Stability of polynomials under coefficient perturbation

Abstract

Let the real polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n-1}x + a_{n}</tex> be stable and let the real numbers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b_{k}, c_{k} \geq 0, 0 \leq k \leq n</tex> , be given. We present a simple determinant criterion for finding the largest <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t_{0} \geq 0</tex> such that the polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_{0}x^{n} + \alpha_{1}x^{n-1}+ ... +\alpha_{n-1}x + \alpha_{n}</tex> is stable for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_{k} \in (a_{k} - b_{k}t_{0}, a_{k} + C_{k}t_{0}) \cup {a_{k}}, 0 \leq k \leq n</tex> . Several further observations allow us to reduce the computational cost considerably.