ρ (ζ) = ζ^{2} - (1 + a) ζ + a = (ζ - 1) (ζ - a), σ (ζ) = \frac{1 - 3 a}{2} ζ + \frac{1 + a}{2} ζ^{2} .

To 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:

ρ (e^{z}) - z σ (e^{z}) = - \frac{1 + 5 a}{12} z^{3} - \frac{3 + 11 a}{24} z^{4} + O (z^{5}) .

(2)

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 \neq = 0

. Hence

p = 3 if a = - \frac{1}{5}, p = 2 otherwise .

(3)

Here 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

ρ^{'} (1)

and

σ (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.