TY - JOUR

T1 - Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation

AU - Fukuda, Akiko

AU - Yamamoto, Yusaku

AU - Iwasaki, Masashi

AU - Ishiwata, Emiko

AU - Nakamura, Yoshimasa

N1 - Funding Information:
This was partially supported by Grants-in-Aid for Young Scientists (B) No. 24740110, Scientific Research (A) No. 20246027 and Scientific Research (C) No. 23540163 from the Japan Society for the Promotion of Science, and Core Research for Evolutional Science and Technology (CREST) Program “Highly Productive, High Performance Application Frameworks for Post Petascale Computing” of Japan Science and Technology Agency (JST).

PY - 2012/10

Y1 - 2012/10

N2 - 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.

AB - 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.

KW - Discrete hungry Toda equation

KW - Mixed stability

KW - Relative perturbation

KW - Shifted LR transformation

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U2 - 10.1007/s11075-012-9606-6

DO - 10.1007/s11075-012-9606-6

M3 - Article

AN - SCOPUS:84866547459

SN - 1017-1398

VL - 61

SP - 243

EP - 260

JO - Numerical Algorithms

JF - Numerical Algorithms

IS - 2

ER -