ord2()

ord2(states: ndarray, dt: float, inputs=None)

Second-order forward, central, and backward differences for estimating the first derivative.

Central differences are used where possible; forward differences are used for the first point and backward differences are used for the last point:

\[\begin{split}\frac{\textup{d}}{\textup{d}t}\q(t)\bigg|_{t = t_0} &\approx \frac{1}{2\delta t}(-3\q(t_0) + 4\q(t_{1}) - \q(t_{2})), \\ \\ \frac{\textup{d}}{\textup{d}t}\q(t)\bigg|_{t = t_j} &\approx \frac{1}{2\delta t}(-\q(t_{j-1}) + \q(t_{j+1})), \quad j = 2,\ldots,k-2, \\ \\ \frac{\textup{d}}{\textup{d}t}\q(t)\bigg|_{t = t_{k-1}} &\approx \frac{1}{2\delta t}(\q(t_{k-3}) - 4\q(t_{k-2}) + 3\q(t_{k-1}))\end{split}\]

where \(\delta t = t_{j+1} - t_j\) for all \(j\).

Parameters:

states(r, k) ndarray

State snapshots: states[:, j] is the state at time \(t_j\).

dtfloat

Time step between snapshots.

inputs(m, k) or (k,) ndarray or None

Inputs corresponding to the states, if applicable.

Returns:

states(r, k) ndarray

State snapshots.

ddts(r, k) ndarray

Time derivative estimates corresponding to the state snapshots.

inputs(m, k) or (k,) ndarray or None

Inputs corresponding to _states, if applicable. Only returned if inputs is not None.