Backward stability of polynomial root-finding using Fiedler companion matrices (Q2794700)

This is an interesting paper on computing roots of scalar polynomials as the eigenvalues of Fiedler companion matrices. The authors analyze backward stability of polynomial root-finding algorithms by studying whether the whole set of computed eigenvalues of the companion matrix, obtained with a backward stable algorithm, for the standard eigenvalue problems, is the set of roots of nearby polynomials. They compare properties, strength and weakness of the Frobenius and Fiedler companion matrices. They quantify stability of the backward eigenvalue algorithms with the norm of the polynomial and discuss balancing and backward errors. Extensive numerical experiments are conducted to illustrate the analysis and effect of balancing the matrices.