= Amplification root
{title2=$\xi(z)$}
An amplification root is a <root of a polynomial> governing a geometric mode $y_n=\xi^n$ of a numerical recurrence on the <Dahlquist test equation>. Its <modulus> measures damping or growth per step. A repeated root produces additional modes such as $n\xi^n$; therefore a <unit-circle root> can be permitted by a bounded-root condition only when it is simple. The principal root approximates $e^z$ near zero, while additional roots can describe <parasitic amplification roots>.
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