Semi-Implicit Trapezoidal Rule

The semi-implicit trapezoidal rule method is given by [558]

$$\underline{\hat{x}}(n) \isdefs \underline{\hat{x}}(n-1) + \frac{T}{2}\, \frac{f[n,\underline{\hat{x}}(n-1),\underline{u}(n)] + f[n-1,\underline{\hat{x}}(n-1),\underline{u}(n-1)]}{1-\frac{T}{2}\,\ddot{\underline{\hat{x}}}(n-1)}. \protect$$ (8.17)