Tighter Bounds of Errors of Numerical Roots

Abstract

Let $P(z)$ be a monic univariate polynomial over $\mathbf{C}$, of degree $n$ and having roots $\zeta_1,\ldots,\zeta_n$. Given approximate roots $z_1,\ldots,z_n$, with $\zeta_i \simeq z_i$ ($i=1,\ldots,n$), we derive a very tight upper bound of $|\zeta_i - z_i|$, by assuming that $\zeta_i$ has no close root. The bound formula has a similarity with Smale's and Smith's formulas. We also derive a lower bound of $|\zeta_i - z_i|$ and a lower bound of $\min\{|\zeta_j - z_i|\mid j \neq i\}$.