Numerical Algorithms

, Volume 70, Issue 3, pp 469–484 | Cite as

A finite-step construction of totally nonnegative matrices with specified eigenvalues

  • Kanae Akaiwa
  • Yoshimasa Nakamura
  • Masashi Iwasaki
  • Hisayoshi Tsutsumi
  • Koichi Kondo
Original Paper

Abstract

Matrices where all minors are nonnegative are said to be totally nonnegative (TN) matrices. In the case of banded TN matrices, which can be expressed by products of several bidiagonal TN matrices, Fukuda et al. (Annal. Mat. Pura Appl. 192, 423–445, 2013) discussed the eigenvalue problem from the viewpoint of the discrete hungry Toda (dhToda) equation. The dhToda equation is a discrete integrable system associated with box and ball systems. In this paper, we consider an inverse eigenvalue problem for such banded TN matrices by examining the properties of the dhToda equation. This problem is a real-valued nonnegative inverse eigenvalue problem. First, we show the determinant solution to the dhToda equation with suitable boundary conditions. Next, we clarify the relationship between the characteristic polynomials of the banded TN matrices and the determinant solution. Finally, taking this relationship into account, we design a finite-step procedure for constructing banded TN matrices with specified eigenvalues. We also present an example to demonstrate this procedure.

Keywords

Finite-step construction Totally nonnegative Inverse eigenvalue problem Discrete hungry Toda equation 

Mathematical Subject Classifications (2010)

65F18 15A48 37N30 

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References

  1. 1.
    Akaiwa, K., Iwasaki, M., Kondo, K., Nakamura, Y.: A tridiagonal matrix construction by the quotient difference recursion formula in the case of multiple eigenvalues. Pacific. J. Math. Indust. 6 (2014)Google Scholar
  2. 2.
    Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)MATHCrossRefGoogle Scholar
  3. 3.
    de Boor, C., Golub, G.H.: The numerically stable reconstruction of a Jacobi matrix from spectral data. Linear Algebra Appl. 21, 245–260 (1978)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chu, M.T., Golub, G.H.: Inverse Eigenvalue Problems: Theory, Algorithms, and Applications. Oxford University Press, New York (2005)CrossRefGoogle Scholar
  5. 5.
    Cryer, C.W.: Some properties of totally positive matrices. Linear Algebra Appl. 15, 1–25 (1976)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fallat, S.: Bidiagonal factorizations of totally nonnegative matrices. Amer. Math. Monthly 108, 697–712 (2001)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton University Press, Princeton (2011)MATHCrossRefGoogle Scholar
  8. 8.
    Fukuda, A., Yamamoto, Y., Iwasaki, M., Ishiwata, E., Nakamura, Y.: Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation. Numer. Algor. 61, 243–260 (2012)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fukuda, A., Ishiwata, E., Yamamoto, Y., Iwasaki, M., Nakamura, Y.: Integrable discrete hungry systems and their related matrix eigenvalues. Annal. Mat. Pura Appl. 192, 423–445 (2013)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fukuda, A., Yamamoto, Y., Iwasaki, M., Ishiwata, E., Nakamura, Y.: On a shifted LR transformation derived from the discrete hungry Toda equation. Monat. Math. 170, 11–26 (2013)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Gasca, M., Micchelli, C.A. (eds.): Total positivity and its applications. Mathematics Applied, vol. 359. Kluwer Academic, Dordrecht (1996)Google Scholar
  12. 12.
    Gladwell, G.M.L.: Inner totally positive matrices. Linear Algebra Appl. 393, 179–195 (2004)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    The GNU Multiple precision arithmetic library, https://gmplib.org
  14. 14.
    The GNU MPFR library, http://www.mpfr.org
  15. 15.
    Gragg, W.B., Harrod, W.J.: The numerically stable reconstruction of Jacobi matrices from spectral data. Numer. Math. 44, 317–335 (1984)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 1. John Wiley, New York (1974)MATHGoogle Scholar
  17. 17.
    Hirota, R.: Nonlinear partial difference equation. II. Discrete-Time Toda equation. J. Phys. Soc. Japan 46, 2074–2078 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Karlin, S.: Total Positivity, vol. 1. Stanford University Press, Stanford (1968)Google Scholar
  19. 19.
    Pinkus, A.: Totally Positive Matrices. Cambridge University Press, New York (2009)CrossRefGoogle Scholar
  20. 20.
    Press, W.H., Vetterling, W.T., Teukolsky, S.A., Flannery, B.P., 2nd ed.: Numerical Recipes in C. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  21. 21.
    Symes. W.W.: The QR algorithm and scattering for the finite nonperiodic Toda lattice. Phys. D 4, 275–280 (1982)MATHCrossRefGoogle Scholar
  22. 22.
    Tokihiro, T., Nagai, A., Satsuma, J.: Proof of solitonical nature of box and ball system by the means of inverse ultra-discretization. Inverse Probl. 15, 1639–1662 (1999)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Kanae Akaiwa
    • 1
  • Yoshimasa Nakamura
    • 1
  • Masashi Iwasaki
    • 2
  • Hisayoshi Tsutsumi
    • 3
  • Koichi Kondo
    • 3
  1. 1.Graduate School of InformaticsKyoto UniversitySakyo-kuJapan
  2. 2.Department of Informatics and Environmental ScienceKyoto Prefectural UniversitySakyo-kuJapan
  3. 3.Graduate School of Science and EngineeringDoshisha UniversityKyotanabe CityJapan

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