Tateaki Sasaki
Abstract
Let P(x) be a univariate polynomial over C, such that P(x) = cnxn + ... + cm+1xm+1 + xm + em-1xm-1 + ... + e0, where max{|cn|, ..., |cm+1|} = 1 and e = max{|em-1|, |em-2|1/2, ..., |e0|1/m} << 1. P(x) has m small roots around the origin so long as e << 1. In 1999, we derived a formula that if e < 1/9 then P(x) has m roots inside a disc Din of radius Rin and other n - m roots outside a disc Dout of radius Rout, located at the origin, where Rin(out) = [1 - (+) √1 - (16e)/(1 + 3e)2] × (1 + 3e)/4. Note that Rin = Rout if e = 1/9. Our formula is essentially the same as that derived independently by Yakoubsohn at almost the same time. In this short article, we introduce the formula and check its sharpness on many polynomials generated randomly.
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Digital Library
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- {SI02} T. Sasaki and D. Inaba: On analytic continuation of algebraic functions. Preprint of Univ. Tsukuba, 14 pages (2002).Google Scholar
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- {TS00} A. Terui and T. Sasaki: "Approximate Zero-points" of Real Univariate Polynomial with Large Error Terms. IPSJ(Information Processing Society of Japan) Journal, 41, 974-989, 2000.Google Scholar
- {Yak00} J.C. Yakoubsohn: Finding a cluster of zeros of univariate polynomials. J. Complexity 16, 603-636 (2000). Google ScholarDigital Library
Index Terms
- A formula for separating small roots of a polynomial
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