Integrating

Complete intervals

We are interested in integrating the smoothed interpolation from the start to some $t > t_1$. To compute this efficiently we need to know the integral of the interpolation over the various intervals. More precisely:

• For the linear sections we obtain

\[\begin{equation} \int_{t_{i-1+\frac{\lambda}{2}}}^{t_{i-\frac{\lambda}{2}}} u_{i-1} + \frac{\Delta u_i}{\Delta t_i}(\tau - t_{i-1}) \text{d}\tau = (1 - \lambda)\Delta t_i \left[ u_{i-1} + \frac{1}{2}(1 - \lambda) \Delta u_i \right]. \end{equation}\]

• For the spline sections we obtain

\[\begin{equation} \begin{aligned} \int_{t_{i-\frac{\lambda}{2}}}^{t_{i + \frac{\lambda}{2}}} u_i(\tau)\text{d}\tau &=& \int_0^1 T'_i(s)u_i(T_i(s))\text{d}s \\ &=& \int_0^1 \left[\lambda(\Delta t_{i+1} - \Delta t_i)s + \lambda\Delta t_i\right] \left[\frac{\lambda}{2}(\Delta u_{i+1} - \Delta u_i)s^2 + \lambda \Delta u_i s + u_{i - \frac{\lambda}{2}}\right] \text{d}s \\ &=& \frac{\lambda^2}{24} \left[ \Delta t_i \left(-3\Delta u_i + \Delta u_{i+1}\right) + \Delta t_{i+1} \left(-\Delta u_i + 3 \Delta u_{i+1}\right) \right] + \frac{\lambda}{2}(\Delta t_i + \Delta t_{i+1})u_i. \end{aligned} \end{equation}\]

Incomplete intervals

We now define the new set of points $(\tilde{\mathbf{p}}_j)_{j=1}^{2n}$ given by all the $\mathbf{p}_{i - \frac{\lambda}{2}}, \mathbf{p}_{i+ \frac{\lambda}{2}}$ and the original boundary points, sorted by $t$. Then for $t \in \left[\tilde{t}_{J-1}, \tilde{t}_J\right]$ we have

\[\begin{equation} \begin{aligned} U(t) = \int_{t_1}^t u(\tau)\text{d}\tau = \sum_{j = 2}^{J-1} \int_{\tilde{t}_{j-1}}^{\tilde{t}_j} u(\tau)\text{d}\tau + \int_{\tilde{t}_{J-1}}^t u(\tau)\text{d}\tau, \end{aligned} \end{equation}\]

Where the summed integrals are given by the values above. For the last integral:

• If $J$ is odd then the last integral is of a linear section:

\[\begin{equation} \int_{\tilde{t}_{J-1}}^t u(\tau)\text{d}\tau = \left((t-t_I) - \frac{\lambda}{2}\Delta t_{I+1}\right)u_I + \frac{1}{2}\frac{\Delta u_{I+1}}{\Delta t_{I+1}}\left[(t-t_I)^2 - \frac{\lambda^2}{4}\Delta t_{I+1}^2\right] \end{equation}\]

where $I = \frac{J-1}{2}$.

• If $J$ is even the last integral is of a spline section:

\[\begin{equation} \begin{aligned} \int_{\tilde{t}_{J-1}}^t u_I(\tau)\text{d}\tau &=& \int_0^{S_I(t)} T'_I(s)u_I\left(T_I(s)\right)\text{d}s \\ &=& \int_0^{S_I(t)} \left[\lambda(\Delta t_{I+1} - \Delta t_I)s + \lambda\Delta t_I\right] \left[\frac{\lambda}{2}(\Delta u_{I+1} - \Delta u_I)s^2 + \lambda \Delta u_I s + u_{I - \frac{\lambda}{2}}\right] \text{d}s \end{aligned} \end{equation}\]

where $I = \frac{J}{2}$.