Amplification polynomial of a multistep method ( $P_{z} (w) = ρ (w) - z σ (w)$ )

Applying a linear multistep method to the Dahlquist test equation with

z = hλ

produces a constant-coefficient recurrence whose modes satisfy

P_{z} (w) = 0

. Absolute stability requires the resulting roots of a polynomial to lie in the unit disk, with unit-modulus roots simple. Thus multistep stability depends on every recurrence mode, rather than a single one-step stability function.

Table of contents
- Amplification root Amplification polynomial of a multistep method
- Exterior roots of the derivative polynomial bound multistep stability Amplification polynomial of a multistep method

Amplification root ( $ξ (z)$ )

 0  0

An amplification root is a root of a polynomial governing a geometric mode

y_{n} = ξ^{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 ξ^{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.

Exterior roots of the derivative polynomial bound multistep stability ( $σ (w_{}) = 0$ , $∣ w_{} ∣ > 1$ )

 0  0

If the derivative characteristic polynomial

σ

of a linear multistep method has a simple root outside the unit disk and keeps its degree in the limit, then its absolute stability region is bounded. Otherwise a stable sequence with

∣ z ∣ \to \infty

would give amplification polynomials

σ - ρ / z

tending coefficientwise to

σ

. A root near the exterior root would persist by continuity, contradicting the root condition for a multistep method. This argument excludes stability at every sufficiently large complex