 Stability function

The stability function $R$ of a one-step numerical method is its amplification factor on the Dahlquist test equation $y^{\prime }=\lambda y$: one step gives $y_{n+1}=R(h\lambda )y_n$. Its absolute stability region is $\{z:|R(z)|\le 1\}$, using boundedness of repeated steps as the convention.

For explicit Euler method, $R(z)=1+z$, so the region is the closed disc centered at $-1$ with radius one. For Backward Euler method, $R(z)=1/(1-z)$, and the entire left half-plane is stable. These calculations explain why a step can be accurate locally yet generate a growing long-time solution. Multistep methods instead have several amplification roots and require a root condition.^[ref]