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

  • Kensuke Aishima
  • Takayasu Matsuo
  • Kazuo Murota
Article

Abstract

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 

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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

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