TY - JOUR

T1 - Integrable discrete hungry systems and their related matrix eigenvalues

AU - Fukuda, Akiko

AU - Ishiwata, Emiko

AU - Yamamoto, Yusaku

AU - Iwasaki, Masashi

AU - Nakamura, Yoshimasa

N1 - Funding Information:
This was partially supported by Grants-in-Aid for Young Scientists (B) No. 20740064, Scientific Research (A) No. 20246027, and Scientific Research (C) No. 20540137 from the Japan Society for the Promotion of Science.

PY - 2013/6

Y1 - 2013/6

N2 - Recently, some of the authors designed an algorithm, named the dhLV algorithm, for computing complex eigenvalues of a certain class of band matrix. The recursion formula of the dhLV algorithm is based on the discrete hungry Lotka-Volterra (dhLV) system, which is an integrable system. One of the authors has proposed an algorithm, named the multiple dqd algorithm, for computing eigenvalues of a totally nonnegative (TN) band matrix. In this paper, by introducing a theorem on matrix eigenvalues, we first show that the eigenvalues of a TN matrix are also computable by the dhLV algorithm. We next clarify the asymptotic behavior of the discrete hungry Toda (dhToda) equation, which is also an integrable system, and show that a similarity transformation for a TN matrix is given through the dhToda equation. Then, by combining these properties of the dhToda equation, we design a new algorithm, named the dhToda algorithm, for computing eigenvalues of a TN matrix. We also describe the close relationship among the above three algorithms and give numerical examples.

AB - Recently, some of the authors designed an algorithm, named the dhLV algorithm, for computing complex eigenvalues of a certain class of band matrix. The recursion formula of the dhLV algorithm is based on the discrete hungry Lotka-Volterra (dhLV) system, which is an integrable system. One of the authors has proposed an algorithm, named the multiple dqd algorithm, for computing eigenvalues of a totally nonnegative (TN) band matrix. In this paper, by introducing a theorem on matrix eigenvalues, we first show that the eigenvalues of a TN matrix are also computable by the dhLV algorithm. We next clarify the asymptotic behavior of the discrete hungry Toda (dhToda) equation, which is also an integrable system, and show that a similarity transformation for a TN matrix is given through the dhToda equation. Then, by combining these properties of the dhToda equation, we design a new algorithm, named the dhToda algorithm, for computing eigenvalues of a TN matrix. We also describe the close relationship among the above three algorithms and give numerical examples.

KW - Discrete hungry Lotka-Volterra system

KW - Discrete hungry Toda equation

KW - Matrix eigenvalue

KW - Similarity transformation

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U2 - 10.1007/s10231-011-0231-0

DO - 10.1007/s10231-011-0231-0

M3 - Article

AN - SCOPUS:84878508153

SN - 0373-3114

VL - 192

SP - 423

EP - 445

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

IS - 3

ER -