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A new measure called min-max elementwise backward error is introduced forapproximate roots of scalar polynomials $p(z)$. Compared with the elementwiserelative backward error, this new measure allows for larger relativeperturbations on the coefficients of $p(z)$ that do not participate much in theoverall backward error. By how much these coefficients can be perturbed isdetermined via an associated max-times polynomial and its tropical roots. Analgorithm is designed for computing the roots of $p(z)$. It uses a companionlinearization $C(z) = A-zB$ of $p(z)$ to which we added an extra zero leadingcoefficient, and an appropriate two-sided diagonal scaling that balances $A$and makes $B$ graded in particular when there is variation in the magnitude ofthe coefficients of $p(z)$. An implementation of the QZ algorithm with a strictdeflation criterion for eigenvalues at infinity is then used to obtainapproximations to the roots of $p(z)$. Under the assumption that thisimplementation of the QZ algorithm exhibits a graded backward error when $B$ isgraded, we prove that our new algorithm is min-max elementwise backward stable.Several numerical experiments show the superior performance of the newalgorithm compared with the MATLAB \texttt{roots} function. Extending thealgorithm to polynomial eigenvalue problems leads to a new polynomialeigensolver that exhibits excellent numerical behaviour compared with otherexisting polynomial eigensolvers, as illustrated by many numerical tests.

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