Characteristic polynomials of a linear multistep method ($\rho ,\sigma$)

For ${\sum }_{j=0}^s{\alpha }_j{y}_{n+j}=h{\sum }_{j=0}^s{\beta }_j{f}_{n+j}$, the characteristic polynomials are $\rho (\zeta )={\sum }_j{\alpha }_j{\zeta }^j$ and $\sigma (\zeta )={\sum }_j{\beta }_j{\zeta }^j$. Applied to $y^{\prime }=\lambda y$, the amplification roots solve $\rho (\zeta )-h\lambda \sigma (\zeta )=0$.

A linear multistep method has order $p$ exactly whenfor $0\le q\le p$, with the right side interpreted as zero for $q=0$, and the identity first fails at $q=p+1$.