Tighter bounds of errors of numerical roots

LetP(z) be a monic univariate polynomial overC, of degreen and having roots ζ1,..., ζn. Given approximate rootsz1,...,zn, with ζi ≃zi (i = 1,...,n), we derive a very tight upper bound of ¦ζi −zi¦, by assuming that ζi has no close root. The bound formula has a similarity with Smale’s and Smith’s formulas. We also derive a lower bound of ¦ζi −zi¦ and a lower bound of min{ζj −zi¦ ¦j ≠i}.