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Development of numeric-algebraic hybrid computation system

Research Project

Project/Area Number 03558008
Research Category

Grant-in-Aid for Developmental Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field Informatics
Research InstitutionUniversity of Tsukuba

Principal Investigator

SASAKI Tateaki  Univ.Tsukuba, Inst.Math., Prof., 数学系, 教授 (80087436)

Co-Investigator(Kenkyū-buntansha) MARITSUGU Shuuichi  Univ.Lib.Inf.Sci., Assistant, 図書館情報学部, 助手 (50220075)
SUZUKI Masayuki  Iwate Univ., Fac.Eng., Assoc.Prof., 工学部, 助教授 (20143365)
KAKO Fujio  Nara Women's Univ., Fac.Sci., Prof., 理学部, 教授 (90152610)
NODA Matu-tarou  Univ.Ehime, Fac.Eng., Prof., 工学部, 教授 (10036402)
KITAMOTO Takuya  Univ.Tsukuba, Inst.Math., Assistant, 数学系, 助手 (30241780)
Project Period (FY) 1991 – 1993
Project Status Completed (Fiscal Year 1993)
Budget Amount *help
¥6,500,000 (Direct Cost: ¥6,500,000)
Fiscal Year 1993: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1992: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1991: ¥2,300,000 (Direct Cost: ¥2,300,000)
Keywordsapproximate algebra / application of approximate algebra / formula manipulation / formula manipulation system / numeric-algebraic hybrid computation / numeric-algebraic hybrid system / 数式処理システム / 近似多項式 / 数値数式融合算法 / ピュイズー級数展開 / 解析接続 / 有理関数近似 / 近似GCD / 近似Grobner基底 / 多項式因数分解 / 近似因数分解
Research Abstract

In this research, we aimed at 1) bringing Japanese computer algebra system GAL up to practically usable level, 2) providing a numeric-algebraic hybrid computation facility to GAL, 3) studyng approximate algebraic algorithms, and 4) finding various applications of approximate algebra. As 1), GAL is now usable as a resesarch tool but it still lacks many facilities for general use ; as for 2), we reformed NSL (Nara Standard-Lisp), the host Lisp system for GAL, so that it can link Lisp and C, enabling us to perform numeric-algebraic hybrid computation now ; was for 3), we developed several important algorithms such as "an approximate Puiseux series expansion algorithm of algebraic functions" and "a hybrid algorithm for determining Riemann surface and performing analytic continuation of algebraic functions" ; as for 4), several interesting applications were found such as "application of approximate GCD to rational function approximation" and "sumoothing of data using the rational function approximation".
Summarizing our research, we have the following conclusions.
1. Approximate algebra can fuse numeric and algebraic conputation in algorithm level, and it has a very good possibility of innovating scientific computation.
2. However, development of approximate algebraic algorithm is not so easy at it looks, requesting us to make many years of efforts. In particular, we must construct mathematical theory of approximate algebra urgently.
3. As for computation system of approximate algebra, we must develop a floating-point computation system with facility of monitoring cancellation of numbers.

Report

(4 results)
  • 1993 Annual Research Report   Final Research Report Summary
  • 1992 Annual Research Report
  • 1991 Annual Research Report
  • Research Products

    (50 results)

All Other

All Publications (50 results)

  • [Publications] M.Ochi,M-T.Noda and T.Sasaki: "Approximate greatest common divisor of multivariate polynomials and its application to ill-conditioned system of algebraic equations" J.Inf.Process.14. 292-300 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M-T.Noda and T.Sasaki: "Approximate GCD and its application to ill-conditioned algebraic equations" J.Comp.Appl.Math.38. 335-351 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki,M.Suzuki,M.Kolar and M.Sasaki: "Approximate factorization of multivariate polynomials and absolute irreducibility testing" Japan J.Indus.Appl.Math.8. 357-375 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 森継修一,佐々木建昭: "浮動小数係数のグレゴナ-基底" 数式処理通信. 2. 91-103 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Kolar and T.Sasaki: "Multivariate quotient by power-series division" SIGSAM Bulletin. 3. 17-20 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M-T.Noda and E.Miyahiro: "A hybrid approach for the integration of a rational function" J.Comp.Appl.Math.40. 259-268 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki and M.Suzuki: "Three new algorithms for multivariate polynomial GCD" J.Symb.Comp.13. 395-411 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki,T.Saito and T.Hilano: "Analysis of approximate factorization algorithm I" Japan J.Indus.Appl.Math.9. 351-368 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 宮広栄一,野田松太郎: "新しい有理関数近似によるハイブリッド積分の拡張について" 日本応用数理学会論文誌. 2. 193-206 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M-T.Noda,E.Miyahiro and H.Kai: "Hybrid rational function approximation and its use in the hybrid integration" Proc.IMACS Conf.on Advance in Computer Methods for Partial Differential Equation. 565-571 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Suzuki: "Improvement of RC-RRS GCD algorithm" Japan J.Indus.Appl.Math.10. 41-67 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki and M.Sasaki: "A unified method for multivariate polynomial factorizations" Japan J.Indus.Appl.Math.10. 21-39 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 甲斐博,野田松太郎: "ハイブリッド有理関数近似とデータの平滑化" 日本応用数理学会論文誌. 3. 323-336 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Kitamoto: "Approximate eigenvalues,eigenvectors and inverse of a matrix with polynomial entries" Japan J.Indus.Appl.Math.11(to appear). (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki: "A theory of approximate polynomials,I-representation and arithmetic" Japan J.Indus.Appl.Math.11(to appear). (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 佐々木建昭,他3名: "岩波講座応用数学「計算代数と計算幾何」" 岩波書店, 166 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Ochi, M-T.Noda and T.Sasaki: "Approximate greatest common divisor of multivariate polynomials and its application to ill-conditioned system of algebraic equations" J.Inf.Proces.Vol.14. 292-300 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M-T.Noda and T.Sasaki: "Approximate GCD and its application to ill-conditioned algebraic equations" J.Comp.Appl.Math.Vol.38. 335-351 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki, M.Suzuki, M.Kolar and M.Sasaki: "Approximate factorization of multivariate polynomials and absolute irreducibility testing" Japan J.Indus.Appl.Math.Vol.8. 357-375 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] S.Moritsugu and T.Sasaki: "Grobner basis with floating-point number coefficients (in Japanese)" Suushiki-shori Tsusin. Vol.2. 91-103 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Kolar and M.Sasaki: "Multivariate quotient by power-series division" SIGSAM Bulletin. Vol.26, No.3. 17-20 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M-T.Noda and E.Miyahiro: "A hybrid approach for the integration of a rational function" J.Comp.Appl.Math.Vol.40. 259-268 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki and M.Suzuki: "Three new algorithms for multivariate polynomial GCD" J.Symb.Comp.Vol.13. 395-411 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki, T.Saito and T.Hilano: "Analysis of approximate factorization algorithm, I" Japan J.Indus.Appl.Math.Vol.9. 351-368 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] E.Miyahiro and M-T.Noda: "An extension of hybrid integration using new rational function approximation (in Japanese)" Trans.JSIAM. Vol.2. 193-206 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M-T.Noda, E.Miyahiro and H.Kai: "Hybrid rational function approximation and its use in the hybrid integration" Proc.IMACS Conf.on Adv.in Comp.Methods for Partial Differential Equation, eds.R.Vichnevetsky et al.565-571 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Suzuki: "Improvements of power-series coefficient polynomial remainder sequence GCD algorithm" Japan J.Indus.Appl.Math.Vol.10. 41-67 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki and M.Sasaki: "A unified method for multivariate polynomial factorizations" Japan J.Indus.Appl.Math.Vol.10. 21-39 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Kai and M-T.Noda: "Hybrid rational function approximation and smoothing of data (in Japanese)" Trans.JSIAM. Vol.3. 323-336 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Kitamoto: "Approximate eigenvalues, eigenvectors and inverse of a matrix with polynomial entries" Japan J.Indus.Appl.Math.Vol.11 (to appear). (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] T.Sasaki: "A theory of approximate polynomials, I-representation and arithmetic-" Japan J.Indus.Appl.Math.Vol.11 (to appear). (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Masayuki Suzuki: "Improvement of PC-PRS GCD algorithm" Japan J.Indus.Appl.Math.10. 41-67 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] T.Sasaki and M.Sasaki: "A unified method for multivariate polynomial factorizations" Japan J.Indus.Appl.Math.10. 21-39 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 甲斐博,野田松太郎: "ハイブリッド有理関数近似とデータの平滑化" 日本応用数理学会論文誌. 3. 323-336 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] Takuya Kitamoto: "Approximate eigenvalues,eigenvectors and invers of a matrix with polynomial entries" Japan J.Indus.Appl.Math.11(to appear). (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] Tateaki Sasaki: "A theory of approximate polynomial,I -representation and arithmetic-" Japan J.Indus.Appl.Math.11(to appear). (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] 佐々木建昭,他3名: "岩波講座応用数学「計算代数と計算幾何」" 岩波書店, 166 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] M-T.Noda & E.Miyahiro: "A hybrid approach for the integration of a rational function" J.Comput.Appl.Math. 40. 259-268 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 宮広 英一,野田 松太郎: "新しい有理関数近似によるハイブリッド積分の活用について" 応用数理学会論文誌. 2. 193-203 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] M.Suzuki: "Improvements of power-series coefficient polynomial remainder sequence GCD algorithm" Japan J.Indus.Appl.Math.10. 41-67 (1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] T.Sasaki,T.Saito & T.Hilano: "Analysis of approximate factorization algorithm,I" Japan J.Indus.Appl.Math.9. 351-36 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] T.Sasaki & M.Sasaki: "A unified method for multivariate polynomial factorization" Japan J.Indus.Appl.Math.10. 21-39 (1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] M-T.Noda,E.Miyahiro,H.Kai: "Hybrid rational function approximation and its use in the hybrid integration" Proc.IMACS Conf.on PDE7. (1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] Tateaki Sasaki,他: "Approximate Factorization of Multivariate Polynomials and Absolute Irreducibility Testing" Japan Journal of Industrial and Applied Mathematics. 8. 357-375 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Tateaki Sasaki: "Formula Manipulation System GAL" Proceedings of Infernational Symposium on Computing in High Energy Physics,'91. 383-389 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Tateaki Sasaki,他: "Three New Algorithms for Multivariate Polynomial GCD" Journal of Symbolic Computation. 12. (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Matu-tarow Noda,他: "Approximate GCD and Its Application to Ill-canditioned Algebraic Equations" Journal of Computational and Applied Mathematics. 38. 335-351 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Matu-tarow Noda,他: "A Hybrid Approach fo the Integration of Rational Function" Journal of Computational and Applied Mathematics. 39. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] 森継 修一,他: "行列のJordan標準形の数式処理による厳密計算法" 日本応用数理学会・論文誌. 2. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] 佐々木 建昭(分担執筆): "現代応用数理科学事典(「計算機代数」の項目)" 大阪書籍, 12 (1991)

    • Related Report
      1991 Annual Research Report

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Published: 1991-04-01   Modified: 2016-04-21  

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