Mathematical background

Let $P$ be a polynomial and be maxroot the maximal modulus of the root of $P$. From $P$ we may derive a the unitary polynomial $Q$ such that the roots of $Q$ have a modulus lower or equal to 1 and if $w$ is a root of $Q$ then maxroot$w$ is a root of $P$.

Let $Q=sum_{i=0}^{i=n} a_i x^i$ which may also be written as $sum_{i=0}^{i=n} b_i(x-a)^i$ where $a$ is some fixed point.

Let a range $[a,b]$ for $x$ and let $z_0$ be the mid point of the range. We consider the square in the complex plane centered at $z_0$ and whose edge length is $b-a$. Let $\delta$ be the length of the half-diagonal of this square. If

$$\vert b_0\vert> \sum_{j=1}^{j=n} \vert b_j\vert\delta^j$$

then the polynomial has no root in the square [1]. This procedure may be used for univariate polynomial, with polynomial with interval coefficients or with parametric poynomial. For example it is very efficient for solging the Wilkinson polynomial.