Approximate Greatest Common Divisor of Multivariate Polynomials and Its Application to III-Conditioned Systems of Algebraic Equations

• Item type: JInfP(1)
• 公開日: 1991-12-31
• タイトル: Approximate Greatest Common Divisor of Multivariate Polynomials and Its Application to III-Conditioned Systems of Algebraic Equations
• タイトル / 言語: en
• タイトル: Approximate Greatest Common Divisor of Multivariate Polynomials and Its Application to III-Conditioned Systems of Algebraic Equations
• 言語: eng
• キーワード / 主題Scheme: Other
• キーワード / 主題: （IPSJ Best Paper Award、論文賞受賞）
• 資源タイプ / 資源タイプ識別子: http://purl.org/coar/resource_type/c_6501
• 資源タイプ: journal article
• 著者所属: Computer Division Information Equipment Sector Matsushita Electric Industrial Co. LTD.
• 著者所属: Department of Computer Science Ehime University
• 著者所属: The Institute of Physical and Chemical Research/Institute of Mathematics University of Tsukuba
• 著者所属(英): en
• 著者所属(英): Computer Division, Information Equipment Sector, Matsushita Electric Industrial Co., LTD.
• 著者所属(英): en
• 著者所属(英): Department of Computer Science, Ehime University
• 著者所属(英): en
• 著者所属(英): The Institute of Physical and Chemical Research/Institute of Mathematics, University of Tsukuba
• 著者名: Masa-AkiOchi; Matu-TarowNoda; Tateaki, Sasaki
• 著者名(英): Masa-Aki, Ochi; Matu-Tarow, Noda; Tateaki, Sasaki
• 論文抄録 / 内容記述タイプ: Other
• 論文抄録 / 内容記述: Let F F and D be multivariate polynomials andεbe a small positive number 0 < ε < < 1. If F=DF+△F where △F is a polynomial with coefficients that are O(ε)-smaller than those of F D is called an approximate divisor of F of accuracy c. Given multivariate polynomials F and G an algorithm is proposed for calculating with accuracyεthe approximate greatest common divisor (GCD) of F and G. The algorithm is a naive extension of the conventional Euclidean algorithm but it is necessary to treat the polynomials carefully. As an application of the approximate GCD of multivariate polynomials the solution of a system of algebraic equations {F_1(x y . . . z)=0 . . . F_r(x y . . . z)=0} is considered where F_i and F_j i≠j have a non-trivial approximately common divisor. Such a system is ill-conditioned for conventional numerical methods and is transformed to a well-con-ditioned system by calculating approximate GCD's. A method is also given for determining the initial approximations of the roots for numerical iterative calculation. The proposed method is tested by using several examples and the results are very good.
• 論文抄録(英) / 内容記述タイプ: Other
• 論文抄録(英) / 内容記述: Let F, F and D be multivariate polynomials andεbe a small positive number,0 < ε < < 1. If F=DF+△F, where △F is a polynomial with coefficients that are O(ε)-smaller than those of F, D is called an approximate divisor of F of accuracy c. Given multivariate polynomials F and G, an algorithm is proposed for calculating with accuracyεthe approximate greatest common divisor (GCD) of F and G. The algorithm is a naive extension of the conventional Euclidean algorithm, but it is necessary to treat the polynomials carefully. As an application of the approximate GCD of multivariate polynomials, the solution of a system of algebraic equations {F_1(x, y, . . . , z)=0,. . . , F_r(x, y, . . ., z)=0} is considered, where F_i and F_j, i≠j, have a non-trivial approximately common divisor. Such a system is ill-conditioned for conventional numerical methods, and is transformed to a well-con-ditioned system by calculating approximate GCD's. A method is also given for determining the initial approximations of the roots for numerical iterative calculation. The proposed method is tested by using several examples, and the results are very good.
• 書誌レコードID / 収録物識別子タイプ: NCID
• 書誌レコードID / 収録物識別子: AA00700121
• 書誌情報: Journal of Information Processing; 巻 14, 号 3, p. 292-300, 発行日 1991-12-31
• ISSN / 収録物識別子タイプ: ISSN
• ISSN / 収録物識別子: 1882-6652
• 出版者 / 言語: ja
• 出版者: 情報処理学会