Approximate Square-free Decomposition and Root-finding of III-conditioned Algebraic Equations

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概要

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• The exact square-free decomposition is generalized to polynomials with coefficients of floating-point numbers, and an algorithm of approximate square-free decomposition is presented. Given a polynomial P(x)and a small positive numbeε, 0<ε<<1, the decomposition algorithm calculates polynomials Q_1, Q_2, . . . , Q_l such that P(x)=Q_l(x)Q^2_2(x)...Q^l_l(x),where each root of Q_m(x)=0 is approximately equal to m multiple root or the average vdue of m close roots of P(x)=0. The decomposition is performed by using a generalized Euclidean algorithm, and the properties of approximate GCD (greatest common divisor), computed by the Euclidean algorithm, is investigated by developing a theory of approximate GCD. The equations Q_i(x)=0, i=1, . . . , l, are much easier to solve numerically than P(x)=0. Hence, we apply the approximate square-free decomposition to solving il1-conditioned algebraic equations and propose an algorithm which finds not only multiple but also close roots nicely.

• 一般社団法人情報処理学会の論文
• 1989-08-30

著者

• Sasaki Tateaki

  The Institute Of Physical And Chemical Research

• Noda Matu-tarow

  Faculty Of Engineering Ehime University

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