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Zh. Vychisl. Mat. Mat. Fiz.:

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Zh. Vychisl. Mat. Mat. Fiz., 2006, Volume 46, Number 7, Pages 1195–1210 (Mi zvmmf438)  

This article is cited in 6 scientific papers (total in 6 papers)

Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions

S. L. Skorokhodova, D. V. Khristoforovb

a Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia
b Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992, Russia

Abstract: A method for calculating eigenvalues $\lambda_{mn}(c)$ corresponding to the wave spheroidal functions in the case of a complex parameter c is proposed, and a comprehensive numerical analysis is performed. It is shown that some points $c_s$ are the branch points of the functions $\lambda_{mn}(c)$ with different indexes $n_1$ and $n_2$ so that the value $\lambda_{mn_1}(c_s)$ is a double one: $\lambda_{mn_1}(c_s)=\lambda_{mn_2}(c_s)$. The numerical analysis suggests that, for each fixed $m$, all the branches of the eigenvalues $\lambda_{mn}(c)$ corresponding to the even spheroidal functions form a complete analytic function of the complex argument $c$. Similarly, all the branches of the eigenvalues $\lambda_{mn}(c)$ corresponding to the odd spheroidal functions form a complete analytic function of $c$. To perform highly accurate calculations of the branch points $c_s$ of the double eigenvalues $\lambda_{mn}(c)$, the Padé approximants, the Hermite–Padé quadratic approximants, and the generalized Newton iterative method are used. A large number of branch points are calculated.

Key words: wave spheroidal functions, computation of eigenvalues, computation of branch points of eigenvalues, Padé approximants, generalized Newton iterative method.

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English version:
Computational Mathematics and Mathematical Physics, 2006, 46:7, 1132–1146

Bibliographic databases:

UDC: 519.6:517.589
Received: 21.12.2005

Citation: S. L. Skorokhodov, D. V. Khristoforov, “Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions”, Zh. Vychisl. Mat. Mat. Fiz., 46:7 (2006), 1195–1210; Comput. Math. Math. Phys., 46:7 (2006), 1132–1146

Citation in format AMSBIB
\by S.~L.~Skorokhodov, D.~V.~Khristoforov
\paper Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2006
\vol 46
\issue 7
\pages 1195--1210
\jour Comput. Math. Math. Phys.
\yr 2006
\vol 46
\issue 7
\pages 1132--1146

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    This publication is cited in the following articles:
    1. S. L. Skorokhodov, D. V. Khristoforov, “Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation”, Comput. Math. Math. Phys., 47:11 (2007), 1802–1818  mathnet  crossref  mathscinet  elib
    2. S. L. Skorokhodov, “Numerical analysis of the spectrum of the Orr–Sommerfeld problem”, Comput. Math. Math. Phys., 47:10 (2007), 1603–1621  mathnet  crossref  mathscinet  elib  elib
    3. Vinitsky S.I., Gerdt V.P., Gusev A.A., Kaschiev M.S., Rostovtsev V.A., Samoilov V.N., Tyupikova T.V., Chuluunbaatar O., “A symbolic-numerical algorithm for the computation of matrix elements in the parametric eigenvalue problem”, Program. Comput. Software, 33:2 (2007), 105–116  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. A. Abramov, E. D. Kalinin, S. V. Kurochkin, “Calculation of the spheroidal functions of the first kind for complex values of the argument and parameters”, Comput. Math. Math. Phys., 55:5 (2015), 788–796  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. S. L. Skorokhodov, “Vychislenie sobstvennykh znachenii i sobstvennykh funktsii kulonovskogo volnovogo sferoidalnogo uravneniya”, Matem. modelirovanie, 27:7 (2015), 111–116  mathnet  mathscinet  elib
    6. Richard-Jung F., Ramis J.-P., Thomann J., Fauvet F., “New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation”, Stud. Appl. Math., 138:1 (2017), 3–42  crossref  mathscinet  zmath  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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