Groups, Comput. Math. Math. Physi., 1998, vol. 38, no. 1, pp. 30–37.
Popov, A.S., Search for the Best Cubature Formulas for the Sphere that are Invariant under the Octahedral Rotation Group, Sib. Zh. Vych. Mat., 2002, vol. 5, no. 4, pp. 367–372.
Popov, A.S., Search for Best Cubature Formulas for a Sphere that are Invariant under the Octahedral Group of Rotations with Inversion, Sib. Zh. Vych. Mat., 2005, vol. 8, no. 2, pp. 143–148.
Popov, A.S., Cubature Formulas on a Sphere Invariant under the Icosahedral Rotation Group, Num. An. Appl., 2008, vol. 1, no. 4, pp. 355–361.
Popov, A.S., New Cubature Formulas on a Sphere that are Invariant under the Icosahedral Group of Rotations, Trudy 10go Mezhdunarodnogo seminara-soveshchaniya (Proc. 10th International Seminar–Conference), Ulan-Ude, 2009, pp. 111–118.
Popov, A.S., Cubature formulas on a Sphere Invariant under the Tetrahedral Group with Inversion, Sib. El. Mat. Izv., 2014, vol. 11, pp. 372–379.
Popov, A.S., Cubature Formulas on a Sphere Invariant under the Symmetry Groups of Regular Polyhedrons, Sib. El. Mat. Izv., 2017, vol. 14, pp. 190–198.
Popov, A.S., Cubature Formulas Invariant under the Icosahedral Group of Rotations with Inversion on a Sphere, Num. An. Appl., 2017, vol. 10, no. 4, pp. 339–346.
Popov, A.S., Cubature Formulas on a Sphere that are Invariant with Respect to a Group of Dihedron Rotations with Inversion \(D_{6h}\), Num. An. Appl., 2013, vol. 6, no. 1, pp. 49–53.
Popov, A.S., Cubature Formulas on a Sphere Invariant under the Dihedral Group of Rotations with Inversion \(D_{4h}\), Sib. El. Mat. Izv., 2015, vol. 12, pp. 457–464.
Popov, A.S., Cubature Formulas on a Sphere Invariant under the Dihedral Group \(D_{2h}\), Sib. El. Mat. Izv., 2016, vol. 13, pp. 252–259.
Popov, A.S., Cubature Formulas on a Sphere Invariant under the Dihedral Group of Rotations with Inversion \(D_{5d}\), Sib. El. Mat. Izv., 2018, vol. 15, pp. 389–396.
Popov, A.S., Cubature Formulas on a Sphere that are Invariant under the Transformations of the Dihedral Group of Rotations with Inversion \(D_{3d}\), Sib. El. Mat. Izv., 2019, vol. 16, pp. 1196–1204.
Popov, A.S., Cubature Formulas on the Sphere that are Invariant under the Transformations of the Dihedral Group of Rotations \(D_4\), Sib. El. Math. Rep., 2020, vol. 17, pp. 964–970.
Popov, A.S., Cubature Formulas on the Sphere that are Invariant Under the Transformations of the Dihedral Groups of Rotations with Inversion, Sib. El. Math. Rep., 2021, vol. 18, no. 1, pp. 703–709.
Landau, L.D. and Lifshitz, E.M. Quantum Mechanics ( Non-Relativistic Theory), New York: Pergamon Press, 1977.
Ditkin, V.A., On Some Approximate Formulas to Calculate Triple Integrals, Dokl. Akad. Nauk SSSR, 1948, vol. 62, no. 4, pp. 445–447.
Ditkin, V.A. and Lyusternik, L.A., On a Method of Practical Harmonic Analysis on a Sphere, Vych. Mat. Vych. Tekh., Moscow: Mashgiz, 1953, no. 1, pp. 3–13.
Dennis, J.E. and Schnabel, R.B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, New Jersey, Englewood Cliffs: Prentice-Hall, 1983.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Zhurnal Vychislitel’noi Matematiki, 2023, Vol. 26, No. 4, pp. 415-430. https://doi.org/10.15372/SJNM20230406.
Publisher’s Note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Popov, A.S. Search for the Best Cubature Formulas on the Sphere Invariant under the Icosahedral Rotation Group. Numer. Analys. Appl. 16, 348–358 (2023). https://doi.org/10.1134/S1995423923040067
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995423923040067