Wilkinson's polynomial

Example

Wilkinson's polynomial is the polynomial $$w(x) = \prod_{i=1}^{20} (x - i) = (x - 1)(x - 2) \cdots (x - 20) = x^{20} - 210 x^{19} + \cdots + 20!.$$ Its roots $1, 2, \ldots, 20$ are simple and well separated, yet they are extremely sensitive to the coefficients. James H. Wilkinson changed the coefficient $-210$ of $x^{19}$ to $-210 - 2^{-23}$, a relative change of about $6 \times 10^{-10}$. The perturbed polynomial has only $10$ real roots, and the roots near $10, \ldots, 19$ turn into five complex conjugate pairs. For example, the roots $16$ and $17$ become approximately $16.73 \pm 2.81\,i$. Check of the sensitivity: if $a$ is the coefficient of $x^{19}$ and $r$ a simple root, differentiating $w(r) = 0$ with respect to $a$ gives $$\frac{\partial r}{\partial a} = -\frac{r^{19}}{w'(r)}.$$ For $r = 20$, $w'(20) = 19!$, so $\abs{\partial r/\partial a} = 20^{19}/19! \approx 4.3 \times 10^{7}$: the root-finding problem has a huge condition number with respect to this coefficient.