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

• Published in: Numerical algorithms Vol. 61; no. 2; pp. 243 - 260
• Main Authors: Fukuda, Akiko, Yamamoto, Yusaku, Iwasaki, Masashi, Ishiwata, Emiko, Nakamura, Yoshimasa
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2012; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computation; Computer Science; Eigenvalues; Error analysis; Floating point arithmetic; Numeric Computing; Numerical Analysis; Numerical stability; Original Paper; Stability analysis; Theory of Computation; 65F50; Shifted; Mixed stability; Relative perturbation; 65F15; transformation; Discrete hungry Toda equation; 37K10
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-012-9606-6

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.