Accurate Algorithms for Bessel Matrices

  • Jorge DelgadoEmail author
  • Héctor Orera
  • J. M. Peña


In this paper, we prove that any collocation matrix of Bessel polynomials at positive points is strictly totally positive, that is, all its minors are positive. Moreover, an accurate method to construct the bidiagonal factorization of these matrices is obtained and used to compute with high relative accuracy the eigenvalues, singular values and inverses. Similar results for the collocation matrices for the reverse Bessel polynomials are also obtained. Numerical examples illustrating the theoretical results are included.


Bessel matrices Totally positive matrices High relative accuracy Bessel polynomials Reverse Bessel polynomials 

Mathematics Subject Classification

65F05 65F15 65G50 33C10 33C45 15A23 



This work was partially supported through the Spanish research grant PGC2018-096321-B-I00 (MCIU/AEI), by Gobierno de Aragón (E41_17R) and by Feder 2014-2020 “Construyendo Europa desde Aragon”.


  1. 1.
    Alonso, P., Delgado, J., Gallego, R., Peña, J.M.: Conditioning and accurate computations with Pascal matrices. J. Comput. Appl. Math. 252, 21–26 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carnicer, J.M., Mainar, E., Peña, J.M.: Critical lengths of cycloidal spaces are zeros of Bessel functions. Calcolo 54, 1521–1531 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Delgado, J., Peña, J.M.: Fast and accurate algorithms for Jacobi–Stirling matrices. Appl. Math. Comput. 236, 253–259 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Delgado, J., Peña, J.M.: Accurate computations with collocation matrices of q-Bernstein polynomials. SIAM J. Matrix Anal. Appl. 36, 880–893 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Demmel, J., Koev, P.: The accurate and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J. Matrix Anal. Appl. 27, 142–152 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton Series in Applied Mathematics, vol. 35. Princeton University Press, Princeton (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gantmacher, F.R., Krein, M.G.: Oszillationsmatrizen, oszillationskerne und kleine schwingungen mechanischer systeme. Akademie, Berlin (1960)zbMATHGoogle Scholar
  9. 9.
    Gasca, M., Peña, J.M.: Total positivity and Neville elimination. Linear Algebra Appl. 165, 25–44 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gasca, M., Peña, J.M.: On factorizations of totally positive matrices. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and Its Applications, pp. 109–130. Kluver Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  11. 11.
    Grosswald, E.: Bessel Polynomials. Springer, New York (1978)CrossRefzbMATHGoogle Scholar
  12. 12.
    Han, H., Seo, S.: Combinatorial proofs of inverse relations and log-concavity for Bessel numbers. Eur. J. Combin. 29, 1544–1554 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27, 1–23 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koev, P.: Accessed November 12th (2018)
  16. 16.
    Krall, H.L., Frink, O.: A new class of orthogonal polynomials: the Bessel polynomials. Trans. Am. Math. Soc. 65, 100–115 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marco, A., Martínez, J.-J.: A fast and accurate algorithm for solving Bernstein–Vandermonde linear systems. Linear Algebra Appl. 422, 616–628 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Marco, A., Martínez, J.-J.: Accurate computations with Said–Ball–Vandermonde matrices. Linear Algebra Appl. 432, 2894–2908 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Marco, A., Martínez, J.-J.: Accurate computations with totally positive Bernstein–Vandermonde matrices. Electron. J. Linear Algebra 26, 357–380 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pasquini, L.: Accurate computation of the zeros of the generalized Bessel polynomials. Numer. Math. 86, 507–538 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pinkus, A.: Totally Positive Matrices. Tracts in Mathematics, vol. 181. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  22. 22.
    Yang, S.L., Qiao, Z.K.: The Bessel numbers and Bessel matrices. J. Math. Res. Expo. 31, 627–636 (2011)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, Escuela Universitaria Politécnica de TeruelUniversidad de ZaragozaTeruelSpain
  2. 2.Departamento de Matemática Aplicada, Facultad de CienciasUniversidad de ZaragozaZaragozaSpain

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