Mathematics for computer algebra
Mathematics for computer algebra
Complexity and real computation
Complexity and real computation
Error Bounds for Zeros of a Polynomial Based Upon Gerschgorin's Theorems
Journal of the ACM (JACM)
Finding a cluster of zeros of univariate polynomials
Journal of Complexity
A theorem for separating close roots of a polynomial and its derivatives
ACM SIGSAM Bulletin
Distance bounds of ε-points on hypersurfaces
Theoretical Computer Science
Computing clustered close-roots of univariate polynomials
Proceedings of the 2009 conference on Symbolic numeric computation
Parametrization of ε-rational curves: extended abstract
Proceedings of the 2009 conference on Symbolic numeric computation
Approximate parametrization of plane algebraic curves by linear systems of curves
Computer Aided Geometric Design
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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} P(x) has m small roots around the origin so long as e e 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.