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Study of Algorithm and Application of Approximate Groebner Basis

Research Project

Project/Area Number 23500003
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Fundamental theory of informatics
Research InstitutionUniversity of Tsukuba

Principal Investigator

SASAKI Tateaki  筑波大学, 名誉教授 (80087436)

Co-Investigator(Kenkyū-buntansha) SAKURAI Tetsuya  筑波大学, システム情報系, 教授 (60187086)
KAKO Fujio  奈良女子大学, 自然科学系, 教授 (90152610)
Research Collaborator INABA Daiju  日本数学検定協会
Project Period (FY) 2011 – 2013
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2012: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2011: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywordsアルゴリズム理論 / 数式処理 / 数値数式融合算法 / 近似グレブナー基底 / 近似特異系と特異化 / 悪条件連立代数方程式 / パラメータ係数線形方程式系 / 疎な線形方程式系 / 近似グレブナー基底の算法 / 近似特異系 / 近似特異系の特異化 / 悪条件多項式系の良条件化 / モデルに基づく開発 / 疎なパラメータ係数線形系 / 疎線形系のパラメータ解析 / 近似線形従属関係 / 多変数代数関数 / 特異点での級数展開 / 近似無平方分解 / 近似因数分解 / ヘンゼル級数 / 近似イデアル / 悪条件方程式系の良条件化 / 特異点での多変数級数展開
Research Abstract

Based on a proposed "approximate ideal", we constructed a theory of approximate Groebner basis, clarified the instability of Buchberger's algorithm on floating point numbbers using a developed subresultnat-like theory, and proposed an algorithm of approximate Groebner basis by stabilizing Buchberger's algorithm.
We also proposed a concept of "approximate singular system" as a multivariate polynomial ideal whose dimension is decreased by a purturbation, and presented an algorithm which recovers the dimension. Applying this operation to algebraic systems of approximately singular type, we proposed a well-conditioning method for such systems. Furthermore, we proposed an error suppressing method and a characteristics extracting method for solving parametric sparse linear systems.

Report

(4 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Research-status Report
  • 2011 Research-status Report
  • Research Products

    (49 results)

All 2014 2013 2012 2011

All Journal Article (11 results) (of which Peer Reviewed: 11 results) Presentation (38 results)

  • [Journal Article] A Subresultant-like Theory of Buchberger's Procedure2014

    • Author(s)
      Tateaki Sasaki
    • Journal Title

      Japan J. Indust. Appl

      Pages: 137-164

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Algebraic Preprocessing of Floating point DAEs for Numerical Model Simulation2014

    • Author(s)
      Tateaki Sasaki and Tetsu Yamaguchi
    • Journal Title

      SYNASC 2013, IEEE Computer Society

      Pages: 81-88

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Algebraic Preprocessing of Floating-point DAEs for Numerical Model Simulation2014

    • Author(s)
      Tateaki Sasaki, Testu Yamaguchi
    • Journal Title

      SYNASC 2013: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

      Volume: なし Pages: 81-89

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Subresultant-like Theory of Buchberger's Procedure2014

    • Author(s)
      Tateaki Sasaki
    • Journal Title

      Japan J. Indust. Appl. Math.

      Volume: 31 Issue: 1 Pages: 137-164

    • DOI

      10.1007/s13160-013-0133-1

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Towards Industrial Application of Approximate Computer Algebra2013

    • Author(s)
      Tateaki Sasaki, Daiju Inaba and Fujio Kako
    • Journal Title

      LNCS 3186 : Computer Algebra in Scientific Computing

      Pages: 315-330

    • DOI

      10.1007/978-3-319-02297-0_26

    • ISBN
      9783319022963, 9783319022970
    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Towards Industrial Application of Approximate Computer Algebra2013

    • Author(s)
      Tateaki Sasaki, Daiju Inaba and Fujio Kako
    • Journal Title

      LNCS 8136: Computer Algebra in Scientific Computing

      Volume: なし Pages: 315-330

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Proposal of singularization of approximately singular polynomial systems2012

    • Author(s)
      Tateaki Sasaki
    • Journal Title

      SYNASC 2012

      Volume: なし Pages: 45-52

    • DOI

      10.1109/synasc.2012.28

    • Related Report
      2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Approximately singular systems and ill-conditioned polynomial systems2012

    • Author(s)
      Tateaki Sasaki, Daiju Inaba
    • Journal Title

      Lecture Notes in Computer Science

      Volume: 7442 Pages: 308-320

    • DOI

      10.1007/978-3-642-32973-9_26

    • ISBN
      9783642329722, 9783642329739
    • Related Report
      2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] A Theory and an Algorithm of Approximate Groebner Basis2012

    • Author(s)
      Tateaki Sasaki
    • Journal Title

      SYNASC 2011, IEEE Computer Society

      Volume: - Pages: 23-40

    • DOI

      10.1109/synasc.2011.12

    • Related Report
      2013 Final Research Report 2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] A Study of Hensel Series in General Case2011

    • Author(s)
      Tateaki Sasaki and Daiju Inaba
    • Journal Title

      SNC'11 : Symbolic Numeric Computation, ACM

      Pages: 34-43

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] A Study of Hensel Series in General Case2011

    • Author(s)
      Tateaki Sasaki and Daiju Inaba
    • Journal Title

      Proceedings of 2011 International Workshop on Symbolic-Numeric Computation

      Volume: - Pages: 34-43

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Presentation] パラメータ係数線形疎方程式系の局所ブロック化による解法2014

    • Author(s)
      佐々木建昭, 加古富志雄, 稲葉大樹
    • Organizer
      Risa/Asir Conference 2014
    • Place of Presentation
      神戸市
    • Related Report
      2013 Final Research Report
  • [Presentation] パラメータ係数線形疎多項式系の局所ブロック化による解法2014

    • Author(s)
      佐々木建昭、加古富志雄、稲葉大樹
    • Organizer
      Risa/Asir Conference/2014
    • Place of Presentation
      神戸市、神戸大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] パラメータ入りの連立線形方程式の誤差低減法2013

    • Author(s)
      佐々木建昭, 山口哲
    • Organizer
      RIMS 研究集会「数式処理とその周辺分野の研究」
    • Place of Presentation
      京都市
    • Related Report
      2013 Final Research Report
  • [Presentation] 産業計算における近似代数の有用性2013

    • Author(s)
      佐々木建昭, 稲葉大樹, 加古富志雄
    • Organizer
      RIMS研究集会「数式処理とその周辺分野の研究」
    • Place of Presentation
      京都市
    • Related Report
      2013 Final Research Report
  • [Presentation] On Algebraic Preprocessing of Floating point DAEs for Numerical Model Simulation2013

    • Author(s)
      Tateaki Sasaki and Tetsu Yamaguchi
    • Organizer
      15th Int'l Symposium on Symbolic and Numeric Algorithms for Scientific Computing
    • Place of Presentation
      Timisoara, Rumania
    • Related Report
      2013 Final Research Report
  • [Presentation] Towards Industrial Application of Approximate Computer Algebra2013

    • Author(s)
      Tateaki Sasaki, Daiju Inaba and Fujio Kako
    • Organizer
      15th Int'l Workshop on Computer Algebra in Scientific Computing
    • Place of Presentation
      Berlin, Germany
    • Related Report
      2013 Final Research Report
  • [Presentation] 摂動で乱れた多変数多項式系の矯正2013

    • Author(s)
      佐々木建昭
    • Organizer
      第42回数値解析シンポジウム
    • Place of Presentation
      松山市
    • Related Report
      2013 Final Research Report
  • [Presentation] 近似代数の産業応用を目指して2013

    • Author(s)
      佐々木建昭, 稲葉大樹, 加古富志雄
    • Organizer
      第42回数値解析シンポジウム
    • Place of Presentation
      松山市
    • Related Report
      2013 Final Research Report
  • [Presentation] Hensel 級数の応用を目指して-- 航空機の制御を例に--2013

    • Author(s)
      佐々木建昭, 稲葉大樹
    • Organizer
      Risa/Asir Conference 2013
    • Place of Presentation
      神戸市
    • Related Report
      2013 Final Research Report
  • [Presentation] Towards Industrial Application of Approximate Computer Algebra2013

    • Author(s)
      Tateaki Sasaki, Daiju Inaba and Fujio Kako
    • Organizer
      14th International Workshop on Computer Algebra in Scientific Computation, CASC 2013
    • Place of Presentation
      Berlin, Germany
    • Related Report
      2013 Annual Research Report
  • [Presentation] On Algebraic Preprocessing of Floating-point DAEs for Numerical Model Simulation2013

    • Author(s)
      Tateaki Sasaki and Tetsu Yamaguchi
    • Organizer
      15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
    • Place of Presentation
      Timisoara, Romania
    • Related Report
      2013 Annual Research Report
  • [Presentation] 摂動で乱れた多変数多項式系の矯正2013

    • Author(s)
      佐々木建昭
    • Organizer
      第42回数値解析シンポジウム
    • Place of Presentation
      松山市、道後温泉道後館
    • Related Report
      2013 Annual Research Report
  • [Presentation] 近似代数の産業応用を目指して2013

    • Author(s)
      佐々木建昭、稲葉大樹、加古富志雄
    • Organizer
      第42回数値解析シンポジウム
    • Place of Presentation
      松山市、道後温泉道後館
    • Related Report
      2013 Annual Research Report
  • [Presentation] パラメータ入りの連立線形方程式の誤差低減法2013

    • Author(s)
      佐々木建昭、山口 哲
    • Organizer
      RIMS研究集会;数式処理とその周辺分野の研究
    • Place of Presentation
      京都市、京都大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] 産業計算における近似代数の有用性2013

    • Author(s)
      佐々木建昭、稲葉大樹、加古富志雄
    • Organizer
      RIMS研究集会;数式処理とその周辺分野の研究
    • Place of Presentation
      京都市、京都大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] Hensel級数の応用を目指して --航空機の制御を例に--2013

    • Author(s)
      佐々木建昭, 稲葉大樹
    • Organizer
      Risa/Asir Conference 2013
    • Place of Presentation
      神戸大学滝川記念会館(神戸市)
    • Related Report
      2012 Research-status Report
  • [Presentation] 過酷事故に陥った航空機の制御を目指して-- 多変数べき級数根の利用 --2012

    • Author(s)
      佐々木建昭, 稲葉大樹
    • Organizer
      RIMS 研究集会「数式処理 -- その研究と目指すもの」
    • Place of Presentation
      京都市
    • Related Report
      2013 Final Research Report
  • [Presentation] 近似特異系の特異化-- 摂動で乱れた系の矯正 --2012

    • Author(s)
      佐々木建昭
    • Organizer
      RIMS 研究集会「数式処理 -- その研究と目指すもの」
    • Place of Presentation
      京都市
    • Related Report
      2013 Final Research Report
  • [Presentation] Proposal of Singularization of Approximately Singular Polynomial Systems2012

    • Author(s)
      Tateaki Sasaki
    • Organizer
      14th Int'l Symposium on Symbolic and Numeric Algorithms for Scientific Computing
    • Place of Presentation
      Timisoara, Rumania
    • Related Report
      2013 Final Research Report
  • [Presentation] Approximately Singular Systems and Ill-conditioned Polynomial Systems2012

    • Author(s)
      Tateaki Sasaki, Daiju Inaba and Fujio Kako
    • Organizer
      14th Int'l Workshop on Computer Algebra in Scientific Computing
    • Place of Presentation
      Maribor, Slovenia
    • Related Report
      2013 Final Research Report
  • [Presentation] 近似特異系と悪条件連立代数方程式2012

    • Author(s)
      佐々木建昭, 稲葉大樹
    • Organizer
      第41回数値解析シンポジウム
    • Place of Presentation
      渋川市
    • Related Report
      2013 Final Research Report
  • [Presentation] 近似特異系と悪条件連立代数方程式2012

    • Author(s)
      佐々木建昭, 稲葉大樹
    • Organizer
      Risa/Adir Conference 2011
    • Place of Presentation
      神戸市
    • Related Report
      2013 Final Research Report
  • [Presentation] Approximately singular systems and ill-conditioned polynomial systems2012

    • Author(s)
      Tateaki Sasaki, Daiju Inaba
    • Organizer
      14th International Workshop on Computer Algebra in Scientific Computing
    • Place of Presentation
      Maribor (Slovenia)
    • Related Report
      2012 Research-status Report
  • [Presentation] Proposal of singularization of approximately singular polynomial systems2012

    • Author(s)
      Tateaki Sasaki
    • Organizer
      14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
    • Place of Presentation
      Timisoara (Romania)
    • Related Report
      2012 Research-status Report
  • [Presentation] 近似特異系と悪条件連立代数方程式2012

    • Author(s)
      佐々木建昭, 稲葉大樹
    • Organizer
      第41回数値解析シンポジウム
    • Place of Presentation
      伊香保温泉よろこびの宿(群馬県渋川市)
    • Related Report
      2012 Research-status Report
  • [Presentation] 過酷事故に陥った航空機の制御を目指して --多変数べき級数根の利用--2012

    • Author(s)
      佐々木建昭, 稲葉大樹
    • Organizer
      RIMS研究集会「数式処理 --その研究と目指すもの--」
    • Place of Presentation
      京都大学数理解析研究所(京都市)
    • Related Report
      2012 Research-status Report
  • [Presentation] 近似特異系の特異化 --摂動で乱れた多項式系の矯正--2012

    • Author(s)
      佐々木建昭
    • Organizer
      RIMS研究集会「数式処理 --その研究と目指すもの--」
    • Place of Presentation
      京都大学数理解析研究所(京都市)
    • Related Report
      2012 Research-status Report
  • [Presentation] 近似特異系と悪条件連立代数方程式2012

    • Author(s)
      佐々木建昭、稲葉大樹
    • Organizer
      Risa/Asir Conference 2012
    • Place of Presentation
      神戸大学理学部(神戸市)
    • Related Report
      2011 Research-status Report
  • [Presentation] 近似グレブナー基底の二つの応用2011

    • Author(s)
      佐々木建昭
    • Organizer
      RIMS 研究集会「数式処理 -- その研究と目指すもの」
    • Place of Presentation
      京都市
    • Related Report
      2013 Final Research Report
  • [Presentation] A Theory and an Algorithm of Approximate Groebner Basis2011

    • Author(s)
      Tateaki Sasaki
    • Organizer
      13th Int'l Symposium on Symbolic and Numeric Algorithms for Scientific Computing
    • Place of Presentation
      Timisoara, Rumania
    • Related Report
      2013 Final Research Report
  • [Presentation] 近似グレブナー基底の理論と算法2011

    • Author(s)
      佐々木建昭
    • Organizer
      第40回数値解析シンポジウム
    • Place of Presentation
      鳥羽市
    • Related Report
      2013 Final Research Report
  • [Presentation] A Study of Hensel Series in General Case2011

    • Author(s)
      Tateaki Sasaki and Daiju Inaba
    • Organizer
      2011 Int'l workshop on Symbolic Numeric Computation
    • Place of Presentation
      San Jose, USA
    • Related Report
      2013 Final Research Report
  • [Presentation] 近似グレブナー基底の理論と算法2011

    • Author(s)
      佐々木建昭
    • Organizer
      第40回数値解析シンポジウム
    • Place of Presentation
      鳥羽シーサイドホテル(鳥羽市)
    • Related Report
      2011 Research-status Report
  • [Presentation] 多変数代数関数の級数展開の収束領域公式2011

    • Author(s)
      佐々木建昭
    • Organizer
      第40回数値解析シンポジウム
    • Place of Presentation
      鳥羽シーサイドホテル(鳥羽市)
    • Related Report
      2011 Research-status Report
  • [Presentation] A Study of Hensel Series in General Case2011

    • Author(s)
      Tateaki Sasaki and Daiju Inaba
    • Organizer
      2011 International Workshop on Symbolic-Numeric Computation
    • Place of Presentation
      San Jose Convention Center (USA)
    • Related Report
      2011 Research-status Report
  • [Presentation] A Theory and An Algorithm of Approximate Groebner Bases2011

    • Author(s)
      Tateaki Sasaki
    • Organizer
      13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
    • Place of Presentation
      West University of Timisoara (Romania)
    • Related Report
      2011 Research-status Report
  • [Presentation] Approximate Groebner Bases and Two Applications2011

    • Author(s)
      Tateaki Sasaki
    • Organizer
      Fields Institute Workshop on Hybrid Methodologies for Symbolic-Numeric Computation(招待講演)
    • Place of Presentation
      The University of Waterloo (Canada)
    • Related Report
      2011 Research-status Report
  • [Presentation] 近似グレブナー基底の二つの応用2011

    • Author(s)
      佐々木建昭
    • Organizer
      数理解析研究所研究集会ー数式処理、その研究と目指すもの
    • Place of Presentation
      京都大学数理解析研究所(京都市)
    • Related Report
      2011 Research-status Report

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Published: 2011-08-05   Modified: 2023-03-16  

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