An efficient approximation to the Cauchy radius

Abstract

AbstractThe Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it is optimal among all bounds depending only on the moduli of the coefficients. It has the disadvantage of being implicit because it requires the solution of a nonlinear equation. In this note, simple and explicit upper bounds are derived that are useful approximations to the Cauchy radius. The general approach to obtain the aforementioned bounds is to embed scalar polynomials into the larger framework of their generalization to matrix polynomials and then use bounds on the eigenvalues of the latter.