Mathematical background

Let a polynomial whose interval coefficients have a fixed sign. Let $a_n$ be the leading coefficient such that $a_n=1$. We define the sequence $A_1=\vert a_{n-1}a_0\vert$, $A_2=\vert a_{n-1}a_1-a0\vert$, $A_j=\vert a_{n-1}a_j-a_{j-1}\vert$. If $A$ is the largest upper bound of the sequence $A_j$, then the modulus $\rho$ of the roots of the polynomial is lower or equal to $(1+\sqrt{1+4A})/2$.