= Necessity of the root condition for multistep convergence
For a <linear multistep method>, convergence for every set of consistent starting values requires the <root condition for a multistep method>. Apply the recurrence to $y'=0$ over $N$ steps. A root $\xi$ with $|\xi|>1$ gives the error mode $e_n=|\xi|^{-N}\xi^n$: its fixed-index starting errors vanish as $N\to\infty$, but $|e_N|=1$. A unit-modulus root of multiplicity $m\geq2$ instead gives $e_n=N^{-(m-1)}n^{m-1}\xi^n$, with the same failure. Repeated roots inside the unit disk are harmless because their polynomial growth is dominated by exponential decay.
Back to article page
