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Research on the influence of perturbation to algebraic problems and coping methodology for them

Research Project

Project/Area Number 18K11172
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 60010:Theory of informatics-related
Research InstitutionTokyo University of Science

Principal Investigator

Sekigawa Hirosi  東京理科大学, 理学部第一部応用数学科, 教授 (00396178)

Co-Investigator(Kenkyū-buntansha) 白柳 潔  東邦大学, 理学部, 教授 (80396176)
Project Period (FY) 2018-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords摂動 / 誤差 / 安定性 / 代数方程式 / 多項式 / 数値数式融合計算 / 安定化理論
Outline of Final Research Achievements

Some algebraic problems can be unstable if they contain errors, for example, in the coefficients of polynomials. In such cases, we carried out research on the influence of perturbations of coefficients, methods for posing stable related problems, and algorithms that efficiently solve the posed problems. Furthermore, we studied the application of the stabilization techniques that make the constructed algorithms more efficient. Our main results are as follows: (1) The continuity of the roots of simultaneous algebraic equations satisfying some conditions with respect to the coefficients. (2) Algorithms for interpolation of black-box polynomials whose values contain errors. (3) Confirmation of the effectiveness of the stabilization techniques and a log method based on the techniques.

Academic Significance and Societal Importance of the Research Achievements

計算機により数学的な計算を行う方法には数値計算と数式処理の二つがある。数値計算は計算が高速かつ、入力データに誤差があっても計算可能だが、結果の保証が必要である。数式処理は結果は信頼できるが、計算が低速であり、誤差のある入力は受け付けない。
本研究の成果は、信頼性の高い数式処理を基礎に、数値計算、数式処理双方の長所を合わせ持つ計算方法を確立する一歩となるものである。こういった計算方法は様々な分野における数学的な計算に利用できる可能性を秘めている。

Report

(4 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (34 results)

All 2021 2020 2019 2018

All Journal Article (15 results) (of which Open Access: 6 results,  Peer Reviewed: 7 results) Presentation (19 results) (of which Int'l Joint Research: 4 results)

  • [Journal Article] Robustな疎多項式補間アルゴリズム2021

    • Author(s)
      近藤和希、関川浩
    • Journal Title

      数式処理

      Volume: 27 Pages: 35-38

    • Related Report
      2020 Annual Research Report
  • [Journal Article] 疎多項式補間に関するrobustな計算手法及びアルゴリズム2020

    • Author(s)
      近藤和希、関川浩
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2159 Pages: 143-148

    • NAID

      40022335059

    • Related Report
      2020 Annual Research Report
    • Open Access
  • [Journal Article] A型量子群の既約最高ウェイト加群の結晶基底とHiveモデル2020

    • Author(s)
      成澤翔大、白柳潔
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2159 Pages: 149-158

    • NAID

      120006956636

    • Related Report
      2020 Annual Research Report
    • Open Access
  • [Journal Article] A complete classification of three-dimensional algebras over R and C ― 温故知新 (visiting old, learn new)2020

    • Author(s)
      Kobayashi Yuji、Shirayanagi Kiyoshi、Tsukada Makoto、Takahasi Sin-Ei
    • Journal Title

      Asian-European Journal of Mathematics

      Volume: 14 Issue: 08 Pages: 2150131-2150131

    • DOI

      10.1142/s179355712150131x

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On parametric border bases2020

    • Author(s)
      Yosuke Sato, Hiroshi Sekigawa, Ryoya Fukasaku, Katsusuke Nabeshima
    • Journal Title

      Lecture Notes in Computer Science

      Volume: 11989 Pages: 10-15

    • DOI

      10.1007/978-3-030-43120-4_2

    • ISBN
      9783030431198, 9783030431204
    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Robust computation methods for sparse interpolation of multivariate polynomials2019

    • Author(s)
      Kondo Kazuki、Sekigawa Hiroshi
    • Journal Title

      ACM Communications in Computer Algebra

      Volume: 53 Issue: 3 Pages: 130-133

    • DOI

      10.1145/3377006.3377018

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Three-dimensional zeropotent algebras over an algebraically closed field of characteristic two2019

    • Author(s)
      Shirayanagi Kiyoshi、Kobayashi Yuji、Takahasi Sin-Ei、Tsukada Makoto
    • Journal Title

      Communications in Algebra

      Volume: 48 Issue: 4 Pages: 1613-1625

    • DOI

      10.1080/00927872.2019.1691579

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] 中国剰余定理を用いたsparse interpolation2019

    • Author(s)
      近藤和希、関川浩
    • Journal Title

      数式処理

      Volume: 25 Pages: 22-25

    • Related Report
      2019 Research-status Report
  • [Journal Article] Robustな疎多項式補間2019

    • Author(s)
      近藤和希、関川浩
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2138 Pages: 73-80

    • NAID

      120006888174

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] 指定された領域に零点を持つ最近接多項式について2019

    • Author(s)
      若月雄麻、関川浩
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2138 Pages: 81-86

    • NAID

      120006888175

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] 複数の点での値が指定された場合の最近接多項式2019

    • Author(s)
      若月雄麻、関川浩
    • Journal Title

      数式処理

      Volume: 25 Pages: 67-70

    • Related Report
      2018 Research-status Report
  • [Journal Article] パラメトリックな連立代数方程式の根の連続性について II2019

    • Author(s)
      佐藤洋祐、深作亮也、関川浩
    • Journal Title

      数式処理

      Volume: 25 Pages: 87-89

    • Related Report
      2018 Research-status Report
  • [Journal Article] On Continuity of the Roots of a Parametric Zero Dimensional Multivariate Polynomial Ideal2018

    • Author(s)
      Sato Yosuke、Fukasaku Ryoya、Sekigawa Hiroshi
    • Journal Title

      Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation

      Volume: 2018 Pages: 359-365

    • DOI

      10.1145/3208976.3209004

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Effect of the interval-symbol method with correct zero rewriting on the δ-LLL algorithm2018

    • Author(s)
      Nagashima Hiroki、Shirayanagi Kiyoshi
    • Journal Title

      ACM Communications in Computer Algebra

      Volume: 52 Issue: 2 Pages: 24-31

    • DOI

      10.1145/3282678.3282679

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Classification of three-dimensional zeropotent algebras over the real number field2018

    • Author(s)
      Shirayanagi Kiyoshi、Takahasi Sin-Ei、Tsukada Makoto、Kobayashi Yuji
    • Journal Title

      Communications in Algebra

      Volume: 46 Issue: 11 Pages: 4663-4681

    • DOI

      10.1080/00927872.2018.1448852

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] 3次元凸包構成の安定化について2020

    • Author(s)
      奥田和樹、白柳潔
    • Organizer
      第29回日本数式処理学会大会
    • Related Report
      2020 Annual Research Report
  • [Presentation] メビウス変換の凸結合2020

    • Author(s)
      若菜魁、関川浩
    • Organizer
      第29回日本数式処理学会大会
    • Related Report
      2020 Annual Research Report
  • [Presentation] ISCZ法の3次元凸包構成への適用2020

    • Author(s)
      奥田和樹、白柳潔
    • Organizer
      RIMS共同研究(公開型)Computer Algebra―Theory and its Applications
    • Related Report
      2020 Annual Research Report
  • [Presentation] A型量子群のExtremal weight moduleの結晶基底とHiveモデル2020

    • Author(s)
      成澤翔大、白柳潔
    • Organizer
      RIMS共同研究(公開型)Computer Algebra―Theory and its Applications
    • Related Report
      2020 Annual Research Report
  • [Presentation] Robust computation methods for sparse interpolation of multivariate polynomials2019

    • Author(s)
      Kazuki Kondo, Hiroshi Sekigawa
    • Organizer
      44th International Symposium on Symbolic and Algebraic Computation (ISSAC2019)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] On Parametric Border Bases2019

    • Author(s)
      Yosuke Sato, Hiroshi Sekigawa, Ryoya Fukasaku, Katsusuke Nabeshima
    • Organizer
      Mathematical Aspects of Computer and Information Sciences 2019 (MACIS2019)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] パラメトリックな代数方程式の根の連続性とBorder Basisとの関係2019

    • Author(s)
      佐藤洋祐、関川浩、深作亮也、鍋島克輔
    • Organizer
      第28回日本数式処理学会大会
    • Related Report
      2019 Research-status Report
  • [Presentation] Robustな疎多項式補間アルゴリズム2019

    • Author(s)
      近藤和希、関川浩
    • Organizer
      第28回日本数式処理学会大会
    • Related Report
      2019 Research-status Report
  • [Presentation] パラメトリックな連立代数方程式の根の連続性とBorderBasisとの関係II2019

    • Author(s)
      佐藤洋祐、関川浩、深作亮也、鍋島克輔
    • Organizer
      日本数式処理学会2019年度基礎理論分科会&システム分科会合同研究会
    • Related Report
      2019 Research-status Report
  • [Presentation] 疎多項式補間に関するrobustな計算手法及びアルゴリズム2019

    • Author(s)
      近藤和希、関川浩
    • Organizer
      RIMS共同研究(公開型)Computer Algebra―Theory and its Applications
    • Related Report
      2019 Research-status Report
  • [Presentation] A型量子群の最高ウェイト既約加群の結晶基底とHiveモデル2019

    • Author(s)
      成澤翔大、白柳潔
    • Organizer
      RIMS共同研究(公開型)Computer Algebra―Theory and its Applications
    • Related Report
      2019 Research-status Report
  • [Presentation] 計算誤差を抑える疎多項式補間アルゴリズム2019

    • Author(s)
      近藤和希、関川浩
    • Organizer
      Risa/Asir Conference 2019
    • Related Report
      2018 Research-status Report
  • [Presentation] 複数の点での値が指定された場合の最近接多項式2018

    • Author(s)
      若月雄麻、関川浩
    • Organizer
      第27回日本数式処理学会大会
    • Related Report
      2018 Research-status Report
  • [Presentation] パラメトリックな連立代数方程式の根の連続性について II2018

    • Author(s)
      佐藤洋祐、深作亮也、関川浩
    • Organizer
      第27回日本数式処理学会大会
    • Related Report
      2018 Research-status Report
  • [Presentation] A canonical representation of continuity of the roots of a parametric zero dimensional multi-variate polynomial ideal2018

    • Author(s)
      Yosuke Sato, Ryoya Fukasaku and Hiroshi Sekigawa
    • Organizer
      24th Conference on Applications of Computer Algebra (ACA2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] On Continuity of the Roots of a Parametric Zero Dimensional Multivariate Polynomial Ideal2018

    • Author(s)
      Yosuke Sato, Ryoya Fukasaku and Hiroshi Sekigawa
    • Organizer
      43rd International Symposium on Symbolic and Algebraic Computation (ISSAC2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] 中国剰余定理を用いた sparse interpolation2018

    • Author(s)
      近藤和希、関川浩
    • Organizer
      日本数式処理学会2018年度基礎理論分科会&システム分科会合同研究会
    • Related Report
      2018 Research-status Report
  • [Presentation] Robustな疎多項式補間2018

    • Author(s)
      近藤和希、関川浩
    • Organizer
      RIMS共同研究(公開型)Computer Algebra―Theory and its Applications
    • Related Report
      2018 Research-status Report
  • [Presentation] 指定された領域に零点を持つ最近接多項式について2018

    • Author(s)
      若月雄麻、関川浩
    • Organizer
      RIMS共同研究(公開型)Computer Algebra―Theory and its Applications
    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2022-01-27  

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