# Source code for landlab.components.flow_director.flow_director_steepest

```
#! /usr/env/python
"""
flow_director_steepest.py: provides the component FlowDirectorSteepest.
This components finds the steepest single-path steepest descent flow
directions. It is equivalent to D4 method in the special case of a raster grid
in that it does not consider diagonal links between nodes. For that capability,
use FlowDirectorD8.
"""
import numpy as np
from landlab import NodeStatus, VoronoiDelaunayGrid
from landlab.components.flow_director import flow_direction_DN
from landlab.components.flow_director.flow_director_to_one import _FlowDirectorToOne
[docs]
class FlowDirectorSteepest(_FlowDirectorToOne):
"""Single-path (steepest direction) flow direction without diagonals.
This components finds the steepest single-path steepest descent flow
directions. It is equivalent to D4 method in the special case of a raster
grid in that it does not consider diagonal links between nodes. For that
capability, use FlowDirectorD8.
Stores as ModelGrid fields:
- Node array of receivers (nodes that receive flow), or ITS OWN ID if
there is no receiver: *'flow__receiver_node'*
- Node array of steepest downhill slopes:
*'topographic__steepest_slope'*
- Node array containing ID of link that leads from each node to its
receiver, or grid.BAD_INDEX if no link:
*'flow__link_to_receiver_node'*
- Boolean node array of all local lows: *'flow__sink_flag'*
- Link array identifing if flow goes with (1) or against (-1) the link
direction: *'flow__link_direction'*
The primary method of this class is :func:`run_one_step`.
Examples
--------
This method works for both raster and irregular grids. First we will look
at a raster example, and then an irregular example.
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowDirectorSteepest
>>> mg = RasterModelGrid((3, 3), xy_spacing=(1, 1))
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> _ = mg.add_field(
... "topographic__elevation",
... mg.node_x + mg.node_y,
... at="node",
... )
>>> fd = FlowDirectorSteepest(mg, "topographic__elevation")
>>> fd.surface_values
array([ 0., 1., 2., 1., 2., 3., 2., 3., 4.])
>>> fd.run_one_step()
>>> mg.at_node["flow__receiver_node"]
array([0, 1, 2, 3, 1, 5, 6, 7, 8])
>>> mg.at_node["topographic__steepest_slope"]
array([ 0., 0., 0., 0., 1., 0., 0., 0., 0.])
>>> mg.at_node["flow__link_to_receiver_node"]
array([-1, -1, -1, -1, 3, -1, -1, -1, -1])
>>> mg.at_node["flow__sink_flag"].astype(int)
array([1, 1, 1, 1, 0, 1, 1, 1, 1])
>>> mg_2 = RasterModelGrid((5, 4), xy_spacing=(1, 1))
>>> topographic__elevation = [
... [0.0, 0.0, 0.0, 0.0],
... [0.0, 21.0, 10.0, 0.0],
... [0.0, 31.0, 20.0, 0.0],
... [0.0, 32.0, 30.0, 0.0],
... [0.0, 0.0, 0.0, 0.0],
... ]
>>> _ = mg_2.add_field(
... "topographic__elevation",
... topographic__elevation,
... at="node",
... )
>>> mg_2.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> fd_2 = FlowDirectorSteepest(mg_2)
>>> fd_2.run_one_step()
>>> mg_2.at_node["flow__receiver_node"].reshape(mg_2.shape)
array([[ 0, 1, 2, 3],
[ 4, 1, 2, 7],
[ 8, 10, 6, 11],
[12, 14, 10, 15],
[16, 17, 18, 19]])
And the at-link field ``'flow__link_direction'`` indicates if the flow along
the link is with or against the direction indicated by ``'link_dirs_at_node'``
(from tail node to head node).
>>> mg_2.at_link["flow__link_direction"]
array([ 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0,
0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], dtype=int8)
This indicates that flow on links 4, 5, 12, and 19 goes against the
topologic ordering -- that is that flow goes from head node to tail node --
and that flow goes with the topologic ordering on links 15 and 22. All other
links have no flow on them.
The FlowDirectorSteepest attribute ``flow_link_direction_at_node`` indicates
the link flow direction (with or against topology directions) for all links
at node. The ordering of links at node mirrors the grid attribute
``links_at_node``.
>>> fd_2.flow_link_direction_at_node()
array([[ 0, 0, 0, 0],
[ 0, -1, 0, 0],
[ 0, -1, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, -1, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 1, 0, 0, 0],
[ 0, -1, 1, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 1, 0, 0, 0],
[ 0, 0, 1, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0]], dtype=int8)
For example, this indicates that node 10 has flow going along three links
that are attached to it. The link to the East has no flow, the link to the
North has flow going against the topologic direction, the link to the West
has flow going with the topologic direction, and the link to the South has
flow going against the topologic direction.
In many use cases, one might want to know which links are bringing flow into
or out of the node. The flow director attribute ``flow_link_incoming_at_node``
provides this information. Here -1 means that flow is outgoing from the node
and 1 means it is incoming.
>>> fd_2.flow_link_incoming_at_node()
array([[ 0, 0, 0, 0],
[ 0, 1, 0, 0],
[ 0, 1, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, 1, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[-1, 0, 0, 0],
[ 0, 1, 1, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[-1, 0, 0, 0],
[ 0, 0, 1, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0]], dtype=int8)
So if one wanted to identify the source nodes at node, you would do the
following:
>>> np.where(
... fd_2.flow_link_incoming_at_node() == 1, mg_2.adjacent_nodes_at_node, -1
... )
array([[-1, -1, -1, -1],
[-1, 5, -1, -1],
[-1, 6, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, 10, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, 14, 9, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, -1, 13, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1],
[-1, -1, -1, -1]])
The flow directors also have the ability to return the flow receiver nodes
>>> receiver = fd.direct_flow()
>>> receiver
array([0, 1, 2,
3, 1, 5,
6, 7, 8])
For the second example we will use a Hexagonal Model Grid, a special type
of Voroni Grid that has regularly spaced hexagonal cells.
>>> from landlab import HexModelGrid
>>> mg = HexModelGrid((5, 3))
>>> _ = mg.add_field(
... "topographic__elevation",
... mg.node_x + np.round(mg.node_y),
... at="node",
... )
>>> fd = FlowDirectorSteepest(mg, "topographic__elevation")
>>> fd.surface_values
array([ 1. , 2. , 3. ,
1.5, 2.5, 3.5, 4.5,
2. , 3. , 4. , 5. , 6. ,
3.5, 4.5, 5.5, 6.5,
4. , 5. , 6. ])
>>> fd.run_one_step()
>>> mg.at_node["flow__receiver_node"]
array([ 0, 1, 2,
3, 0, 1, 6,
7, 3, 4, 5, 11,
12, 8, 9, 15,
16, 17, 18])
>>> mg.at_node["topographic__steepest_slope"]
array([ 0. , 0. , 0. ,
0. , 1.5, 1.5, 0. ,
0. , 1.5, 1.5, 1.5, 0. ,
0. , 1.5, 1.5, 0. ,
0. , 0. , 0. ])
>>> mg.at_node["flow__link_to_receiver_node"]
array([-1, -1, -1,
-1, 3, 5, -1,
-1, 12, 14, 16, -1,
-1, 25, 27, -1,
-1, -1, -1])
>>> mg.at_node["flow__sink_flag"].astype(int)
array([1, 1, 1,
1, 0, 0, 1,
1, 0, 0, 0, 1,
1, 0, 0, 1,
1, 1, 1])
>>> receiver = fd.direct_flow()
>>> receiver
array([ 0, 1, 2,
3, 0, 1, 6,
7, 3, 4, 5, 11,
12, 8, 9, 15,
16, 17, 18])
References
----------
**Required Software Citation(s) Specific to this Component**
None Listed
**Additional References**
None Listed
"""
_name = "FlowDirectorSteepest"
_unit_agnostic = True
_info = {
"flow__link_direction": {
"dtype": np.int8,
"intent": "out",
"optional": False,
"units": "-",
"mapping": "link",
"doc": (
"Direction of flow on link. A value of -1 indicates that "
"water flow goes from head node to tail node, while a value "
"of 1 indicates that water flow goes from tail node to head node."
),
},
"flow__link_to_receiver_node": {
"dtype": int,
"intent": "out",
"optional": False,
"units": "-",
"mapping": "node",
"doc": "ID of link downstream of each node, which carries the discharge",
},
"flow__receiver_node": {
"dtype": int,
"intent": "out",
"optional": False,
"units": "-",
"mapping": "node",
"doc": "Node array of receivers (node that receives flow from current node)",
},
"flow__sink_flag": {
"dtype": bool,
"intent": "out",
"optional": False,
"units": "-",
"mapping": "node",
"doc": "Boolean array, True at local lows",
},
"topographic__elevation": {
"dtype": float,
"intent": "in",
"optional": True,
"units": "m",
"mapping": "node",
"doc": "Land surface topographic elevation",
},
"topographic__steepest_slope": {
"dtype": float,
"intent": "out",
"optional": False,
"units": "-",
"mapping": "node",
"doc": "The steepest *downhill* slope",
},
}
[docs]
def __init__(self, grid, surface="topographic__elevation"):
"""
Parameters
----------
grid : ModelGrid
A grid.
surface : field name at node or array of length node, optional
The surface to direct flow across, default is field at node:
topographic__elevation,.
"""
self._method = "D4"
super().__init__(grid, surface)
self._is_Voroni = isinstance(self._grid, VoronoiDelaunayGrid)
# get 'flow__link_direction' field
self._flow_link_direction = grid.at_link["flow__link_direction"]
self.updated_boundary_conditions()
[docs]
def updated_boundary_conditions(self):
"""Method to update FlowDirectorSteepest when boundary conditions
change.
Call this if boundary conditions on the grid are updated after
the component is instantiated.
"""
self._active_links = self._grid.active_links
self._activelink_tail = self._grid.node_at_link_tail[self._grid.active_links]
self._activelink_head = self._grid.node_at_link_head[self._grid.active_links]
[docs]
def run_one_step(self):
"""Find flow directions and save to the model grid.
run_one_step() checks for updated boundary conditions, calculates
slopes on links, finds baselevel nodes based on the status at node,
calculates flow directions, and saves results to the grid.
An alternative to direct_flow() is run_one_step() which does the same
things but also returns the receiver nodes not return values.
"""
self.direct_flow()
[docs]
def direct_flow(self):
"""Find flow directions, save to the model grid, and return receivers.
direct_flow() checks for updated boundary conditions, calculates
slopes on links, finds baselevel nodes based on the status at node,
calculates flow directions, saves results to the grid, and returns a
at-node array of receiver nodes. This array is stored in the grid at:
grid['node']['flow__receiver_node']
An alternative to direct_flow() is run_one_step() which does the same
things but also returns a at-node array of receiver nodes. This array
is stored in the grid at:
grid['node']['flow__receiver_node']
"""
self._check_updated_bc()
# update the surface, if it was provided as a model grid field.
self._changed_surface()
# step 1. Calculate link slopes at active links only.
all_grads = -self._grid.calc_grad_at_link(self._surface_values)
link_slope = all_grads[self._grid.active_links]
# Step 2. Find and save base level nodes.
(baselevel_nodes,) = np.where(
np.logical_or(
self._grid.status_at_node == NodeStatus.FIXED_VALUE,
self._grid.status_at_node == NodeStatus.FIXED_GRADIENT,
)
)
# Calculate flow directions
receiver, steepest_slope, sink, recvr_link = flow_direction_DN.flow_directions(
self._surface_values,
self._active_links,
self._activelink_tail,
self._activelink_head,
link_slope,
grid=self._grid,
baselevel_nodes=baselevel_nodes,
)
# Save the four ouputs of this component.
self._grid["node"]["flow__receiver_node"][:] = receiver
self._grid["node"]["topographic__steepest_slope"][:] = steepest_slope
self._grid["node"]["flow__link_to_receiver_node"][:] = recvr_link
self._grid["node"]["flow__sink_flag"][:] = np.zeros_like(receiver, dtype=bool)
self._grid["node"]["flow__sink_flag"][sink] = True
# determine link directions
self._determine_link_directions()
return receiver
def _determine_link_directions(self):
"""Determine link directions and set flow_link_direction field.
This routine is slightly different between the route-to-one and
route-to-many methods.
It works when DepressionFinderAndRouter is run.
"""
# start by re-setting all links to zero.
self._flow_link_direction[:] = 0
# identify where flow is active on links
is_active_flow_link = self._links_to_receiver != self._grid.BAD_INDEX
# make an array that says which link ID is active
active_flow_links = self._links_to_receiver[is_active_flow_link]
# for each of those links, the position is the upstream node
upstream_node_of_active_flow_link = np.where(is_active_flow_link)[0]
# get the head node
head_node_at_active_flow_link = self._grid.node_at_link_head[active_flow_links]
# if head node is upstream node = -1, else 1
self._flow_link_direction[
active_flow_links[
head_node_at_active_flow_link == upstream_node_of_active_flow_link
]
] = -1
self._flow_link_direction[
active_flow_links[
head_node_at_active_flow_link != upstream_node_of_active_flow_link
]
] = 1
[docs]
def flow_link_direction_at_node(self):
"""Return array of flow link direction at node.
This property mirrors links_at_node and indicates the relationship
between the flow direction (determined based on the elevation of nodes)
and the topologic link direction (in which the head and tail nodes are
defined based on relative position in x-y space).
It has the shape (number of nodes, maximum number of links at node).
Recall that the standard landlab link direction goes from the tail node
to the head node.
A value of zero indicates that the link does not exist or is not
active.
A value of -1 indicates that water flow based on
``flow__link_to_receiver_node`` goes from head node to tail node, while
a value of 1 indicates that water flow goes from tail node to head
node.
Examples
--------
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowDirectorSteepest
>>> mg = RasterModelGrid((3, 3))
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> _ = mg.add_field(
... "topographic__elevation",
... mg.node_x + mg.node_y,
... at="node",
... )
>>> fd = FlowDirectorSteepest(mg, "topographic__elevation")
>>> fd.run_one_step()
>>> fd.flow_link_direction_at_node()
array([[ 0, 0, 0, 0],
[ 0, -1, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0]], dtype=int8)
This method will be updated when the DepressionFinderAndRouter is run.
First, without DepressionFinderAndRouter:
>>> from landlab.components import FlowAccumulator
>>> mg1 = RasterModelGrid((5, 5))
>>> z1 = mg1.add_field(
... "topographic__elevation",
... mg1.x_of_node + 2 * mg1.y_of_node,
... at="node",
... )
>>> z1[12] -= 5
>>> mg1.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> fa1 = FlowAccumulator(mg1, flow_director="Steepest")
>>> fa1.run_one_step()
>>> fa1.flow_director.links_to_receiver
array([-1, -1, -1, -1, -1,
-1, 5, 15, 7, -1,
-1, 19, -1, 20, -1,
-1, 23, 24, 25, -1,
-1, -1, -1, -1, -1])
>>> fa1.flow_director.flow_link_direction_at_node()
array([[ 0, 0, 0, 0],
[ 0, -1, 0, 0],
[ 0, 0, 0, 0],
[ 0, -1, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, 1, 0, 0],
[ 0, 0, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 1, -1, 0, 0],
[-1, -1, 1, 1],
[ 0, -1, -1, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, 0, 0, -1],
[ 0, 0, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0]], dtype=int8)
Next with DepressionFinderAndRouter:
>>> mg2 = RasterModelGrid((5, 5))
>>> z2 = mg2.add_field(
... "topographic__elevation",
... mg2.x_of_node + 2 * mg2.y_of_node,
... at="node",
... )
>>> z2[12] -= 5
>>> mg2.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> fa2 = FlowAccumulator(
... mg2,
... flow_director="Steepest",
... depression_finder="DepressionFinderAndRouter",
... routing="D4",
... )
>>> fa2.run_one_step()
>>> fa2.flow_director.links_to_receiver
array([-1, -1, -1, -1, -1,
-1, 5, 6, 7, -1,
-1, 19, 15, 20, -1,
-1, 23, 24, 25, -1,
-1, -1, -1, -1, -1])
>>> fa2.flow_director.flow_link_direction_at_node()
array([[ 0, 0, 0, 0],
[ 0, -1, 0, 0],
[ 0, -1, 0, 0],
[ 0, -1, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, -1, 0, -1],
[ 0, 0, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 1, -1, 0, 0],
[-1, -1, 1, -1],
[ 0, -1, -1, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, 0, 0, -1],
[ 0, 0, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0]], dtype=int8)
"""
flow_link_direction_at_node = self._flow_link_direction[
self._grid.links_at_node
]
flow_to_bad = self._grid.links_at_node == self._grid.BAD_INDEX
flow_link_direction_at_node[flow_to_bad] = 0
return flow_link_direction_at_node
[docs]
def flow_link_incoming_at_node(self):
"""Return array that mirrors links at node and indicates incoming flow.
This array has the shape
(number of nodes, maximum number of links at node).
Incoming flow is indicated as 1 and outgoing as -1. 0 indicates
that no flow moves along the link or that the link does not exist.
Examples
--------
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowDirectorSteepest
>>> mg = RasterModelGrid((3, 3))
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> _ = mg.add_field(
... "topographic__elevation",
... mg.node_x + mg.node_y,
... at="node",
... )
>>> fd = FlowDirectorSteepest(mg, "topographic__elevation")
>>> fd.run_one_step()
>>> fd.flow_link_incoming_at_node()
array([[ 0, 0, 0, 0],
[ 0, 1, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, -1],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0],
[ 0, 0, 0, 0]], dtype=int8)
"""
incoming_at_node = (
self.flow_link_direction_at_node() * self._grid.link_dirs_at_node
)
return incoming_at_node
@property
def flow_link_direction(self):
"""Return array of flow link direction.
This property indicates the relationship between the flow direction
(determined based on the elevation of nodes) and the topologic link
direction (in which the head and tail nodes are defined based on
relative position in x-y space).
It has the shape (number_of_links,).
Recall that the standard landlab link direction goes from the tail node
to the head node.
A value of zero indicates that the link does not exist or is not
active.
A value of -1 indicates that water flow based on
``flow__link_to_receiver_node`` goes from head node to tail node, while
a value of 1 indicates that water flow goes from tail node to head
node.
Examples
--------
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowDirectorSteepest
>>> mg = RasterModelGrid((3, 3))
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> _ = mg.add_field(
... "topographic__elevation",
... mg.node_x + mg.node_y,
... at="node",
... )
>>> fd = FlowDirectorSteepest(mg, "topographic__elevation")
>>> fd.run_one_step()
>>> fd.flow_link_direction
array([ 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], dtype=int8)
"""
return self._flow_link_direction
[docs]
def upstream_node_at_link(self):
"""At-link array of the upstream node based on flow direction.
BAD_INDEX_VALUE is given if no upstream node is defined.
Examples
--------
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowDirectorSteepest
>>> mg = RasterModelGrid((3, 3))
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> _ = mg.add_field(
... "topographic__elevation",
... mg.node_x + mg.node_y,
... at="node",
... )
>>> fd = FlowDirectorSteepest(mg, "topographic__elevation")
>>> fd.run_one_step()
>>> fd.upstream_node_at_link()
array([-1, -1, -1, 4, -1, -1, -1, -1, -1, -1, -1, -1])
"""
out = -1 * self._grid.ones(at="link", dtype=int)
out[self._flow_link_direction == 1] = self._grid.node_at_link_tail[
self._flow_link_direction == 1
]
out[self._flow_link_direction == -1] = self._grid.node_at_link_head[
self._flow_link_direction == -1
]
return out
[docs]
def downstream_node_at_link(self):
"""At-link array of the downstream node based on flow direction.
BAD_INDEX_VALUE is given if no downstream node is defined.
Examples
--------
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowDirectorSteepest
>>> mg = RasterModelGrid((3, 3))
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, False)
>>> _ = mg.add_field(
... "topographic__elevation",
... mg.node_x + mg.node_y,
... at="node",
... )
>>> fd = FlowDirectorSteepest(mg, "topographic__elevation")
>>> fd.run_one_step()
>>> fd.downstream_node_at_link()
array([-1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1])
"""
out = -1 * self._grid.ones(at="link", dtype=int)
out[self._flow_link_direction == 1] = self._grid.node_at_link_head[
self._flow_link_direction == 1
]
out[self._flow_link_direction == -1] = self._grid.node_at_link_tail[
self._flow_link_direction == -1
]
return out
if __name__ == "__main__": # pragma: no cover
import doctest
doctest.testmod()
```