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

概要

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Let F, F and D be multivariate polynomials andεbe a small positive number,0 < ε < < 1. If F=DF+ΔF, where ΔF 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.
一般社団法人情報処理学会の論文
1991-12-31