Consider the multistep method for numerical solution of the differential equation $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$ :

$$\sum_{l=0}^{s} \rho_{l} \mathbf{y}_{n+l}=h \sum_{l=0}^{s} \sigma_{l} \mathbf{f}\left(t_{n+l}, \mathbf{y}_{n+l}\right), \quad n=0,1, \ldots$$

What does it mean to say that the method is of order $p$, and that the method is convergent?

Show that the method is of order $p$ if

$$\sum_{l=0}^{s} \rho_{l}=0, \quad \sum_{l=0}^{s} l^{k} \rho_{l}=k \sum_{l=0}^{s} l^{k-1} \sigma_{l}, \quad k=1,2, \ldots, p$$

and give the conditions on $\rho(w)=\sum_{l=0}^{s} \rho_{l} w^{l}$ that ensure convergence.

Hence determine for what values of $\theta$ and the $\sigma_{i}$ the two-step method

$$\mathbf{y}_{n+2}-(1-\theta) \mathbf{y}_{n+1}-\theta \mathbf{y}_{n}=h\left[\sigma_{0} \mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)+\sigma_{1} \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)+\sigma_{2} \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)\right]$$

is (a) convergent, and (b) of order 3 .