# Source code for geomstats.numerics.finite_differences

```
"""Finite differences machinery."""
import geomstats.backend as gs
[docs]
def forward_difference(array, delta=None, axis=-1):
"""Forward difference in a Euclidean space.
Points live in R^m, but are a k-dim embedding (e.g. a curve).
Assumes points are in correspondence for cases higher than dim=1.
Parameters
----------
array : array-like
Values of a function.
delta : float
Spacing between points.
axis : int
Axis in which perform the difference.
Must be given backwards.
Returns
-------
forward_diff : array-like
Shape in the specified axis reduces by one.
"""
n = array.shape[axis]
if delta is None:
delta = 1 / (n - 1)
point_ndim_slc = (slice(None),) * (abs(axis) - 1)
slc = (..., slice(1, n)) + point_ndim_slc
forward = array[slc]
slc = (..., slice(0, n - 1)) + point_ndim_slc
center = array[slc]
return (forward - center) / delta
[docs]
def centered_difference(array, delta=None, axis=-1, endpoints=False):
"""Centered difference in a Euclidean space.
Points live in R^m, but are a k-dim embedding (e.g. a curve).
Assumes points are in correspondence for cases higher than dim=1.
Parameters
----------
array : array-like
Values of a function.
delta : float
Spacing between points.
axis : int
Axis in which perform the difference.
Must be given backwards.
endpoints : bool
If True, endpoints are computed by backward and forward differences,
respectively.
Returns
-------
centered_diff : array-like
Same shape as array.
"""
n = array.shape[axis]
if delta is None:
delta = 1 / (n - 1)
point_ndim_slc = (slice(None),) * (abs(axis) - 1)
slc = (..., slice(2, n)) + point_ndim_slc
forward = array[slc]
slc = (..., slice(0, n - 2)) + point_ndim_slc
backward = array[slc]
diff = (forward - backward) / (2 * delta)
if endpoints:
slc_left = (..., [0]) + point_ndim_slc
slc_left_forward = (..., [1]) + point_ndim_slc
diff_left = (array[slc_left_forward] - array[slc_left]) / delta
slc_right = (..., [-1]) + point_ndim_slc
slc_right_backward = (..., [-2]) + point_ndim_slc
diff_right = (array[slc_right] - array[slc_right_backward]) / delta
slc_right = (..., [-1]) + point_ndim_slc
return gs.concatenate((diff_left, diff, diff_right), axis=axis)
return diff
[docs]
def second_centered_difference(array, delta=None, axis=-1):
"""Second centered difference in a Euclidean space.
Points live in R^m, but are a k-dim embedding (e.g. a curve).
Assumes points are in correspondence for cases higher than dim=1.
Parameters
----------
array : array-like
Values of a function.
delta : float
Spacing between points.
axis : int
Axis in which perform the difference.
Must be given backwards.
Returns
-------
second_centered_diff : array-like
Shape in the specified axis reduces by two (endpoints).
"""
n = array.shape[axis]
if delta is None:
delta = 1 / (n - 1)
point_ndim_slc = (slice(None),) * (abs(axis) - 1)
slc = (..., slice(2, n)) + point_ndim_slc
forward = array[slc]
slc = (..., slice(0, n - 2)) + point_ndim_slc
backward = array[slc]
slc = (..., slice(1, n - 1)) + point_ndim_slc
central = array[slc]
return (forward + backward - 2 * central) / (delta**2)
```