Application of discrete and ultradiscrete integrable systems of hungry type to eigenvalue problem

• Project/Area Number: 16K21368
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis; Foundations of mathematics/Applied mathematics
• Research Institution: Shibaura Institute of Technology
• Principal Investigator: Fukuda Akiko 芝浦工業大学, システム理工学部, 准教授 (70609297)
• Research Collaborator: Watanabe Sennosuke
• Project Period (FY): 2016-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 離散可積分系 / 超離散可積分系 / 超離散化 / 固有値 / min-plus代数 / 箱玉系 / 保存量 / totally nonnegative行列 / 離散戸田方程式 / 超離散戸田方程式 / 超離散ロトカ・ボルテラ系 / Max-Plus代数 / トロピカル代数 / 可積分系 / Min-Plus代数 / 有向グラフ / 固有値問題 / 固有値計算

Outline of Final Research Achievements

In this project, we first developed a new algorithm to compute eigenvalues of matrices with high accuracy, using equations that generalize existing integrable systems. Furthermore, we clarified that the limit procedure, called ultradiscretization, can be applied to compute eigenvalues on min-plus algebra, which is different from those of ordinary linear algebra. The eigenvalues on the min-plus algebra correspond to the minimum average weights of circuits in weighted directed graphs, and it is expected that it is applied for the shortest path problem on graphs.

Academic Significance and Societal Importance of the Research Achievements

行列の固有値問題は振動解析や情報検索，主成分分析，各種コンピュータシミュレーションなど実用上様々な応用があり，高速高精度な計算が必要とされている。しかし，一般の大規模で非対称な行列の固有値を精度良く求めることは非常に難しい。本研究開発した新しいアルゴリズムは，対象となる行列のクラスは多少限定されるものの，相対誤差の意味で高精度に固有値を計算できる。また，超離散化と呼ばれるテクニックを用いて，min-plus代数上の固有値を計算する新しいアルゴリズムを開発した。本研究は超離散可積分系と固有値計算アルゴリズムの対応を示した初の結果である。

Research Products

• [Journal Article] Ultradiscrete Lotka-Volterra system computes tropical eigenvalue of symmetric tridiagonal matrices (2019)
  • Author(s): Akiko Fukuda, Sennosuke Watanabe, Ayumi Hanaoka, Masashi Iwasaki
  • Journal Title: IOP Conference Series Volume: accepted
• [Journal Article] Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation (2018)
  • Author(s): Watanabe Sennosuke、Fukuda Akiko、Shigitani Hitomi、Iwasaki Masashi
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 51 Issue: 44 Pages: 444001-444001
  • DOI: 10.1088/1751-8121/aae325
  • Peer Reviewed / Open Access
• [Journal Article] Periodic convergence in the discrete hungry Toda equation (2018)
  • Author(s): You Takahashi, Masashi Iwasaki, Akiko Fukuda, Emiko Ishiwata, Yoshimasa Nakamura
  • Journal Title: Journal of Physics A-Mathematical and Theoretical Volume: 51 Issue: 34 Pages: 344001-344001
  • DOI: 10.1088/1751-8121/aaccb5
  • Peer Reviewed
• [Journal Article] Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete Lotka-Volterra system (2017)
  • Author(s): Tobita Akihiko, Akiko Fukuda, Emiko Ishiwata, Masashi Iwasaki, Yoshimasa Nakamura
  • Journal Title: Lecture Notes in Computational Science and Engineering Volume: 117 Pages: 157-169
  • DOI: 10.1007/978-3-319-62426-6_11
  • ISBN: 9783319624242, 9783319624266
  • Peer Reviewed
• [Journal Article] Computation of Eigenvectors for a Specially Structured Banded Matrix (2017)
  • Author(s): Takeuchi Hiroshi, Aihara Kensuke, Fukuda Akiko, Ishiwata Emiko
  • Journal Title: Lecture Notes in Computational Science and Engineering Volume: 117 Pages: 143-155
  • DOI: 10.1007/978-3-319-62426-6_10
  • ISBN: 9783319624242, 9783319624266
  • Peer Reviewed / Open Access
• [Journal Article] Computation of eigenvectors for a specially structured banded matrix (2017)
  • Author(s): Hiroshi Takeuchi, Kensuke Aihara, Akiko Fukuda, Emiko Ishiwata
  • Journal Title: Lecture Notes in Computational Science and Engineering (LNCSE) Volume: -
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete Lotka-Volterra system (2017)
  • Author(s): Tobita Akihiko, Akiko Fukuda, Emiko Ishiwata, Masashi Iwasaki, Yoshimasa Nakamura
  • Journal Title: Lecture Notes in Computational Science and Engineering (LNCSE) Volume: -
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] 箱と玉の両方に番号が付いた箱玉系について (2017)
  • Author(s): 福田 亜希子，山本 有作，岩崎 雅史，石渡 恵美子，中村 佳正
  • Journal Title: 九州大学応用力学研究所研究集会「非線形波動研究の深化と展開」研究集会報告 Volume: -
  • Peer Reviewed / Open Access
• [Presentation] 超離散可積分系とMin-Plus 行列の固有値 (2019)
  • Author(s): 福田 亜希子
  • Organizer: 津田塾大学数学・計算機科学研究所研究集会 離散力学系と組合せ論
• [Presentation] Ultradiscrete integrable systems and eigenvalues over min-plus algebra (2019)
  • Author(s): Akiko Fukuda
  • Organizer: The 3rd UOG-SIT Workshop in Pure/Applied Mathematics and Computer Science
  • Int'l Joint Research
• [Presentation] Enumeration of fuzzy cellular automata with prescribed fixed point using Groebner basis (2018)
  • Author(s): Akiko Fukuda, Sennosuke Watanabe
  • Organizer: Symmetries and Integrability of Difference Equations (SIDE13)
  • Int'l Joint Research
• [Presentation] Ultradiscrete Lotka-Volterra system computes tropical eigenvalue of symmetric tridiagonal matrices (2018)
  • Author(s): Akiko Fukuda, Sennosuke Watanabe, Ayumi Hanaoka, Masashi Iwasaki
  • Organizer: International Conference on Mathematics: Pure, Applied and Computation, Mathematics for Better Future Life
  • Int'l Joint Research
• [Presentation] Ultra-discrete analogue of the qd algorithm for min-plus tridiagonal matrix (2018)
  • Author(s): Akiko Fukuda, Sennosuke Watanabe, Hitomi Shigitani, Masashi Iwasaki
  • Organizer: SIAM Conference on Applied Linear Algebra (SIAM-ALA18)
  • Int'l Joint Research
• [Presentation] グラフの最大平均閉路重みを不変量とするmax-plus ウォーク (2018)
  • Author(s): 渡邉 扇之介，福田 亜希子，瀬川 悦生，佐藤 宏平，佐藤 巌
  • Organizer: 2018年度応用数学合同研究集会
• [Presentation] グレブナー基底を用いた黄金比に収束するファジーセルオートマトンの列挙 (2018)
  • Author(s): 福田 亜希子, 渡邉 扇之介
  • Organizer: 日本応用数理学会 2018年度 年会
• [Presentation] Discrete integrable systems related to eigenvalue algorithms (2017)
  • Author(s): Akiko Fukuda
  • Organizer: The 1st UOG-SIT Research Workshop in Mathematics and Computer Science in Guam
  • Place of Presentation: University of Guam
  • Year and Date: 2017-02-24
  • Int'l Joint Research
• [Presentation] Integrable eigenvalue algorithms for totally nonnegative matrices (2017)
  • Author(s): Akiko Fukuda
  • Organizer: International Conference on Matrix Analysis and its Applications
  • Int'l Joint Research
• [Presentation] 超離散戸田方程式に基づくMin-Plus行列の固有値計算 (2017)
  • Author(s): 渡邉 扇之介, 福田 亜希子, 鴫谷 瞳, 岩崎 雅史
  • Organizer: 日本応用数理学会 2017年度 年会
• [Presentation] 超離散戸田方程式に付随するMin-Plus 行列の固有多項式の根とグラフの閉路 (2017)
  • Author(s): 渡邉 扇之介, 福田 亜希子
  • Organizer: 2017年度応用数学合同研究集会
• [Presentation] トロピカル3重対角行列の固有値を計算する超離散ロトカ・ボルテラ系 (2017)
  • Author(s): 福田 亜希子, 渡邉 扇之介, 花岡 歩, 岩崎 雅史
  • Organizer: 日本応用数理学会2018年研究部会連合発表会
• [Presentation] 箱と玉の両方に番号が付いた箱玉系について (2016)
  • Author(s): 福田 亜希子, 山本 有作, 岩崎 雅史, 石渡 恵美子, 中村 佳正
  • Organizer: 平成28年度 九州大学応用力学研究所 共同利用研究集会「非線形波動研究の深化と展開」
  • Place of Presentation: 九州大学
  • Year and Date: 2016-11-03
• [Presentation] Extensions of the qd algorithm for totally nonnegative matrices (2016)
  • Author(s): Akiko Fukuda, Hiroshi Takeuchi, Kensuke Aihara and Emiko Ishiwata
  • Organizer: 20th Conference of the International Linear Algebra Society (ILAS2016)
  • Place of Presentation: KU Leuven, Belgium
  • Year and Date: 2016-07-11
  • Int'l Joint Research
• [Presentation] ある非対称帯行列の固有値を計算するqd型dhLVアルゴリズムの拡張 (2016)
  • Author(s): 福田 亜希子, 竹内 弘史, 石渡 恵美子
  • Organizer: 研究会「応用解析研究会 ～可積分系から計算数学まで～」
  • Place of Presentation: 旧桜宮公会堂
  • Year and Date: 2016-05-19
• [Remarks] Akiko Fukuda's HP
  • URL: http://www.sic.shibaura-it.ac.jp/~afukuda/