Abstract
Recently, some of the authors designed an algorithm, named the dhLV algorithm, for computing complex eigenvalues of a certain class of band matrix. The recursion formula of the dhLV algorithm is based on the discrete hungry Lotka–Volterra (dhLV) system, which is an integrable system. One of the authors has proposed an algorithm, named the multiple dqd algorithm, for computing eigenvalues of a totally nonnegative (TN) band matrix. In this paper, by introducing a theorem on matrix eigenvalues, we first show that the eigenvalues of a TN matrix are also computable by the dhLV algorithm. We next clarify the asymptotic behavior of the discrete hungry Toda (dhToda) equation, which is also an integrable system, and show that a similarity transformation for a TN matrix is given through the dhToda equation. Then, by combining these properties of the dhToda equation, we design a new algorithm, named the dhToda algorithm, for computing eigenvalues of a TN matrix. We also describe the close relationship among the above three algorithms and give numerical examples.
Article PDF
Similar content being viewed by others
References
Ando T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)
Bogoyavlensky O.I.: Integrable discretizations of the KdV equation. Phys. Lett. A 134, 34–38 (1988)
Brenti F.: Combinatrics and total positivity. J. Combin. Theory Ser. A 71, 175–218 (1995)
Chu M.T.: Linear algebra algorithm as dynamical systems. Acta Numer. 17, 1–86 (2008)
Fukuda A., Ishiwata E., Iwasaki M., Nakamura Y.: The discrete hungry Lotka-Volterra system and a new algorithm for computing matrix eigenvalues. Inverse Probl. 25, 015007 (2009)
Gantmacher F., Krein M.: Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, revised edition. AMS Chelsea, Providence (2002)
Gasca, M., Micchelli, C.A. (eds.): Total Positivity and Its Applications, Math. Appl. 359. Kluwer, Dordrecht (1996)
Geist G.A., Howell G.W., Watkins D.S.: The BR eigenvalue algorithm. SIAM J. Mat. Anal. Appl. 20, 1083–1098 (1999)
Hirota R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50, 3785–3791 (1981)
Hirota R.: Conserved quantities of a random-time Toda equation. J. Phys. Soc. Jpn. 66, 283–284 (1997)
Itoh Y.: Integrals of a Lotka-Volterra system of odd number of variables. Prog. Theor. Phys. 78, 507–510 (1987)
Iwasaki M., Nakamura Y.: On the convergence of a solution of the discrete Lotka-Volterra system. Inverse Probl. 18, 1569–1578 (2002)
Iwasaki M., Nakamura Y.: An application of the discrete Lotka-Volterra system with variable step-size to singular value computation. Inverse Probl. 20, 553–563 (2004)
Iwasaki M., Nakamura Y.: Accurate computation of singular values in terms of shifted integrable schemes. Jpn. J. Indust. Appl. Math. 23, 239–259 (2006)
Karlin S.: Total Positivity, vol. I. Stanford University Press, Stanford (1968)
Koev P.: Accurate eigenvalue and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27, 1–23 (2005)
LAPACK: http://www.netlib.org/lapack/
Nagai A., Tokihiro T., Satsuma J.: The Toda molecule equation and the ε-algorithm. Math. Comp. 67–224, 1565–1575 (1998)
Nakamura Y.: Calculating Laplace transforms in terms of the Toda molecule. SIAM J. Sci. Comput. 20, 306–317 (1999)
Nakamura Y., Mukaihira A.: Dynamics of the finite Toda molecule over finite fields and a decoding algorithm. Phys. Lett. A 249, 295–302 (1998)
Nakamura, Y. (ed.): Applied Integrable Systems (in Japanese). Shokabo, Tokyo (2000)
Parlett B.N.: The new qd algorithms. Acta Numer. 4, 459–491 (1995)
Pinkus A.: Totally Positive Matrices. Cambridge University Press, New York (2010)
Takahashi D., Matsukidaira J.: Box and ball system with a carrier and ultradiscrete modified KdV equation. J. Phys. A 30–21, L733–L739 (1997)
Tokihiro T., Nagai A., Satsuma J.: Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization. Inverse Probl. 15, 1639–1662 (1999)
Rutishauser H.: Ein infinitesimales Analogon zum Quotienten-Differenzen-Algorithmus. Arch. Math. 5, 132–137 (1954)
Rutishauser H.: Solution of eigenvalue problems with the LR transformation. Nat. Bur. Stand. Appl. Math. Ser. 49, 47–81 (1958)
Rutishauser, H.: Lectures on Numerical Mathematics, Birkhäuser, Boston (1990)
Symes W.W.: The QR algorithm and scattering for the finite nonperiodic Toda lattice. Physica D 4, 275–280 (1982)
Tsujimoto S., Hirota R., Oishi S.: An extension and discretization of Volterra equation I (in Japanese). Tech. Rep. Proc. IEICE NLP 92–90, 1–3 (1993)
Watkins D.S.: Product eigenvalue problems. SIAM Rev. 47, 3–40 (2005)
Yamamoto Y., Fukaya T.: Differential qd algorithm for totally nonnegative band matrices: convergence properties and error analysis. JSIAM Lett. 1, 56–59 (2009)
Yamazaki S.: On the system of non-linear differential equations ẏ k = y k (y k+1 − y k-1). J. Phys. A: Math. Gen. 20, 6237–6241 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
This was partially supported by Grants-in-Aid for Young Scientists (B) No. 20740064, Scientific Research (A) No. 20246027, and Scientific Research (C) No. 20540137 from the Japan Society for the Promotion of Science.
Rights and permissions
About this article
Cite this article
Fukuda, A., Ishiwata, E., Yamamoto, Y. et al. Integrable discrete hungry systems and their related matrix eigenvalues. Annali di Matematica 192, 423–445 (2013). https://doi.org/10.1007/s10231-011-0231-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-011-0231-0
Keywords
- Discrete hungry Lotka–Volterra system
- Discrete hungry Toda equation
- Matrix eigenvalue
- Similarity transformation