 Root sensitivity in power-sum reconstruction

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 $\delta =2{\epsilon }^2>0$, the second power sum changes by $2{\epsilon }^2$ while each polynomial root moves by $\epsilon$. The ratio $1/(2\epsilon )$ grows without bound as $\epsilon$ 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.