(i) A general multistep method for the numerical approximation to the scalar differential equation y′=f(t,y) is given by
ℓ=0∑sρℓyn+ℓ=hℓ=0∑sσℓfn+ℓ,n=0,1,…
where fn+ℓ=f(tn+ℓ,yn+ℓ). Show that this method is of order p⩾1 if and only if
ρ(ez)−zσ(ez)=O(zp+1) as z→0
where
ρ(w)=ℓ=0∑sρℓwℓ and σ(w)=ℓ=0∑sσℓwℓ
(ii) A particular three-step implicit method is given by
yn+3+(a−1)yn+1−ayn=h(fn+3+ℓ=0∑2σℓfn+ℓ)
where the σℓ are chosen to make the method third order. [The σℓ need not be found.] For what values of a is the method convergent?