## Min-Max Elementwise Backward Error for Roots of Polynomials and a Corresponding Backward Stable Root Finder KU Leuven

A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials $p(z)$. Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on the coefficients of $p(z)$ that do not participate much in the overall backward error. By how much these coefficients can be perturbed is determined via an associated max-times polynomial and its tropical roots. ...