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2006 Fiscal Year Final Research Report Summary

Study of Algorithms and Applications of Approximate Algebra

Research Project

Project/Area Number 15300002
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Fundamental theory of informatics
Research InstitutionUniversity of Tsukuba

Principal Investigator

SASAKI Tateaki  University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院数理物質科学研究科, 教授 (80087436)

Co-Investigator(Kenkyū-buntansha) TERUI Akira  University of Tsukuba, Graduate School of Pure and Applied Sciences, Instructor, 大学院数理物質科学研究科, 助手 (80323260)
KAI Hiroshi  Ehime University, Faculty of Engineering, Assistant Professor, 工学部, 講師 (10274341)
NODA Matu-tarou  Ehime University, Professor Emeritus, 名誉教授 (10036402)
KAKO Fujio  Nara Women's University, Faculty of Science, Professor, 理学部, 教授 (90152610)
FUKUI Tetsuo  Mukogawa Women's University, School of Human Environmental Sciences, Professor, 生活環境学部, 教授 (70218890)
Project Period (FY) 2003 – 2006
Keywordsapproximate algebra / algebraic computation / algebraic-numeric computation / approximate GCD / approximate factorization / extended Hensel construction / approximate Grobner base / eight-line arrangement
Research Abstract

The purposes of this research are, 1) to establish and analyze new concepts, 2) to develop new algorithms, 3) to stabilize existing algorithms, and 4) applications to science and engineering.
Results on 1) Concepts of approximately singular and approximate non-conjugateness are introduced (Sasaki). The denominator factors appearing in extended Hensel factors are clarified, and the properties of convergence and many-valuedness of extended Hensel series are investigated numerically (Sasaki & student).
Results on 2) A stable method for computing Grobner bases with floating-point numbers is proposed (Sasaki-Kako). A new method is found for multivariate polynomial factorization (Sasaki & student). A semi-algebraic method is proposed to separate close-root clusters of univariate polynomial (Sasaki-Kako). A very tight error-bound formula is derived for numerical roots of univariate polynomial (Sasaki). A theory of recursive subresultants is developed for separating the real roots of univariate … More polynomial (Terui). A simultaneous iterative formula of arbitrary degree of convergence is derived for symbolic Newton's method (Terui). A method of drawing two-dimensional pseudovariety is developed (Kai et al.). A method of constructing nearest polynomials of degrees up to 4 is developed (Noda-Kai et al.). A method for approximate indefinite integral of rational functions with parameters is proposed (Kai-Noda & student).
Results on 3) The reason of occurrence of large errors in the computation of floating-point Grobner base is clarified (Sasaki-Kako). As for ill-conditioned cases in computing approximate GCDs of multivariate as well as univariate polynomials, several techniques to stabilize PRS-type algorithms are proposed (Sasaki & student). By analyzing the univariate rational-function approximation with floating-point numbers, clarified is the reason of appearance of unnecessary poles and a stabilization method based on the Pade approximation is proposed (Kai-Noda & students).
Results on 4) As for the 8-line arrangement problem, a complete classification of the arrangements is attained, after many experiments of generating 8-line arrangements (Fukui et al.). Less

  • Research Products

    (41 results)

All 2007 2006 2005 2004 2003

All Journal Article (41 results)

  • [Journal Article] Tighter bounds of errors of numerical roots2007

    • Author(s)
      T.Sasaki
    • Journal Title

      Japan J. Indus. Appl. Math. Vol. 24(to appaer)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Extended Hensel construction and multivariate algebraic functions2007

    • Author(s)
      T.Sasaki, D.Inaba
    • Journal Title

      Commni. Comp. Algebra (to appaer)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Ill-conditioned properties and hybrid computations2007

    • Author(s)
      M-T.Noda
    • Journal Title

      Symbolic Numeric Computation (Birkhauser)

      Pages: 17-46

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Rational interpolation and its ill-conditioned property2007

    • Author(s)
      H.Kai
    • Journal Title

      Symbolic Numeric Computation (Birkhauser)

      Pages: 47-53

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An algebraic method for separating close-root clusters and the minimum root separation2007

    • Author(s)
      T.Sasaki, F.Kako
    • Journal Title

      Symbolic Numeric Computation (Birkhauser)

      Pages: 149-166

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Tighter bounds of errors of numerical roots2007

    • Author(s)
      T.Sasaki
    • Journal Title

      Japan J.Indus.Appl.Math. Vol.24 (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Extended Hensel construction and multivariate algebraic functions2007

    • Author(s)
      T.Sasaki, D.Inaba
    • Journal Title

      Communi.Comp.Algebra (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Ill-conditioned properties and hybrid computation2007

    • Author(s)
      M-T.Noda
    • Journal Title

      Symbolic Numeric Computation Birkhauser

      Pages: 17-45

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Rational interpolation and its ill-conditioned property2007

    • Author(s)
      H.Kai
    • Journal Title

      Symbolic Numeric Computation Birkhauser

      Pages: 47-53

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An algebraic method for separating close-root clusters and the minimum root separation2007

    • Author(s)
      T.Sasaki, F.Kako
    • Journal Title

      Symbolic Numeric Computation Birkhauser

      Pages: 149-166

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Computing determinants of rational functions2006

    • Author(s)
      Y.Umeda, T.Sasaki
    • Journal Title

      Commin. Comp. Algebra 40

      Pages: 2-8

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Computing determinants of rational functions2006

    • Author(s)
      Y.Umeda, T.Sasaki
    • Journal Title

      Communi.Comp.Algebra Vol.40

      Pages: 2-8

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Approximate GCD and its applications2005

    • Author(s)
      M-T.Noda, H.Kai
    • Journal Title

      Proc. Asian Symp. Comp. Math. (Seoul)

      Pages: 215-218

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A remarkable simple eight-line arrangement on a real projective plane2005

    • Author(s)
      T.Fukui, J.Sekiguchi
    • Journal Title

      J. Comp. Numer. Anal. Appl. Vol. 5

      Pages: 361-386

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Recursive polynomial remainder sequence and the nested subresultants2005

    • Author(s)
      A.Terui
    • Journal Title

      Lec. Notes Comp. Sci. Vol. 3718

      Pages: 445-456

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] ハイブリッド有理関数近似と悪条件問題2005

    • Author(s)
      中島裕美, 甲斐博, 野田松太郎
    • Journal Title

      数式処理 Vol. 11

      Pages: 141-152

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Approximate GCD and its applications2005

    • Author(s)
      M-T.Noda, H.Kai
    • Journal Title

      Proc.ASCM2005, World Sci.

      Pages: 215-218

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A remarkable simple eight-line arrangement on a real projective plane2005

    • Author(s)
      T.Fukui, J.Sekiguchi
    • Journal Title

      J.Comp.Numer.Anal.Appl. Vol.5

      Pages: 361-386

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Recursive polynomial remainder sequence and the nested subresultants2005

    • Author(s)
      A.Terui
    • Journal Title

      Lec.Nores Comp.Sci. Vol.3718

      Pages: 445-456

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hybrid rational function approximation and ill-conditioned problem (in Japanese)2005

    • Author(s)
      H.Nakajima, H.Kai, M-T.Noda
    • Journal Title

      Sushiki-Shori Vol.11

      Pages: 141-152

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Certification of analytic continuation of algebraic function2004

    • Author(s)
      D.Inaba, T.Sasaki
    • Journal Title

      Proc. CASC 2004 (St. Petersburg)

      Pages: 249-260

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Approximately singular multivariate polynomials2004

    • Author(s)
      T.Sasaki
    • Journal Title

      Proc. CASC 2004 (St. Petersburg)

      Pages: 399-407

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A theorem for separating close roots of a polynomial and its derivatives2004

    • Author(s)
      T.Sasaki
    • Journal Title

      ACM SIGSAM Bulletin Vol. 38

      Pages: 85-92

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Ill-conditioned properties and hybrid rational function approximation2004

    • Author(s)
      H.Kai
    • Journal Title

      Proc. ACA2004 (Beaumont, Texas)

      Pages: 85-92

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Experimental computation of eight-line arrangements generated by all possible transversals on real projective plane for image production2004

    • Author(s)
      T.Fukui, J.Sekiguchi, K.Ohta
    • Journal Title

      Kansai Eng. Intern. Vol. 4

      Pages: 1-10

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hybrid method for computing the nearest singular polynomials2004

    • Author(s)
      L.Zhi, M.T.Noda, H.Kai, W.Wu
    • Journal Title

      Japan J. Indus. App. Math. Vol. 21

      Pages: 149-162

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Certification of analytic continuation of algebraic function2004

    • Author(s)
      D.Inaba, T.Sasaki
    • Journal Title

      Proc.of CASC2004

      Pages: 249-260

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Approximately singular multivariate polynomials2004

    • Author(s)
      T.Sasaki
    • Journal Title

      Proc.of CASC2004

      Pages: 399-407

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A theorem for separating close roots of a polynomial and its derivatives2004

    • Author(s)
      T.Sasaki
    • Journal Title

      ACM SIGSAM Bulletin Vol.38

      Pages: 85-92

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Ill-conditioned properties and hybrid rational function approximation2004

    • Author(s)
      H.Kai
    • Journal Title

      Proc.of ACA2004

      Pages: 85-92

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Experimental computation of eight-line arrangements generated by all possible transversals on real projective plane for image production2004

    • Author(s)
      T.Fukui, J.Sekiguchi, K.Ohta
    • Journal Title

      Kansei Eng.Intern. Vol.4

      Pages: 1-10

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hybrid method for computing the nearest singular polynomial2004

    • Author(s)
      L.Zhi, M-T.Noda, H.Kai, W.Wu
    • Journal Title

      Japan J.Indus.Appl.Math. Vol.21

      Pages: 149-162

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The subresultant and clusters of close roots2003

    • Author(s)
      T.Sasaki
    • Journal Title

      Proc. ISSAC2003 (ACM)

      Pages: 232-239

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A program converter for algorithm stabilization technique2003

    • Author(s)
      Y.Kondoh, M-T.Noda
    • Journal Title

      Proc. of 8th Asian Tech. Conf. Math. (Bangkok)

      Pages: 134-143

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Lupin : towards the framework of web-based problem solving environments2003

    • Author(s)
      K.Li, M.Sakai, Y.Morizane, K.Kono, M.T.Noda
    • Journal Title

      Proc. of 8th Asian Tech. Conf. Math. (Bangkok)

      Pages: 276-285

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Approximate computation of pseudovarieties2003

    • Author(s)
      R.M.Corless, H.Kai, S.M.Watt
    • Journal Title

      ACM SIGSAM Bulletin Vol. 37

      Pages: 67-71

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Subresultants in recursive polynomial remainder sequence2003

    • Author(s)
      A.Terui
    • Journal Title

      Proc. of CASC2003 (Passau, Germany)

      Pages: 363-375

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The subresultant and clusters of close roots2003

    • Author(s)
      T.Sasaki
    • Journal Title

      Proc.ISSAC2003 ACM

      Pages: 232-239

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A program converter for algorithm stabilization technique2003

    • Author(s)
      Y.Kondoh, M-T.Noda
    • Journal Title

      Proc.of 8th ATCM World Sci.

      Pages: 134-143

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Lupin : towards the framework of web-based problem solving environments2003

    • Author(s)
      K.Li, M.Sakai, Y.Morizane, K.Kono, M.T.Noda
    • Journal Title

      Proc.of 8th ATCM World Sci.

      Pages: 276-285

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Subresultants in recursive polynomial remainder sequence2003

    • Author(s)
      A.Terui
    • Journal Title

      Proc.of CASC2003

      Pages: 363-375

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2008-05-27  

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