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Insert an exact smooth solution into the linear multistep method and define its unscaled local truncation error byIts Taylor expansion has coefficientsThe method has order at least $p$ exactly when $c_0=\cdots =c_p=0$, since then ${\mathcal{R}}_h[y]=O(h^{p+1})$ for every sufficiently smooth $y$. Necessity can be checked by inserting polynomials of successive degrees. This also corresponds to the usual truncation error ${\mathcal{R}}_h/h=O(h^p)$.

On the other hand, expansion of the characteristic polynomials of a linear multistep method givesThus the exponential-symbol order criterion for a multistep method isIf “order $p$” is meant to be the exact rather than guaranteed order, one additionally requires $c_{p+1}\ne 0$. The first two vanishing conditions are the usual consistency of a numerical method relations $\rho (1)=0$, ${\rho }^{\prime }(1)=\sigma (1)$.

The characteristic polynomials of a linear multistep method areTo determine formal order, substitute a smooth exact solution and expand about the first time level. The exponential-symbol order criterion for a multistep method collects precisely the same coefficients:The constant, linear and quadratic coefficients vanish for every $a$. The cubic coefficient vanishes only at $a=-1/5$, where the quartic coefficient is $-1/30\ne 0$. HenceHere order means the exact-solution step residual is $O(h^{p+1})$. It is a formal consistency result, not a convergence assertion. In particular at $a=1$ both ${\rho }^{\prime }(1)$ and $\sigma (1)$ vanish, and the double root at one destroys zero-stability; cancelling its common factor gives a different, first-order recurrence with an additional integration constant left unspecified by the original formula.