from __future__ import division, absolute_import, print_function
import sys
import math
import numpy.core.numeric as _nx
from numpy.core.numeric import (
asarray, ScalarType, array, alltrue, cumprod, arange
)
from numpy.core.numerictypes import find_common_type, issubdtype
from . import function_base
import numpy.matrixlib as matrixlib
from .function_base import diff
from numpy.core.multiarray import ravel_multi_index, unravel_index
from numpy.lib.stride_tricks import as_strided
__all__ = [
'ravel_multi_index', 'unravel_index', 'mgrid', 'ogrid', 'r_', 'c_',
's_', 'index_exp', 'ix_', 'ndenumerate', 'ndindex', 'fill_diagonal',
'diag_indices', 'diag_indices_from'
]
def ix_(*args):
"""
Construct an open mesh from multiple sequences.
This function takes N 1-D sequences and returns N outputs with N
dimensions each, such that the shape is 1 in all but one dimension
and the dimension with the non-unit shape value cycles through all
N dimensions.
Using `ix_` one can quickly construct index arrays that will index
the cross product. ``a[np.ix_([1,3],[2,5])]`` returns the array
``[[a[1,2] a[1,5]], [a[3,2] a[3,5]]]``.
Parameters
----------
args : 1-D sequences
Each sequence should be of integer or boolean type.
Boolean sequences will be interpreted as boolean masks for the
corresponding dimension (equivalent to passing in
``np.nonzero(boolean_sequence)``).
Returns
-------
out : tuple of ndarrays
N arrays with N dimensions each, with N the number of input
sequences. Together these arrays form an open mesh.
See Also
--------
ogrid, mgrid, meshgrid
Examples
--------
>>> a = np.arange(10).reshape(2, 5)
>>> a
array([[0, 1, 2, 3, 4],
[5, 6, 7, 8, 9]])
>>> ixgrid = np.ix_([0, 1], [2, 4])
>>> ixgrid
(array([[0],
[1]]), array([[2, 4]]))
>>> ixgrid[0].shape, ixgrid[1].shape
((2, 1), (1, 2))
>>> a[ixgrid]
array([[2, 4],
[7, 9]])
>>> ixgrid = np.ix_([True, True], [2, 4])
>>> a[ixgrid]
array([[2, 4],
[7, 9]])
>>> ixgrid = np.ix_([True, True], [False, False, True, False, True])
>>> a[ixgrid]
array([[2, 4],
[7, 9]])
"""
out = []
nd = len(args)
for k, new in enumerate(args):
new = asarray(new)
if new.ndim != 1:
raise ValueError("Cross index must be 1 dimensional")
if new.size == 0:
# Explicitly type empty arrays to avoid float default
new = new.astype(_nx.intp)
if issubdtype(new.dtype, _nx.bool_):
new, = new.nonzero()
new = new.reshape((1,)*k + (new.size,) + (1,)*(nd-k-1))
out.append(new)
return tuple(out)
class nd_grid(object):
"""
Construct a multi-dimensional "meshgrid".
``grid = nd_grid()`` creates an instance which will return a mesh-grid
when indexed. The dimension and number of the output arrays are equal
to the number of indexing dimensions. If the step length is not a
complex number, then the stop is not inclusive.
However, if the step length is a **complex number** (e.g. 5j), then the
integer part of its magnitude is interpreted as specifying the
number of points to create between the start and stop values, where
the stop value **is inclusive**.
If instantiated with an argument of ``sparse=True``, the mesh-grid is
open (or not fleshed out) so that only one-dimension of each returned
argument is greater than 1.
Parameters
----------
sparse : bool, optional
Whether the grid is sparse or not. Default is False.
Notes
-----
Two instances of `nd_grid` are made available in the NumPy namespace,
`mgrid` and `ogrid`::
mgrid = nd_grid(sparse=False)
ogrid = nd_grid(sparse=True)
Users should use these pre-defined instances instead of using `nd_grid`
directly.
Examples
--------
>>> mgrid = np.lib.index_tricks.nd_grid()
>>> mgrid[0:5,0:5]
array([[[0, 0, 0, 0, 0],
[1, 1, 1, 1, 1],
[2, 2, 2, 2, 2],
[3, 3, 3, 3, 3],
[4, 4, 4, 4, 4]],
[[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4]]])
>>> mgrid[-1:1:5j]
array([-1. , -0.5, 0. , 0.5, 1. ])
>>> ogrid = np.lib.index_tricks.nd_grid(sparse=True)
>>> ogrid[0:5,0:5]
[array([[0],
[1],
[2],
[3],
[4]]), array([[0, 1, 2, 3, 4]])]
"""
def __init__(self, sparse=False):
self.sparse = sparse
def __getitem__(self, key):
try:
size = []
typ = int
for k in range(len(key)):
step = key[k].step
start = key[k].start
if start is None:
start = 0
if step is None:
step = 1
if isinstance(step, complex):
size.append(int(abs(step)))
typ = float
else:
size.append(
int(math.ceil((key[k].stop - start)/(step*1.0))))
if (isinstance(step, float) or
isinstance(start, float) or
isinstance(key[k].stop, float)):
typ = float
if self.sparse:
nn = [_nx.arange(_x, dtype=_t)
for _x, _t in zip(size, (typ,)*len(size))]
else:
nn = _nx.indices(size, typ)
for k in range(len(size)):
step = key[k].step
start = key[k].start
if start is None:
start = 0
if step is None:
step = 1
if isinstance(step, complex):
step = int(abs(step))
if step != 1:
step = (key[k].stop - start)/float(step-1)
nn[k] = (nn[k]*step+start)
if self.sparse:
slobj = [_nx.newaxis]*len(size)
for k in range(len(size)):
slobj[k] = slice(None, None)
nn[k] = nn[k][slobj]
slobj[k] = _nx.newaxis
return nn
except (IndexError, TypeError):
step = key.step
stop = key.stop
start = key.start
if start is None:
start = 0
if isinstance(step, complex):
step = abs(step)
length = int(step)
if step != 1:
step = (key.stop-start)/float(step-1)
stop = key.stop + step
return _nx.arange(0, length, 1, float)*step + start
else:
return _nx.arange(start, stop, step)
def __len__(self):
return 0
mgrid = nd_grid(sparse=False)
ogrid = nd_grid(sparse=True)
mgrid.__doc__ = None # set in numpy.add_newdocs
ogrid.__doc__ = None # set in numpy.add_newdocs
class AxisConcatenator(object):
"""
Translates slice objects to concatenation along an axis.
For detailed documentation on usage, see `r_`.
"""
# allow ma.mr_ to override this
concatenate = staticmethod(_nx.concatenate)
makemat = staticmethod(matrixlib.matrix)
def __init__(self, axis=0, matrix=False, ndmin=1, trans1d=-1):
self.axis = axis
self.matrix = matrix
self.trans1d = trans1d
self.ndmin = ndmin
def __getitem__(self, key):
# handle matrix builder syntax
if isinstance(key, str):
frame = sys._getframe().f_back
mymat = matrixlib.bmat(key, frame.f_globals, frame.f_locals)
return mymat
if not isinstance(key, tuple):
key = (key,)
# copy attributes, since they can be overridden in the first argument
trans1d = self.trans1d
ndmin = self.ndmin
matrix = self.matrix
axis = self.axis
objs = []
scalars = []
arraytypes = []
scalartypes = []
for k, item in enumerate(key):
scalar = False
if isinstance(item, slice):
step = item.step
start = item.start
stop = item.stop
if start is None:
start = 0
if step is None:
step = 1
if isinstance(step, complex):
size = int(abs(step))
newobj = function_base.linspace(start, stop, num=size)
else:
newobj = _nx.arange(start, stop, step)
if ndmin > 1:
newobj = array(newobj, copy=False, ndmin=ndmin)
if trans1d != -1:
newobj = newobj.swapaxes(-1, trans1d)
elif isinstance(item, str):
if k != 0:
raise ValueError("special directives must be the "
"first entry.")
if item in ('r', 'c'):
matrix = True
col = (item == 'c')
continue
if ',' in item:
vec = item.split(',')
try:
axis, ndmin = [int(x) for x in vec[:2]]
if len(vec) == 3:
trans1d = int(vec[2])
continue
except Exception:
raise ValueError("unknown special directive")
try:
axis = int(item)
continue
except (ValueError, TypeError):
raise ValueError("unknown special directive")
elif type(item) in ScalarType:
newobj = array(item, ndmin=ndmin)
scalars.append(len(objs))
scalar = True
scalartypes.append(newobj.dtype)
else:
newobj = item
if ndmin > 1:
tempobj = array(newobj, copy=False, subok=True)
newobj = array(newobj, copy=False, subok=True,
ndmin=ndmin)
if trans1d != -1 and tempobj.ndim < ndmin:
k2 = ndmin-tempobj.ndim
if (trans1d < 0):
trans1d += k2 + 1
defaxes = list(range(ndmin))
k1 = trans1d
axes = defaxes[:k1] + defaxes[k2:] + \
defaxes[k1:k2]
newobj = newobj.transpose(axes)
del tempobj
objs.append(newobj)
if not scalar and isinstance(newobj, _nx.ndarray):
arraytypes.append(newobj.dtype)
# Ensure that scalars won't up-cast unless warranted
final_dtype = find_common_type(arraytypes, scalartypes)
if final_dtype is not None:
for k in scalars:
objs[k] = objs[k].astype(final_dtype)
res = self.concatenate(tuple(objs), axis=axis)
if matrix:
oldndim = res.ndim
res = self.makemat(res)
if oldndim == 1 and col:
res = res.T
return res
def __len__(self):
return 0
# separate classes are used here instead of just making r_ = concatentor(0),
# etc. because otherwise we couldn't get the doc string to come out right
# in help(r_)
class RClass(AxisConcatenator):
"""
Translates slice objects to concatenation along the first axis.
This is a simple way to build up arrays quickly. There are two use cases.
1. If the index expression contains comma separated arrays, then stack
them along their first axis.
2. If the index expression contains slice notation or scalars then create
a 1-D array with a range indicated by the slice notation.
If slice notation is used, the syntax ``start:stop:step`` is equivalent
to ``np.arange(start, stop, step)`` inside of the brackets. However, if
``step`` is an imaginary number (i.e. 100j) then its integer portion is
interpreted as a number-of-points desired and the start and stop are
inclusive. In other words ``start:stop:stepj`` is interpreted as
``np.linspace(start, stop, step, endpoint=1)`` inside of the brackets.
After expansion of slice notation, all comma separated sequences are
concatenated together.
Optional character strings placed as the first element of the index
expression can be used to change the output. The strings 'r' or 'c' result
in matrix output. If the result is 1-D and 'r' is specified a 1 x N (row)
matrix is produced. If the result is 1-D and 'c' is specified, then a N x 1
(column) matrix is produced. If the result is 2-D then both provide the
same matrix result.
A string integer specifies which axis to stack multiple comma separated
arrays along. A string of two comma-separated integers allows indication
of the minimum number of dimensions to force each entry into as the
second integer (the axis to concatenate along is still the first integer).
A string with three comma-separated integers allows specification of the
axis to concatenate along, the minimum number of dimensions to force the
entries to, and which axis should contain the start of the arrays which
are less than the specified number of dimensions. In other words the third
integer allows you to specify where the 1's should be placed in the shape
of the arrays that have their shapes upgraded. By default, they are placed
in the front of the shape tuple. The third argument allows you to specify
where the start of the array should be instead. Thus, a third argument of
'0' would place the 1's at the end of the array shape. Negative integers
specify where in the new shape tuple the last dimension of upgraded arrays
should be placed, so the default is '-1'.
Parameters
----------
Not a function, so takes no parameters
Returns
-------
A concatenated ndarray or matrix.
See Also
--------
concatenate : Join a sequence of arrays along an existing axis.
c_ : Translates slice objects to concatenation along the second axis.
Examples
--------
>>> np.r_[np.array([1,2,3]), 0, 0, np.array([4,5,6])]
array([1, 2, 3, 0, 0, 4, 5, 6])
>>> np.r_[-1:1:6j, [0]*3, 5, 6]
array([-1. , -0.6, -0.2, 0.2, 0.6, 1. , 0. , 0. , 0. , 5. , 6. ])
String integers specify the axis to concatenate along or the minimum
number of dimensions to force entries into.
>>> a = np.array([[0, 1, 2], [3, 4, 5]])
>>> np.r_['-1', a, a] # concatenate along last axis
array([[0, 1, 2, 0, 1, 2],
[3, 4, 5, 3, 4, 5]])
>>> np.r_['0,2', [1,2,3], [4,5,6]] # concatenate along first axis, dim>=2
array([[1, 2, 3],
[4, 5, 6]])
>>> np.r_['0,2,0', [1,2,3], [4,5,6]]
array([[1],
[2],
[3],
[4],
[5],
[6]])
>>> np.r_['1,2,0', [1,2,3], [4,5,6]]
array([[1, 4],
[2, 5],
[3, 6]])
Using 'r' or 'c' as a first string argument creates a matrix.
>>> np.r_['r',[1,2,3], [4,5,6]]
matrix([[1, 2, 3, 4, 5, 6]])
"""
def __init__(self):
AxisConcatenator.__init__(self, 0)
r_ = RClass()
class CClass(AxisConcatenator):
"""
Translates slice objects to concatenation along the second axis.
This is short-hand for ``np.r_['-1,2,0', index expression]``, which is
useful because of its common occurrence. In particular, arrays will be
stacked along their last axis after being upgraded to at least 2-D with
1's post-pended to the shape (column vectors made out of 1-D arrays).
See Also
--------
column_stack : Stack 1-D arrays as columns into a 2-D array.
r_ : For more detailed documentation.
Examples
--------
>>> np.c_[np.array([1,2,3]), np.array([4,5,6])]
array([[1, 4],
[2, 5],
[3, 6]])
>>> np.c_[np.array([[1,2,3]]), 0, 0, np.array([[4,5,6]])]
array([[1, 2, 3, 0, 0, 4, 5, 6]])
"""
def __init__(self):
AxisConcatenator.__init__(self, -1, ndmin=2, trans1d=0)
c_ = CClass()
class ndenumerate(object):
"""
Multidimensional index iterator.
Return an iterator yielding pairs of array coordinates and values.
Parameters
----------
arr : ndarray
Input array.
See Also
--------
ndindex, flatiter
Examples
--------
>>> a = np.array([[1, 2], [3, 4]])
>>> for index, x in np.ndenumerate(a):
... print(index, x)
(0, 0) 1
(0, 1) 2
(1, 0) 3
(1, 1) 4
"""
def __init__(self, arr):
self.iter = asarray(arr).flat
def __next__(self):
"""
Standard iterator method, returns the index tuple and array value.
Returns
-------
coords : tuple of ints
The indices of the current iteration.
val : scalar
The array element of the current iteration.
"""
return self.iter.coords, next(self.iter)
def __iter__(self):
return self
next = __next__
class ndindex(object):
"""
An N-dimensional iterator object to index arrays.
Given the shape of an array, an `ndindex` instance iterates over
the N-dimensional index of the array. At each iteration a tuple
of indices is returned, the last dimension is iterated over first.
Parameters
----------
`*args` : ints
The size of each dimension of the array.
See Also
--------
ndenumerate, flatiter
Examples
--------
>>> for index in np.ndindex(3, 2, 1):
... print(index)
(0, 0, 0)
(0, 1, 0)
(1, 0, 0)
(1, 1, 0)
(2, 0, 0)
(2, 1, 0)
"""
def __init__(self, *shape):
if len(shape) == 1 and isinstance(shape[0], tuple):
shape = shape[0]
x = as_strided(_nx.zeros(1), shape=shape,
strides=_nx.zeros_like(shape))
self._it = _nx.nditer(x, flags=['multi_index', 'zerosize_ok'],
order='C')
def __iter__(self):
return self
def ndincr(self):
"""
Increment the multi-dimensional index by one.
This method is for backward compatibility only: do not use.
"""
next(self)
def __next__(self):
"""
Standard iterator method, updates the index and returns the index
tuple.
Returns
-------
val : tuple of ints
Returns a tuple containing the indices of the current
iteration.
"""
next(self._it)
return self._it.multi_index
next = __next__
# You can do all this with slice() plus a few special objects,
# but there's a lot to remember. This version is simpler because
# it uses the standard array indexing syntax.
#
# Written by Konrad Hinsen <hinsen@cnrs-orleans.fr>
# last revision: 1999-7-23
#
# Cosmetic changes by T. Oliphant 2001
#
#
class IndexExpression(object):
"""
A nicer way to build up index tuples for arrays.
.. note::
Use one of the two predefined instances `index_exp` or `s_`
rather than directly using `IndexExpression`.
For any index combination, including slicing and axis insertion,
``a[indices]`` is the same as ``a[np.index_exp[indices]]`` for any
array `a`. However, ``np.index_exp[indices]`` can be used anywhere
in Python code and returns a tuple of slice objects that can be
used in the construction of complex index expressions.
Parameters
----------
maketuple : bool
If True, always returns a tuple.
See Also
--------
index_exp : Predefined instance that always returns a tuple:
`index_exp = IndexExpression(maketuple=True)`.
s_ : Predefined instance without tuple conversion:
`s_ = IndexExpression(maketuple=False)`.
Notes
-----
You can do all this with `slice()` plus a few special objects,
but there's a lot to remember and this version is simpler because
it uses the standard array indexing syntax.
Examples
--------
>>> np.s_[2::2]
slice(2, None, 2)
>>> np.index_exp[2::2]
(slice(2, None, 2),)
>>> np.array([0, 1, 2, 3, 4])[np.s_[2::2]]
array([2, 4])
"""
def __init__(self, maketuple):
self.maketuple = maketuple
def __getitem__(self, item):
if self.maketuple and not isinstance(item, tuple):
return (item,)
else:
return item
index_exp = IndexExpression(maketuple=True)
s_ = IndexExpression(maketuple=False)
# End contribution from Konrad.
# The following functions complement those in twodim_base, but are
# applicable to N-dimensions.
def fill_diagonal(a, val, wrap=False):
"""Fill the main diagonal of the given array of any dimensionality.
For an array `a` with ``a.ndim >= 2``, the diagonal is the list of
locations with indices ``a[i, ..., i]`` all identical. This function
modifies the input array in-place, it does not return a value.
Parameters
----------
a : array, at least 2-D.
Array whose diagonal is to be filled, it gets modified in-place.
val : scalar
Value to be written on the diagonal, its type must be compatible with
that of the array a.
wrap : bool
For tall matrices in NumPy version up to 1.6.2, the
diagonal "wrapped" after N columns. You can have this behavior
with this option. This affects only tall matrices.
See also
--------
diag_indices, diag_indices_from
Notes
-----
.. versionadded:: 1.4.0
This functionality can be obtained via `diag_indices`, but internally
this version uses a much faster implementation that never constructs the
indices and uses simple slicing.
Examples
--------
>>> a = np.zeros((3, 3), int)
>>> np.fill_diagonal(a, 5)
>>> a
array([[5, 0, 0],
[0, 5, 0],
[0, 0, 5]])
The same function can operate on a 4-D array:
>>> a = np.zeros((3, 3, 3, 3), int)
>>> np.fill_diagonal(a, 4)
We only show a few blocks for clarity:
>>> a[0, 0]
array([[4, 0, 0],
[0, 0, 0],
[0, 0, 0]])
>>> a[1, 1]
array([[0, 0, 0],
[0, 4, 0],
[0, 0, 0]])
>>> a[2, 2]
array([[0, 0, 0],
[0, 0, 0],
[0, 0, 4]])
The wrap option affects only tall matrices:
>>> # tall matrices no wrap
>>> a = np.zeros((5, 3),int)
>>> fill_diagonal(a, 4)
>>> a
array([[4, 0, 0],
[0, 4, 0],
[0, 0, 4],
[0, 0, 0],
[0, 0, 0]])
>>> # tall matrices wrap
>>> a = np.zeros((5, 3),int)
>>> fill_diagonal(a, 4, wrap=True)
>>> a
array([[4, 0, 0],
[0, 4, 0],
[0, 0, 4],
[0, 0, 0],
[4, 0, 0]])
>>> # wide matrices
>>> a = np.zeros((3, 5),int)
>>> fill_diagonal(a, 4, wrap=True)
>>> a
array([[4, 0, 0, 0, 0],
[0, 4, 0, 0, 0],
[0, 0, 4, 0, 0]])
"""
if a.ndim < 2:
raise ValueError("array must be at least 2-d")
end = None
if a.ndim == 2:
# Explicit, fast formula for the common case. For 2-d arrays, we
# accept rectangular ones.
step = a.shape[1] + 1
#This is needed to don't have tall matrix have the diagonal wrap.
if not wrap:
end = a.shape[1] * a.shape[1]
else:
# For more than d=2, the strided formula is only valid for arrays with
# all dimensions equal, so we check first.
if not alltrue(diff(a.shape) == 0):
raise ValueError("All dimensions of input must be of equal length")
step = 1 + (cumprod(a.shape[:-1])).sum()
# Write the value out into the diagonal.
a.flat[:end:step] = val
def diag_indices(n, ndim=2):
"""
Return the indices to access the main diagonal of an array.
This returns a tuple of indices that can be used to access the main
diagonal of an array `a` with ``a.ndim >= 2`` dimensions and shape
(n, n, ..., n). For ``a.ndim = 2`` this is the usual diagonal, for
``a.ndim > 2`` this is the set of indices to access ``a[i, i, ..., i]``
for ``i = [0..n-1]``.
Parameters
----------
n : int
The size, along each dimension, of the arrays for which the returned
indices can be used.
ndim : int, optional
The number of dimensions.
See also
--------
diag_indices_from
Notes
-----
.. versionadded:: 1.4.0
Examples
--------
Create a set of indices to access the diagonal of a (4, 4) array:
>>> di = np.diag_indices(4)
>>> di
(array([0, 1, 2, 3]), array([0, 1, 2, 3]))
>>> a = np.arange(16).reshape(4, 4)
>>> a
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
>>> a[di] = 100
>>> a
array([[100, 1, 2, 3],
[ 4, 100, 6, 7],
[ 8, 9, 100, 11],
[ 12, 13, 14, 100]])
Now, we create indices to manipulate a 3-D array:
>>> d3 = np.diag_indices(2, 3)
>>> d3
(array([0, 1]), array([0, 1]), array([0, 1]))
And use it to set the diagonal of an array of zeros to 1:
>>> a = np.zeros((2, 2, 2), dtype=int)
>>> a[d3] = 1
>>> a
array([[[1, 0],
[0, 0]],
[[0, 0],
[0, 1]]])
"""
idx = arange(n)
return (idx,) * ndim
def diag_indices_from(arr):
"""
Return the indices to access the main diagonal of an n-dimensional array.
See `diag_indices` for full details.
Parameters
----------
arr : array, at least 2-D
See Also
--------
diag_indices
Notes
-----
.. versionadded:: 1.4.0
"""
if not arr.ndim >= 2:
raise ValueError("input array must be at least 2-d")
# For more than d=2, the strided formula is only valid for arrays with
# all dimensions equal, so we check first.
if not alltrue(diff(arr.shape) == 0):
raise ValueError("All dimensions of input must be of equal length")
return diag_indices(arr.shape[0], arr.ndim)