def _mean(a, axis=None, dtype=None, out=None, keepdims=False):
arr = asanyarray(a)
# Upgrade bool, unsigned int, and int to float64
if dtype is None and arr.dtype.kind in ['b', 'u', 'i']:
ret = um.add.reduce(arr,
axis=axis,
dtype='f8',
out=out,
keepdims=keepdims)
else:
ret = um.add.reduce(arr,
axis=axis,
dtype=dtype,
out=out,
keepdims=keepdims)
rcount = _count_reduce_items(arr, axis)
if isinstance(ret, mu.ndarray):
ret = um.true_divide(ret,
rcount,
out=ret,
casting='unsafe',
subok=False)
else:
ret = ret / float(rcount)
return ret
def atleast_2d(*arys):
"""
View inputs as arrays with at least two dimensions.
Parameters
----------
arys1, arys2, ... : array_like
One or more array-like sequences. Non-array inputs are converted
to arrays. Arrays that already have two or more dimensions are
preserved.
Returns
-------
res, res2, ... : ndarray
An array, or tuple of arrays, each with ``a.ndim >= 2``.
Copies are avoided where possible, and views with two or more
dimensions are returned.
See Also
--------
atleast_1d, atleast_3d
Examples
--------
>>> np.atleast_2d(3.0)
array([[ 3.]])
>>> x = np.arange(3.0)
>>> np.atleast_2d(x)
array([[ 0., 1., 2.]])
>>> np.atleast_2d(x).base is x
True
>>> np.atleast_2d(1, [1, 2], [[1, 2]])
[array([[1]]), array([[1, 2]]), array([[1, 2]])]
"""
res = []
for ary in arys:
ary = asanyarray(ary)
if len(ary.shape) == 0 :
result = ary.reshape(1, 1)
elif len(ary.shape) == 1 :
result = ary[newaxis, :]
else :
result = ary
res.append(result)
if len(res) == 1:
return res[0]
else:
return res
def atleast_1d(*arys):
"""
Convert inputs to arrays with at least one dimension.
Scalar inputs are converted to 1-dimensional arrays, whilst
higher-dimensional inputs are preserved.
Parameters
----------
arys1, arys2, ... : array_like
One or more input arrays.
Returns
-------
ret : ndarray
An array, or sequence of arrays, each with ``a.ndim >= 1``.
Copies are made only if necessary.
See Also
--------
atleast_2d, atleast_3d
Examples
--------
>>> np.atleast_1d(1.0)
array([ 1.])
>>> x = np.arange(9.0).reshape(3,3)
>>> np.atleast_1d(x)
array([[ 0., 1., 2.],
[ 3., 4., 5.],
[ 6., 7., 8.]])
>>> np.atleast_1d(x) is x
True
>>> np.atleast_1d(1, [3, 4])
[array([1]), array([3, 4])]
"""
res = []
for ary in arys:
ary = asanyarray(ary)
if len(ary.shape) == 0 :
result = ary.reshape(1)
else :
result = ary
res.append(result)
if len(res) == 1:
return res[0]
else:
return res
def atleast_1d(*arys):
"""
Convert inputs to arrays with at least one dimension.
Scalar inputs are converted to 1-dimensional arrays, whilst
higher-dimensional inputs are preserved.
Parameters
----------
arys1, arys2, ... : array_like
One or more input arrays.
Returns
-------
ret : ndarray
An array, or sequence of arrays, each with ``a.ndim >= 1``.
Copies are made only if necessary.
See Also
--------
atleast_2d, atleast_3d
Examples
--------
>>> np.atleast_1d(1.0)
array([ 1.])
>>> x = np.arange(9.0).reshape(3,3)
>>> np.atleast_1d(x)
array([[ 0., 1., 2.],
[ 3., 4., 5.],
[ 6., 7., 8.]])
>>> np.atleast_1d(x) is x
True
>>> np.atleast_1d(1, [3, 4])
[array([1]), array([3, 4])]
"""
res = []
for ary in arys:
ary = asanyarray(ary)
if len(ary.shape) == 0:
result = ary.reshape(1)
else:
result = ary
res.append(result)
if len(res) == 1:
return res[0]
else:
return res
def atleast_2d(*arys):
"""
View inputs as arrays with at least two dimensions.
Parameters
----------
arys1, arys2, ... : array_like
One or more array-like sequences. Non-array inputs are converted
to arrays. Arrays that already have two or more dimensions are
preserved.
Returns
-------
res, res2, ... : ndarray
An array, or tuple of arrays, each with ``a.ndim >= 2``.
Copies are avoided where possible, and views with two or more
dimensions are returned.
See Also
--------
atleast_1d, atleast_3d
Examples
--------
>>> np.atleast_2d(3.0)
array([[ 3.]])
>>> x = np.arange(3.0)
>>> np.atleast_2d(x)
array([[ 0., 1., 2.]])
>>> np.atleast_2d(x).base is x
True
>>> np.atleast_2d(1, [1, 2], [[1, 2]])
[array([[1]]), array([[1, 2]]), array([[1, 2]])]
"""
res = []
for ary in arys:
ary = asanyarray(ary)
if len(ary.shape) == 0:
result = ary.reshape(1, 1)
elif len(ary.shape) == 1:
result = ary[newaxis, :]
else:
result = ary
res.append(result)
if len(res) == 1:
return res[0]
else:
return res
def _mean(a, axis=None, dtype=None, out=None, keepdims=False):
arr = asanyarray(a)
# Upgrade bool, unsigned int, and int to float64
if dtype is None and arr.dtype.kind in ['b','u','i']:
ret = um.add.reduce(arr, axis=axis, dtype='f8',
out=out, keepdims=keepdims)
else:
ret = um.add.reduce(arr, axis=axis, dtype=dtype,
out=out, keepdims=keepdims)
rcount = _count_reduce_items(arr, axis)
if isinstance(ret, mu.ndarray):
ret = um.true_divide(ret, rcount,
out=ret, casting='unsafe', subok=False)
else:
ret = ret / float(rcount)
return ret
def _var(a, axis=None, dtype=None, out=None, ddof=0, keepdims=False):
arr = asanyarray(a)
# First compute the mean, saving 'rcount' for reuse later
if dtype is None and arr.dtype.kind in ['b', 'u', 'i']:
arrmean = um.add.reduce(arr, axis=axis, dtype='f8', keepdims=True)
else:
arrmean = um.add.reduce(arr, axis=axis, dtype=dtype, keepdims=True)
rcount = _count_reduce_items(arr, axis)
if isinstance(arrmean, mu.ndarray):
arrmean = um.true_divide(arrmean,
rcount,
out=arrmean,
casting='unsafe',
subok=False)
else:
arrmean = arrmean / float(rcount)
# arr - arrmean
x = arr - arrmean
# (arr - arrmean) ** 2
if arr.dtype.kind == 'c':
x = um.multiply(x, um.conjugate(x), out=x).real
else:
x = um.multiply(x, x, out=x)
# add.reduce((arr - arrmean) ** 2, axis)
ret = um.add.reduce(x, axis=axis, dtype=dtype, out=out, keepdims=keepdims)
# add.reduce((arr - arrmean) ** 2, axis) / (n - ddof)
if not keepdims and isinstance(rcount, mu.ndarray):
rcount = rcount.squeeze(axis=axis)
rcount -= ddof
if isinstance(ret, mu.ndarray):
ret = um.true_divide(ret,
rcount,
out=ret,
casting='unsafe',
subok=False)
else:
ret = ret / float(rcount)
return ret
def _var(a, axis=None, dtype=None, out=None, ddof=0,
keepdims=False):
arr = asanyarray(a)
# First compute the mean, saving 'rcount' for reuse later
if dtype is None and arr.dtype.kind in ['b','u','i']:
arrmean = um.add.reduce(arr, axis=axis, dtype='f8', keepdims=True)
else:
arrmean = um.add.reduce(arr, axis=axis, dtype=dtype, keepdims=True)
rcount = _count_reduce_items(arr, axis)
if isinstance(arrmean, mu.ndarray):
arrmean = um.true_divide(arrmean, rcount,
out=arrmean, casting='unsafe', subok=False)
else:
arrmean = arrmean / float(rcount)
# arr - arrmean
x = arr - arrmean
# (arr - arrmean) ** 2
if arr.dtype.kind == 'c':
x = um.multiply(x, um.conjugate(x), out=x).real
else:
x = um.multiply(x, x, out=x)
# add.reduce((arr - arrmean) ** 2, axis)
ret = um.add.reduce(x, axis=axis, dtype=dtype, out=out, keepdims=keepdims)
# add.reduce((arr - arrmean) ** 2, axis) / (n - ddof)
if not keepdims and isinstance(rcount, mu.ndarray):
rcount = rcount.squeeze(axis=axis)
rcount -= ddof
if isinstance(ret, mu.ndarray):
ret = um.true_divide(ret, rcount,
out=ret, casting='unsafe', subok=False)
else:
ret = ret / float(rcount)
return ret
def atleast_3d(*arys):
"""
View inputs as arrays with at least three dimensions.
Parameters
----------
arys1, arys2, ... : array_like
One or more array-like sequences. Non-array inputs are converted to
arrays. Arrays that already have three or more dimensions are
preserved.
Returns
-------
res1, res2, ... : ndarray
An array, or tuple of arrays, each with ``a.ndim >= 3``. Copies are
avoided where possible, and views with three or more dimensions are
returned. For example, a 1-D array of shape ``(N,)`` becomes a view
of shape ``(1, N, 1)``, and a 2-D array of shape ``(M, N)`` becomes a
view of shape ``(M, N, 1)``.
See Also
--------
atleast_1d, atleast_2d
Examples
--------
>>> np.atleast_3d(3.0)
array([[[ 3.]]])
>>> x = np.arange(3.0)
>>> np.atleast_3d(x).shape
(1, 3, 1)
>>> x = np.arange(12.0).reshape(4,3)
>>> np.atleast_3d(x).shape
(4, 3, 1)
>>> np.atleast_3d(x).base is x
True
>>> for arr in np.atleast_3d([1, 2], [[1, 2]], [[[1, 2]]]):
... print arr, arr.shape
...
[[[1]
[2]]] (1, 2, 1)
[[[1]
[2]]] (1, 2, 1)
[[[1 2]]] (1, 1, 2)
"""
res = []
for ary in arys:
ary = asanyarray(ary)
if len(ary.shape) == 0:
result = ary.reshape(1,1,1)
elif len(ary.shape) == 1:
result = ary[newaxis,:,newaxis]
elif len(ary.shape) == 2:
result = ary[:,:,newaxis]
else:
result = ary
res.append(result)
if len(res) == 1:
return res[0]
else:
return res
def atleast_3d(*arys):
"""
View inputs as arrays with at least three dimensions.
Parameters
----------
arys1, arys2, ... : array_like
One or more array-like sequences. Non-array inputs are converted to
arrays. Arrays that already have three or more dimensions are
preserved.
Returns
-------
res1, res2, ... : ndarray
An array, or tuple of arrays, each with ``a.ndim >= 3``. Copies are
avoided where possible, and views with three or more dimensions are
returned. For example, a 1-D array of shape ``(N,)`` becomes a view
of shape ``(1, N, 1)``, and a 2-D array of shape ``(M, N)`` becomes a
view of shape ``(M, N, 1)``.
See Also
--------
atleast_1d, atleast_2d
Examples
--------
>>> np.atleast_3d(3.0)
array([[[ 3.]]])
>>> x = np.arange(3.0)
>>> np.atleast_3d(x).shape
(1, 3, 1)
>>> x = np.arange(12.0).reshape(4,3)
>>> np.atleast_3d(x).shape
(4, 3, 1)
>>> np.atleast_3d(x).base is x
True
>>> for arr in np.atleast_3d([1, 2], [[1, 2]], [[[1, 2]]]):
... print arr, arr.shape
...
[[[1]
[2]]] (1, 2, 1)
[[[1]
[2]]] (1, 2, 1)
[[[1 2]]] (1, 1, 2)
"""
res = []
for ary in arys:
ary = asanyarray(ary)
if len(ary.shape) == 0:
result = ary.reshape(1, 1, 1)
elif len(ary.shape) == 1:
result = ary[newaxis, :, newaxis]
elif len(ary.shape) == 2:
result = ary[:, :, newaxis]
else:
result = ary
res.append(result)
if len(res) == 1:
return res[0]
else:
return res
def sort(a, axis=-1, kind='quicksort', order=None):
"""Return copy of 'a' sorted along the given axis.
*Description*
Perform an inplace sort along the given axis using the algorithm specified
by the kind keyword.
*Parameters*:
a : array type
Array to be sorted.
axis : integer
Axis to be sorted along. None indicates that the flattened array
should be used. Default is -1.
kind : string
Sorting algorithm to use. Possible values are 'quicksort',
'mergesort', or 'heapsort'. Default is 'quicksort'.
order : list type or None
When a is an array with fields defined, this argument specifies
which fields to compare first, second, etc. Not all fields need be
specified.
*Returns*:
sorted_array : type is unchanged.
*SeeAlso*:
argsort
Indirect sort
lexsort
Indirect stable sort on multiple keys
searchsorted
Find keys in sorted array
*Notes*
The various sorts are characterized by average speed, worst case
performance, need for work space, and whether they are stable. A stable
sort keeps items with the same key in the same relative order. The
three available algorithms have the following properties:
+-----------+-------+-------------+------------+-------+
| kind | speed | worst case | work space | stable|
+===========+=======+=============+============+=======+
| quicksort | 1 | O(n^2) | 0 | no |
+-----------+-------+-------------+------------+-------+
| mergesort | 2 | O(n*log(n)) | ~n/2 | yes |
+-----------+-------+-------------+------------+-------+
| heapsort | 3 | O(n*log(n)) | 0 | no |
+-----------+-------+-------------+------------+-------+
All the sort algorithms make temporary copies of the data when the sort
is not along the last axis. Consequently, sorts along the last axis are
faster and use less space than sorts along other axis.
"""
if axis is None:
a = asanyarray(a).flatten()
axis = 0
else:
a = asanyarray(a).copy()
a.sort(axis, kind, order)
return a
def matrix_power(M,n):
"""
Raise a square matrix to the (integer) power n.
For positive integers n, the power is computed by repeated matrix
squarings and matrix multiplications. If n=0, the identity matrix
of the same type as M is returned. If n<0, the inverse is computed
and raised to the exponent.
Parameters
----------
M : array_like
Must be a square array (that is, of dimension two and with
equal sizes).
n : integer
The exponent can be any integer or long integer, positive
negative or zero.
Returns
-------
M to the power n
The return value is a an array the same shape and size as M;
if the exponent was positive or zero then the type of the
elements is the same as those of M. If the exponent was negative
the elements are floating-point.
Raises
------
LinAlgException
If the matrix is not numerically invertible, an exception is raised.
See Also
--------
The matrix() class provides an equivalent function as the exponentiation
operator.
Examples
--------
>>> np.linalg.matrix_power(np.array([[0,1],[-1,0]]),10)
array([[-1, 0],
[ 0, -1]])
"""
M = asanyarray(M)
if len(M.shape) != 2 or M.shape[0] != M.shape[1]:
raise ValueError("input must be a square array")
if not issubdtype(type(n),int):
raise TypeError("exponent must be an integer")
from numpy.linalg import inv
if n==0:
M = M.copy()
M[:] = identity(M.shape[0])
return M
elif n<0:
M = inv(M)
n *= -1
result = M
if n <= 3:
for _ in range(n-1):
result=N.dot(result,M)
return result
# binary decomposition to reduce the number of Matrix
# multiplications for n > 3.
beta = binary_repr(n)
Z,q,t = M,0,len(beta)
while beta[t-q-1] == '0':
Z = N.dot(Z,Z)
q += 1
result = Z
for k in range(q+1,t):
Z = N.dot(Z,Z)
if beta[t-k-1] == '1':
result = N.dot(result,Z)
return result
def matrix_power(M, n):
"""
Raise a square matrix to the (integer) power n.
For positive integers n, the power is computed by repeated matrix
squarings and matrix multiplications. If n=0, the identity matrix
of the same type as M is returned. If n<0, the inverse is computed
and raised to the exponent.
Parameters
----------
M : array_like
Must be a square array (that is, of dimension two and with
equal sizes).
n : integer
The exponent can be any integer or long integer, positive
negative or zero.
Returns
-------
M to the power n
The return value is a an array the same shape and size as M;
if the exponent was positive or zero then the type of the
elements is the same as those of M. If the exponent was negative
the elements are floating-point.
Raises
------
LinAlgException
If the matrix is not numerically invertible, an exception is raised.
See Also
--------
The matrix() class provides an equivalent function as the exponentiation
operator.
Examples
--------
>>> np.linalg.matrix_power(np.array([[0,1],[-1,0]]),10)
array([[-1, 0],
[ 0, -1]])
"""
M = asanyarray(M)
if len(M.shape) != 2 or M.shape[0] != M.shape[1]:
raise ValueError("input must be a square array")
if not issubdtype(type(n), int):
raise TypeError("exponent must be an integer")
from numpy.linalg import inv
if n == 0:
M = M.copy()
M[:] = identity(M.shape[0])
return M
elif n < 0:
M = inv(M)
n *= -1
result = M
if n <= 3:
for _ in range(n - 1):
result = N.dot(result, M)
return result
# binary decomposition to reduce the number of Matrix
# multiplications for n > 3.
beta = binary_repr(n)
Z, q, t = M, 0, len(beta)
while beta[t - q - 1] == '0':
Z = N.dot(Z, Z)
q += 1
result = Z
for k in range(q + 1, t):
Z = N.dot(Z, Z)
if beta[t - k - 1] == '1':
result = N.dot(result, Z)
return result