Mathematical background

Let $P(x)$ be an univariate polynomial of degree $n$:

$$P(x)= a_n x^n + a_{n-1} x^{n-1}+.....a_0=0$$

with $a_0a_n \not=0$. Let

$$A= max\{\mid a_{n-1}\mid ,\mid a_{n-2}\mid ,...,\mid a_0\mid... ...ime=max\{\mid a_n\mid ,\mid a_{n-1}\mid ,.....,\mid a_1\mid \}$$

Then the modulus of any root $x_k$ of $P(x)=0$ verify:

$$\frac{1}{1+\frac{A^\prime}{\mid a_0\mid }}<\mid x_k\mid < 1 + \frac{A}{\mid a_n\mid }$$