Rigorous proof of cubic convergence for the dqds algorithm for singular values

  • Kensuke Aishima
  • Takayasu Matsuo
  • Kazuo Murota


Fernando and Parlett observed that the dqds algorithm for singular values can be made extremely efficient with Rutishauser’s choice of shift; in particular it enjoys “local” (or one-step) cubic convergence at the final stage of iteration, where a certain condition is to be satisfied. Their analysis is, however, rather heuristic and what has been shown is not sufficient to ensure asymptotic cubic convergence in the strict sense of the word. The objective of this paper is to specify a concrete procedure for the shift strategy and to prove with mathematical rigor that the algorithm with this shift strategy always reaches the “final stage” and enjoys asymptotic cubic convergence.

Key words

singular value bidiagonal matrix dqds algorithm 


  1. [1]
    K. Aishima, T. Matsuo, K. Murota and M. Sugihara, On Convergence of the dqds Algorithm for Singular Value Computation. SIAM Journal on Matrix Analysis and Applications, to appear.Google Scholar
  2. [2]
    K. Aishima, T. Matsuo, K. Murota and M. Sugihara, A Shift Strategy for Superquadratic Convergence in the dqds Algorithm for Singular Values. Mathematical Engineering Technical Reports, METR 2007-12, University of Tokyo, March 2007.Google Scholar
  3. [3]
    E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenny and D. Sorensen, LAPACK Users’ Guide, Third Edition. SIAM, 1999.Google Scholar
  4. [4]
    J. Demmel, Applied Numerical Linear Algebra. SIAM, Philadelphia, 1997.MATHGoogle Scholar
  5. [5]
    J. Demmel and W. Kahan, Accurate Singular Values of Bidiagonal Matrices. SIAM Journal on Scientific Computing,11 (1990), 873–912.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    I.S. Dhillon, A New O(n2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem. Ph.D. Thesis, Computer Science Division, University of California, Berkeley, California, 1997.Google Scholar
  7. [7]
    I.S. Dhillon and B.N. Parlett, Multiple Representations to Compute Orthogonal Eigenvectors of Symmetric Tridiagonal Matrices. Linear Algebra and Its Applications,387 (2004), 1–28.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    I.S. Dhillon and B.N. Parlett, Orthogonal Eigenvectors and Relative Gaps. SIAM Journal on Matrix Analysis and Applications,25 (2004), 858–899.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    K.V. Fernando and B.N. Parlett, Accurate Singular Values and Differential qd Algorithms. Numerische Mathematik,67 (1994), 191–229.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. Henrici, Applied and Computational Complex Analysis, Vol. 1. Wiley, New York, 1974.MATHGoogle Scholar
  11. [11]
    LAPACK, Scholar
  12. [12]
    B.N. Parlett, The New qd Algorithm. Acta Numerica, 1995, 459–491.Google Scholar
  13. [13]
    B.N. Parlett, The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, New Jersey, 1980; SIAM, Philadelphia, 1998.MATHGoogle Scholar
  14. [14]
    B.N. Parlett and O. Marques, An Implementation of the dqds Algorithm (Positive Case). Linear Algebra and Its Applications,309 (2000), 217–259.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    H. Rutishauser, Über eine kubisch konvergente Variante der LR-Transformation. Zeitschrift für Angewandte Mathematik und Mechanik,11 (1960), 49–54.Google Scholar
  16. [16]
    J.H. Wilkinson, The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1965.MATHGoogle Scholar

Copyright information

© JJIAM Publishing Committee 2008

Authors and Affiliations

  • Kensuke Aishima
    • 1
  • Takayasu Matsuo
    • 1
  • Kazuo Murota
    • 1
  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoJapan

Personalised recommendations