"Approximate Zero-points" of Real Univariate Polynomial with Large Error Terms

概要

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Let P(x) be a given real univariate polynomial and let P^^〜(x)=P(x)+Δ(x), while Δ(x) is the sum of error terms, that is, a polynomial with small real unknown but bonded coefficients. We first consider specifying the "existence domain" of the values of P^^〜(x), or the domain in which the value of P^^〜(x) exists for any real number x, by the coefficient bounds for Δ(x), and then introduce a concept of an "approximate real zero-point" of P^^〜(x). We present a practical method for estimating the existence domain of zero-points of P^^〜(x) by applying Smith's celebrated theorem. We next consider counting the number of real zero-points of P^^〜(x). If all the zero-points are sufficiently far apart from each other, the number of real zero-points of P^^〜(x) is the same as that of P(x), and we derive a condition for which we can assert that P(x) and P^^〜(x) have the same number of real zero-points. We calculate the actual number of real zero-points by Sturm's method, which encounters the so-called small leading coefficient problem. For this problem, we show that, under some conditions, small leading terms can be discarded. Furthermore, we investigate four methods for evaluating the effect of error terms on the elements of the Sturm sequence.
一般社団法人情報処理学会の論文
2000-04-15