Applications of Mathematics

, Volume 62, Issue 4, pp 319–331

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Article
  • 21 Downloads

Abstract

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ Rm×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A−1) and Tr(A−2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A−1) and Tr(A−2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A−1)}2/Tr(A−2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.

Keywords

eigenvalue bound symmetric positive definite matrix Laguerre bound singular value computation dqds algorithm 

MSC 2010

15A18 15A42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    K. Aishima, T. Matsuo, K. Murota, M. Sugihara: A survey on convergence theorems of the dqds algorithm for computing singular values. J. Math-for-Ind. 2 (2010), 1–11.MathSciNetMATHGoogle Scholar
  2. [2]
    G. Alefeld: On the convergence of Halley’s method. Am. Math. Monthly 88 (1981), 530–536.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    K. V. Fernando, B. N. Parlett: Accurate singular values and differential qd algorithms. Numer. Math. 67 (1994), 191–229.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    G. H. Golub, C. F. Van Loan: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, 2013.Google Scholar
  5. [5]
    A. S. Householder: The Numerical Treatment of a Single Nonlinear Equation. International Series in Pure and Applied Mathematics, McGraw-Hill Book, New York, 1970.Google Scholar
  6. [6]
    M. Iwasaki, Y. Nakamura: Accurate computation of singular values in terms of shifted integrable schemes. Japan J. Ind. Appl. Math. 23 (2006), 239–259.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    C. R. Johnson: A Gersgorin-type lower bound for the smallest singular value. Linear Algebra Appl. 112 (1989), 1–7.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    C. R. Johnson, T. Szulc: Further lower bounds for the smallest singular value. Linear Algebra Appl. 272 (1998), 169–179.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    K. Kimura, T. Yamashita, Y. Nakamura: Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices. J. Phys. A, Math. Theor. 44 (2011), Article ID 285207, 12 pages.Google Scholar
  10. [10]
    U. von Matt: The orthogonal qd-algorithm. SIAM J. Sci. Comput. 18 (1997), 1163–1186.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    T. Yamashita, K. Kimura, Y. Nakamura: Subtraction-free recurrence relations for lower bounds of the minimal singular value of an upper bidiagonal matrix. J. Math-for-Ind. 4 (2012), 55–71.MathSciNetMATHGoogle Scholar
  12. [12]
    T. Yamashita, K. Kimura, M. Takata, Y. Nakamura: An application of the Kato-Temple inequality on matrix eigenvalues to the dqds algorithm for singular values. JSIAMLett. 5 (2013), 21–24.MathSciNetGoogle Scholar
  13. [13]
    T. Yamashita, K. Kimura, Y. Yamamoto: A new subtraction-free formula for lower bounds of the minimal singular value of an upper bidiagonal matrix. Numer. Algorithms 69 (2015), 893–912.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. H. Wilkinson: The Algebraic Eigenvalue Problem. Monographs on Numerical Analysis, Oxford Science Publications, Clarendon Press, Oxford, 1988.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.The University of Electro-CommunicationsChofu, TokyoJapan

Personalised recommendations