# Source code for pylops.basicoperators.firstderivative

```__all__ = ["FirstDerivative"]

from typing import Callable, Union

import numpy as np
from numpy.core.multiarray import normalize_axis_index

from pylops import LinearOperator
from pylops.utils._internal import _value_or_sized_to_tuple
from pylops.utils.backend import get_array_module
from pylops.utils.decorators import reshaped
from pylops.utils.typing import DTypeLike, InputDimsLike, NDArray

[docs]class FirstDerivative(LinearOperator):
r"""First derivative.

Apply a first derivative using a multiple-point stencil finite-difference
approximation along ``axis``.

Parameters
----------
dims : :obj:`list` or :obj:`int`
Number of samples for each dimension
axis : :obj:`int`, optional

Axis along which derivative is applied.
sampling : :obj:`float`, optional
Sampling step :math:`\Delta x`.
kind : :obj:`str`, optional
Derivative kind (``forward``, ``centered``, or ``backward``).
edge : :obj:`bool`, optional
Use reduced order derivative at edges (``True``) or
ignore them (``False``). This is currently only available
for centered derivative
order : :obj:`int`, optional

Derivative order (``3`` or ``5``). This is currently only available
for centered derivative
dtype : :obj:`str`, optional
Type of elements in input array.
name : :obj:`str`, optional

Name of operator (to be used by :func:`pylops.utils.describe.describe`)

Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly (``True``) or
not (``False``)

Notes
-----
The FirstDerivative operator applies a first derivative to any chosen
direction of a multi-dimensional array using either a second- or third-order
centered stencil or first-order forward/backward stencils.

For simplicity, given a one dimensional array, the second-order centered
first derivative is:

.. math::
y[i] = (0.5x[i+1] - 0.5x[i-1]) / \Delta x

while the first-order forward stencil is:

.. math::
y[i] = (x[i+1] - x[i]) / \Delta x

and the first-order backward stencil is:

.. math::
y[i] = (x[i] - x[i-1]) / \Delta x

Formulas for the third-order centered stencil can be found at

"""

def __init__(
self,
dims: Union[int, InputDimsLike],
axis: int = -1,
sampling: float = 1.0,
kind: str = "centered",
edge: bool = False,
order: int = 3,
dtype: DTypeLike = "float64",
name: str = "F",
) -> None:
dims = _value_or_sized_to_tuple(dims)
super().__init__(dtype=np.dtype(dtype), dims=dims, dimsd=dims, name=name)

self.axis = normalize_axis_index(axis, len(self.dims))
self.sampling = sampling
self.kind = kind
self.edge = edge
self.order = order
self._register_multiplications(self.kind, self.order)

def _register_multiplications(
self,
kind: str,
order: int,
) -> None:
# choose _matvec and _rmatvec kind
self._matvec: Callable
self._rmatvec: Callable
if kind == "forward":
self._matvec = self._matvec_forward
self._rmatvec = self._rmatvec_forward
elif kind == "centered":
if order == 3:
self._matvec = self._matvec_centered3
self._rmatvec = self._rmatvec_centered3
elif order == 5:
self._matvec = self._matvec_centered5
self._rmatvec = self._rmatvec_centered5
else:
raise NotImplementedError("'order' must be '3, or '5'")
elif kind == "backward":
self._matvec = self._matvec_backward
self._rmatvec = self._rmatvec_backward
else:
raise NotImplementedError(
"'kind' must be 'forward', 'centered', or 'backward'"
)

@reshaped(swapaxis=True)
def _matvec_forward(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., :-1] = (x[..., 1:] - x[..., :-1]) / self.sampling
return y

@reshaped(swapaxis=True)
def _rmatvec_forward(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., :-1] -= x[..., :-1]
y[..., 1:] += x[..., :-1]
y /= self.sampling
return y

@reshaped(swapaxis=True)
def _matvec_centered3(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., 1:-1] = 0.5 * (x[..., 2:] - x[..., :-2])
if self.edge:
y[..., 0] = x[..., 1] - x[..., 0]
y[..., -1] = x[..., -1] - x[..., -2]
y /= self.sampling
return y

@reshaped(swapaxis=True)
def _rmatvec_centered3(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., :-2] -= 0.5 * x[..., 1:-1]
y[..., 2:] += 0.5 * x[..., 1:-1]
if self.edge:
y[..., 0] -= x[..., 0]
y[..., 1] += x[..., 0]
y[..., -2] -= x[..., -1]
y[..., -1] += x[..., -1]
y /= self.sampling
return y

@reshaped(swapaxis=True)
def _matvec_centered5(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., 2:-2] = (
x[..., :-4] / 12.0
- 2 * x[..., 1:-3] / 3.0
+ 2 * x[..., 3:-1] / 3.0
- x[..., 4:] / 12.0
)
if self.edge:
y[..., 0] = x[..., 1] - x[..., 0]
y[..., 1] = 0.5 * (x[..., 2] - x[..., 0])
y[..., -2] = 0.5 * (x[..., -1] - x[..., -3])
y[..., -1] = x[..., -1] - x[..., -2]
y /= self.sampling
return y

@reshaped(swapaxis=True)
def _rmatvec_centered5(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., :-4] += x[..., 2:-2] / 12.0
y[..., 1:-3] -= 2.0 * x[..., 2:-2] / 3.0
y[..., 3:-1] += 2.0 * x[..., 2:-2] / 3.0
y[..., 4:] -= x[..., 2:-2] / 12.0
if self.edge:
y[..., 0] -= x[..., 0] + 0.5 * x[..., 1]
y[..., 1] += x[..., 0]
y[..., 2] += 0.5 * x[..., 1]
y[..., -3] -= 0.5 * x[..., -2]
y[..., -2] -= x[..., -1]
y[..., -1] += 0.5 * x[..., -2] + x[..., -1]
y /= self.sampling
return y

@reshaped(swapaxis=True)
def _matvec_backward(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., 1:] = (x[..., 1:] - x[..., :-1]) / self.sampling
return y

@reshaped(swapaxis=True)
def _rmatvec_backward(self, x: NDArray) -> NDArray:
ncp = get_array_module(x)
y = ncp.zeros(x.shape, self.dtype)
y[..., :-1] -= x[..., 1:]
y[..., 1:] += x[..., 1:]
y /= self.sampling
return y
```