On computing minimal realizable spectral radii of non‐negative matrices

For decades considerable efforts have been exerted to resolve the inverse eigenvalue problem for non‐negative matrices. Yet fundamental issues such as the theory of existence and the practice of computation remain open. Recently, it has been proved that, given an arbitrary (n–1)‐tuple ?? = (λ₂,…,λ_n) ∈ ?^n–1 whose components are closed under complex conjugation, there exists a unique positive real number ?(??), called the minimal realizable spectral radius of ??, such that the set {λ₁,…,λ_n} is precisely the spectrum of a certain n × n non‐negative matrix with λ₁ as its spectral radius if and only if λ₁ ? ?(??). Employing any existing necessary conditions as a mode of checking criteria, this paper proposes a simple bisection procedure to approximate the location of ?(??). As an immediate application, it offers a quick numerical way to check whether a given n‐tuple could be the spectrum of a certain non‐negative matrix. Copyright © 2004 John Wiley & Sons, Ltd.