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  "body": "\n## Example\n\n*Wilkinson's polynomial* is the [[polynomial|1G0B73WPX8]]\n$$w(x) = \\prod_{i=1}^{20} (x - i) = (x - 1)(x - 2) \\cdots (x - 20) = x^{20} - 210 x^{19} + \\cdots + 20!.$$\nIts [[roots|019MA1VTR2]] $1, 2, \\ldots, 20$ are simple and well separated, yet they are extremely sensitive to the coefficients.\nJames H. Wilkinson changed the coefficient $-210$ of $x^{19}$ to $-210 - 2^{-23}$, a relative change of about $6 \\times 10^{-10}$.\nThe perturbed polynomial has only $10$ real roots, and the roots near $10, \\ldots, 19$ turn into five complex conjugate pairs.\nFor example, the roots $16$ and $17$ become approximately $16.73 \\pm 2.81\\,i$.\nCheck 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\n$$\\frac{\\partial r}{\\partial a} = -\\frac{r^{19}}{w'(r)}.$$\nFor $r = 20$, $w'(20) = 19!$, so $\\abs{\\partial r/\\partial a} = 20^{19}/19! \\approx 4.3 \\times 10^{7}$: the root-finding problem has a huge [[condition number|AB3S6MHJ8X]] with respect to this coefficient.\n\n## Remarks\n\nThe difficulty lies in the problem, not in the algorithm: any method that works with the coefficients in floating-point arithmetic, where they are rounded, inherits this sensitivity.\nThe same roots are perfectly well conditioned as functions of the factors $(x - i)$, which shows that the choice of representation of a polynomial matters.\n\n## Explanations\n\n### L1 human\n\nThe polynomial with roots $1, 2, \\ldots, 20$ looks harmless, since its roots are evenly spaced whole numbers.\nBut if one changes one coefficient by about one part in a billion, several roots move far away and even become complex numbers.\nThis shows that finding roots from the coefficients of a polynomial can be a badly conditioned problem.\n",
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