Characteristic polynomials of a linear multistep method
= Characteristic polynomials of a linear multistep method
{title2=$\rho,\sigma$}
For $\sum_{j=0}^s\alpha_jy_{n+j}=h\sum_{j=0}^s\beta_jf_{n+j}$, the characteristic polynomials are $\rho(\zeta)=\sum_j\alpha_j\zeta^j$ and $\sigma(\zeta)=\sum_j\beta_j\zeta^j$. Applied to $y'=\lambda y$, the amplification roots solve $\rho(\zeta)-h\lambda\sigma(\zeta)=0$.