Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation

Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Based on the integrable discrete hungry Toda (dhToda) equation, the authors designed an algorithm for computing eigenvalues of a class of totally nonnegative matrices (Ann Mat Pura Appl, doi: 10. 1007/s10231-011-0231-0). This is named the dhToda algorithm, and can be regarded as an extension of the well-known qd algorithm. The shifted dhToda algorithm has been also designed by introducing the origin shift in order to accelerate the convergence. In this paper, we first propose the differential form of the shifted dhToda algorithm, by referring to that of the qds (dqds) algorithm. The number of subtractions is then reduced and the effect of cancellation in floating point arithmetic is minimized. Next, from the viewpoint of mixed error analysis, we investigate numerical stability of the proposed algorithm in floating point arithmetic. Based on this result, we give a relative perturbation bound for eigenvalues computed by the new algorithm. Thus it is verified that the eigenvalues computed by the proposed algorithm have high relative accuracy. Numerical examples agree with our error analysis for the algorithm.

Original languageEnglish
Pages (from-to)243-260
Number of pages18
JournalNumerical Algorithms
Volume61
Issue number2
DOIs
Publication statusPublished - 2012 Oct
Externally publishedYes

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Error analysis
Digital arithmetic
Convergence of numerical methods

Keywords

  • Discrete hungry Toda equation
  • Mixed stability
  • Relative perturbation
  • Shifted LR transformation

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation. / Fukuda, Akiko; Yamamoto, Yusaku; Iwasaki, Masashi; Ishiwata, Emiko; Nakamura, Yoshimasa.

In: Numerical Algorithms, Vol. 61, No. 2, 10.2012, p. 243-260.

Research output: Contribution to journalArticle

Fukuda, Akiko ; Yamamoto, Yusaku ; Iwasaki, Masashi ; Ishiwata, Emiko ; Nakamura, Yoshimasa. / Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation. In: Numerical Algorithms. 2012 ; Vol. 61, No. 2. pp. 243-260.
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