Error Bounds for Zeros of a Polynomial Based Upon Gerschgorin's Theorems

Abstract

Given N approximations to the zeros of an N th-degree polynomial, N circular regions in the complex z -plane are determined whose union contains all the zeros, and each connected component of this union consisting of K such circular regions contains exactly K zeros. The bounds for the zeros provided by these circular regions are not excessively pessimistic; that is, whenever the approximations are sufficiently well separated and sufficiently close to the zeros of this polynomial, the radii of these circular regions are shown to overestimate the errors by at most a modest factor simply related to the configuration of the approximations. A few numerical examples are included.