This is a useful method to force periodic boundary conditions in a `numpy`

array. This is perhaps of interest for any sort of model where a continuum is required and can be approximated by a torus.

The basic idea is to follow the official `numpy`

guide for overloading the operators which is found here.

# Assumptions / Simplifications

The key assumptions are that we will only require periodic boundary conditions where individual points in the array are selected. This is a sensible assumption and in fact if we don’t do this it creates **CHAOS** when printing the overloaded arrays by causing infinite recursions.

# Wrap function

A simple function can be written with the `mod`

function, `%`

in basic python and generalised to operate on an `n`

-dimensional tuple given a specific shape.

```
def latticeWrapIdx(index, lattice_shape):
"""returns periodic lattice index
for a given iterable index
Required Inputs:
index :: iterable :: one integer for each axis
lattice_shape :: the shape of the lattice to index to
"""
if not hasattr(index, '__iter__'): return index # handle integer slices
if len(index) != len(lattice_shape): return index # must reference a scalar
if any(type(i) == slice for i in index): return index # slices not supported
if len(index) == len(lattice_shape): # periodic indexing of scalars
mod_index = tuple(( (i%s + s)%s for i,s in zip(index, lattice_shape)))
return mod_index
raise ValueError('Unexpected index: {}'.format(index))
```

This is tested as:

```
arr = np.array([[ 11., 12., 13., 14.],
[ 21., 22., 23., 24.],
[ 31., 32., 33., 34.],
[ 41., 42., 43., 44.]])
test_vals = [[(1,1), 22.], [(3,3), 44.], [( 4, 4), 11.], # [index, expected value]
[(3,4), 41.], [(4,3), 14.], [(10,10), 33.]]
passed = all([arr[latticeWrapIdx(idx, (4,4))] == act for idx, act in test_vals])
print "Iterating test values. Result: {}".format(passed)
```

and gives the output of,

```
Iterating test values. Result: True
```

# Subclassing `numpy`

The wrapping function can be incorporated into a subclassed `np.ndarray`

as described in the link in the introduction:

```
class Periodic_Lattice(np.ndarray):
"""Creates an n-dimensional ring that joins on boundaries w/ numpy
Required Inputs
array :: np.array :: n-dim numpy array to use wrap with
Only currently supports single point selections wrapped around the boundary
"""
def __new__(cls, input_array, lattice_spacing=None):
"""__new__ is called by numpy when and explicit constructor is used:
obj = MySubClass(params) otherwise we must rely on __array_finalize
"""
# Input array is an already formed ndarray instance
# We first cast to be our class type
obj = np.asarray(input_array).view(cls)
# add the new attribute to the created instance
obj.lattice_shape = input_array.shape
obj.lattice_dim = len(input_array.shape)
obj.lattice_spacing = lattice_spacing
# Finally, we must return the newly created object:
return obj
def __getitem__(self, index):
index = self.latticeWrapIdx(index)
return super(Periodic_Lattice, self).__getitem__(index)
def __setitem__(self, index, item):
index = self.latticeWrapIdx(index)
return super(Periodic_Lattice, self).__setitem__(index, item)
def __array_finalize__(self, obj):
""" ndarray.__new__ passes __array_finalize__ the new object,
of our own class (self) as well as the object from which the view has been taken (obj).
See
http://docs.scipy.org/doc/numpy/user/basics.subclassing.html#simple-example-adding-an-extra-attribute-to-ndarray
for more info
"""
# ``self`` is a new object resulting from
# ndarray.__new__(Periodic_Lattice, ...), therefore it only has
# attributes that the ndarray.__new__ constructor gave it -
# i.e. those of a standard ndarray.
#
# We could have got to the ndarray.__new__ call in 3 ways:
# From an explicit constructor - e.g. Periodic_Lattice():
# 1. obj is None
# (we're in the middle of the Periodic_Lattice.__new__
# constructor, and self.info will be set when we return to
# Periodic_Lattice.__new__)
if obj is None: return
# 2. From view casting - e.g arr.view(Periodic_Lattice):
# obj is arr
# (type(obj) can be Periodic_Lattice)
# 3. From new-from-template - e.g lattice[:3]
# type(obj) is Periodic_Lattice
#
# Note that it is here, rather than in the __new__ method,
# that we set the default value for 'spacing', because this
# method sees all creation of default objects - with the
# Periodic_Lattice.__new__ constructor, but also with
# arr.view(Periodic_Lattice).
#
# These are in effect the default values from these operations
self.lattice_shape = getattr(obj, 'lattice_shape', obj.shape)
self.lattice_dim = getattr(obj, 'lattice_dim', len(obj.shape))
self.lattice_spacing = getattr(obj, 'lattice_spacing', None)
pass
def latticeWrapIdx(self, index):
"""returns periodic lattice index
for a given iterable index
Required Inputs:
index :: iterable :: one integer for each axis
This is NOT compatible with slicing
"""
if not hasattr(index, '__iter__'): return index # handle integer slices
if len(index) != len(self.lattice_shape): return index # must reference a scalar
if any(type(i) == slice for i in index): return index # slices not supported
if len(index) == len(self.lattice_shape): # periodic indexing of scalars
mod_index = tuple(( (i%s + s)%s for i,s in zip(index, self.lattice_shape)))
return mod_index
raise ValueError('Unexpected index: {}'.format(index))
```

# Testing

Testing demonstrates the lattice overloads correctly,

```
arr = np.array([[ 11., 12., 13., 14.],
[ 21., 22., 23., 24.],
[ 31., 32., 33., 34.],
[ 41., 42., 43., 44.]])
test_vals = [[(1,1), 22.], [(3,3), 44.], [( 4, 4), 11.], # [index, expected value]
[(3,4), 41.], [(4,3), 14.], [(10,10), 33.]]
periodic_arr = Periodic_Lattice(arr)
passed = (periodic_arr == arr).all()
passed *= all([periodic_arr[idx] == act for idx, act in test_vals])
print "Iterating test values. Result: {}".format(passed)
```

and gives the output of,

```
Iterating test values. Result: True
```

Finally, using the code provided in the initial problem we obtain:

```
True
error
error
```

# Performance Gains

For highly optimised code, where the bulk computation is offloaded to the `C++`

side of `numpy`

or some low-level library, you may notice that a large amount of computation time is spent in `__array_finalize__`

.

In this high performance context, every line of `python`

code will have an impact on the runtime and as so a good way of skipping several lines is with the following hack:

```
def __array_finalize__(self, obj):
""" ndarray.__new__ passes __array_finalize__ the new object,
of our own class (self) as well as the object from which the view has been taken (obj).
"""
if obj is None: return
try: # this is a way faster method of doing this
self.lattice_shape = obj.lattice_shape # getattr(obj, 'lattice_shape', obj.shape)
self.lattice_dim = obj.lattice_dim # getattr(obj, 'lattice_dim', len(obj.shape))
self.lattice_spacing = obj.lattice_spacing # getattr(obj, 'lattice_spacing', None)
except:
self.lattice_shape = obj.shape
self.lattice_dim = len(self.lattice_shape)
self.lattice_spacing = None
pass
```

This won’t be needed in 99% of usage cases though and and argument could be made that if this is having such an impact then the whole array should be in `C++`

itself.

If you have any ideas on improving this last section then see my StackOverflow post which is currently unanswered at the time of writing!