2022-01-29T05:08:22Zhttps://ipsj.ixsq.nii.ac.jp/ej/?action=repository_oaipmh
oai:ipsj.ixsq.nii.ac.jp:000596902017-03-31T05:36:57Z05471:05480:05482
Approximate Greatest Common Divisor of Multivariate Polynomials and Its Application to III-Conditioned Systems of Algebraic EquationsApproximate Greatest Common Divisor of Multivariate Polynomials and Its Application to III-Conditioned Systems of Algebraic Equationseng（IPSJ Best Paper Award、論文賞受賞）http://id.nii.ac.jp/1001/00059690/Articlehttps://ipsj.ixsq.nii.ac.jp/ej/?action=repository_action_common_download&item_id=59690&item_no=1&attribute_id=1&file_no=1Copyright (c) 1991 by the Information Processing Society of JapanComputer Division Information Equipment Sector Matsushita Electric Industrial Co. LTD.Department of Computer Science Ehime UniversityThe Institute of Physical and Chemical Research/Institute of Mathematics University of TsukubaMasa-AkiOchiMatu-TarowNodaTateaki, SasakiLet 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.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.AA00700121Journal of Information Processing 1432923001991-12-311882-66522009-06-30