Exact numerical solution

Starting with $u^0=I$, the simple recursion (28) can be applied repeatedly $n$ times, with the result that $$\begin{equation} u^{n} = IA^n,\quad A = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t} \tag{29} \end{equation}$$

Such an exact discrete solution is unusual, but very handy for analysis.