= Root sensitivity in power-sum reconstruction
{title2=$p_1=2,\ p_2=2+\delta\implies x_{1,2}=1\pm\sqrt{\delta/2}$}
Exact reconstruction does not promise that small errors in <power sums> produce proportionally small errors in individual <polynomial roots>. For two roots, let $p_1=2$ and $p_2=2+\delta$. <Newton's identities> give $e_1=2$ and $e_2=1-\delta/2$, so the <monic polynomial> is $(z-1)^2-\delta/2$. Its <polynomial roots> are $1\pm\sqrt{\delta/2}$. A zero value of $\delta$ gives a <multiple root>; positive $\delta$ splits it into two real roots, and negative $\delta$ splits it into a complex-conjugate pair.
With $\delta=2\varepsilon^2>0$, the second <power sum> changes by $2\varepsilon^2$ while each <polynomial root> moves by $\varepsilon$. The ratio $1/(2\varepsilon)$ grows without bound as $\varepsilon$ tends to zero. Thus <coefficients> can stay close while individual root locations are more sensitive. This is a reason to distinguish recovering <coefficients> by <Newton's identities> from subsequently approximating the <polynomial roots>.
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