# Wilkinson's polynomial

The conditioning of polynomial root-finding is notoriously bad. One famous example is Wilkinson's polynomial: \begin{align*} P(x) &= \prod_{i=1}^{20}(x-i) = (x-1)(x-2)\cdots(x-20)\\ &= x^{20} - 210x^{19} + 20615x^{18} + \cdots + 2432902008176640000 \end{align*} By inspection, the roots are simply $1,2,\cdots, 20$. However, Wilkinson showed that a decreasing the coefficient of $x^{19}$ from $-210$ to $-210 - 2^{-23} \approx -210.000000119209$ had a drastic effect on many of the roots, some of which become complex. For example, the root at $x=20$ moves to $x=20.8$, a change of 4 % on a perturbation of one coefficient by less than one part in a billion (see also Problem 9.2.2).