Challenge to Breakdown of Integrable Algorithms

• Project/Area Number: 15K13457
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyoto University
• Principal Investigator: Nakamura Yoshimasa 京都大学, 情報学研究科, 教授 (50172458)
• Co-Investigator(Kenkyū-buntansha): 關戸 啓人 京都大学, 国際高等教育院, 特定講師 (40718235)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 可積分アルゴリズム / ブレイクダウン / ランチョス法 / 連立１次方程式 / 特異点閉じ込め / 離散戸田方程式 / ブレークダウン / 連立１次方程式の反復法 / 応用可積分系 / 直交多項式 / 相対精度 / 固有値計算 / 特異値計算 / 原点シフト / 高次収束

Outline of Final Research Achievements

A multistep extension of the progressive algorithm by Lanczos named the MPA is introduced and its application to linear systems is discussed. It is well-known that the Lanczos algorithm may easily breakdown caused by a division by zero of the Lanczos parameters. In this research project the Lanczos parameters are computed very accurately by using the four basic operations of arithmetic through the discrete Toda equation. Then even an ill-conditioned linear system having the Hilbert matrix as coefficient matrix is shown to be solved by using the MPA without any breakdown.

Research Products

• [Journal Article] An application of the discrete-time Toda lattice to the progressive algorithm by Lanczos and related problems (2018)
  • Author(s): Nakamura Yoshimasa、Sekido Hiroto
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 51 Issue: 17 Pages: 174001-174001
  • DOI: 10.1088/1751-8121/aab7fe
  • Peer Reviewed / Open Access
• [Journal Article] An extended Fibonacci sequence associated with the discrete hungry Lotka-Volterra system (2017)
  • Author(s): Masato Shinjo, Kanae Akaiwa, Masashi Iwasaki, Yoshimasa Nakamura
  • Journal Title: International Journal of Biomathematics Volume: 10 Issue: 03 Pages: 1750043-1750043
  • DOI: 10.1142/s1793524517500437
  • 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): Akihiko Tobita, Akiko Fukuda, Emiko Ishiwata, Masashi Iwasaki and Yoshimasa Nakamura
  • Journal Title: Proceedings of International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2015) Volume: -
  • Peer Reviewed
• [Journal Article] A parallel solver based on bisection and inverse iteration for computing a subset of eigenpair of symmetric band matrices (2017)
  • Author(s): Hiroyuki Ishigami, Hidehiko Hasegawa, Kinji Kimura, and Yoshimasa Nakamura
  • Journal Title: Proceedings of International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2015) Volume: -
  • Peer Reviewed
• [Journal Article] An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues (2016)
  • Author(s): Kanae Akaiwa, Yoshimasa Nakamura, Masashi Iwasaki, Akira Yoshida, Koichi Kondo
  • Journal Title: Numer. Algor. Volume: － Issue: 4 Pages: 1079-1101
  • DOI: 10.1007/s11075-016-0231-7
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Performance evaluation of Golub-Kahan-Lanczos algorithm with reorthogonalization by classical Gram-Schmidt algorithm and OpenMP (2016)
  • Author(s): Masami Takata, Hiroyuki Ishigami, Kinji Kimura, Yuki Fujii, Hiroki Tanaka, Yoshimasa Nakamura
  • Journal Title: 情報処理学会研究報告 数理モデル化と問題解決(MPS) Volume: 2015-MPS-104 Pages: 243-249
  • Peer Reviewed
• [Journal Article] Eigenvalue clustering of coefficient matrices in the iterative stride reductions for linear systems (2016)
  • Author(s): Munehiro Nagata, Masatsugu Hada, Masashi Iwasaki and Yoshimasa Nakamura
  • Journal Title: Computers Math. Appl. Volume: 71 Issue: 1 Pages: 349-355
  • DOI: 10.1016/j.camwa.2015.11.022
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Accurate numerical integration algorithm for the Kepler motion (2016)
  • Author(s): Yosuke Nakanishi, Yukitaka Minesaki and Yoshimasa Nakamura
  • Journal Title: Josai Mathematical Monographs Volume: 9 Pages: 135-145
  • NAID: 120005744553
  • Peer Reviewed
• [Journal Article] An asymptotic analysis for an integrable variant of the Lotka-Volterra prey-predator model via a determinant expansion technique (2015)
  • Author(s): Masato Shinjo, Masashi Iwasaki, Akiko Fukuda, Emiko Ishiwata, Yusaku Yamamoto and Yoshimasa Nakamura
  • Journal Title: Cogent Mathematics Volume: 2 Pages: 1046538-1046538
  • Peer Reviewed
• [Journal Article] A finite-step construction of totally nonnegative matrices with specified eigenvalues (2015)
  • Author(s): Kanae Akaiwa, Yoshimasa Nakamura, Masashi Iwasaki, Hisayoshi Tsutsumi, Koichi Kondo
  • Journal Title: Numerical Algorithms Volume: 未定 Issue: 3 Pages: 469-484
  • DOI: 10.1007/s11075-015-9957-x
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] ランチョス・フィリップスアルゴリズムの可積分性とその直交多項式への応用 (2018)
  • Author(s): 澤薫、中村佳正
  • Organizer: 第14回応用数理学会研究部会連合発表会
• [Presentation] 離散戸田方程式（qd 法）によるランチョスの前進アルゴリズムの多段拡張とその応用 (2017)
  • Author(s): 中村佳正, 關戸啓人
  • Organizer: 日本応用数理学会研究部会連合発表会
  • Place of Presentation: 電気通信大学（東京・調布市）
  • Year and Date: 2017-03-06
• [Presentation] 応用可積分系分野の基礎方程式としての戸田格子方程式, (2017)
  • Author(s): 中村佳正
  • Organizer: 平成29年度応用数理学会年会
• [Presentation] 固有ベクトルを求めるためのツイスト分解の計算量削減, (2017)
  • Author(s): 新庄雅斗、岩崎雅史、中村佳正
  • Organizer: 第46回数値解析シンポジウム(NAS2017)
• [Presentation] Toward computational science for Big Data by accurate SVD code (2016)
  • Author(s): Yoshimasa Nakamura
  • Organizer: Soul National University - Kyoto University Joint Workshop in Informatics
  • Place of Presentation: Soul National University (Korea)
  • Year and Date: 2016-09-20
  • Int'l Joint Research
• [Presentation] Recent developments in integrable algorithms (2016)
  • Author(s): Yoshimasa Nakamura
  • Organizer: 日本応用数理学会年会JSIAM-ANZIAM 特別セッション
  • Place of Presentation: 北九州国際会議場(福岡・北九州市）
  • Year and Date: 2016-09-12
• [Presentation] Aspects of integrable numerical algorithms (2016)
  • Author(s): Yoshimasa Nakamura
  • Organizer: The Third China-Japan Joint Workshop on Integrable Systems 2016
  • Place of Presentation: Xian（China）
  • Year and Date: 2016-08-19
  • Int'l Joint Research
• [Presentation] dLVアルゴリズムに対する陽的な原点シフト導入について (2016)
  • Author(s): 岩崎雅史, 中村佳正
  • Organizer: 日本応用数理学会
  • Place of Presentation: 神戸学院大学（兵庫県・神戸市）
  • Year and Date: 2016-03-04
• [Presentation] Implementation of computing singular pairs for large scale matrices using ARPACK (2016)
  • Author(s): Masami Takata, Sho Araki, Kinji Kimura, Yuki Fujii, and Yoshimasa Nakamura
  • Organizer: The 2016 International Conference on Parallel and Distributed Processing Techniques and Applications(PDPTA2016)
  • Place of Presentation: Las Vegas（USA）
  • Int'l Joint Research
• [Presentation] 実対称 5 重対角行列の固有値に収束する解をもつ力学系とその離散化 (2015)
  • Author(s): 石川貴弘, 新庄雅斗, 岩崎雅史, 中村佳正
  • Organizer: 日本応用数理学会
  • Place of Presentation: 金沢大学（石川県・金沢市）
  • Year and Date: 2015-09-09
• [Book] 解析学百科II 可積分系の数理 (2018)
  • Author(s): 中村佳正、高崎金久、辻本諭、尾角正人、井ノ口順一
  • Total Pages: 448
  • Publisher: 朝倉書店
  • ISBN: 9784254117271
• [Remarks] LAPROGNC Library
  • URL: http://www-is.amp.i.kyoto-u.ac.jp/LAPROGNC/