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aoc-2022/venv/Lib/site-packages/numpy/core/numeric.py

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Give it up for day 1 of smooth brain shenanigans! https://pbs.twimg.com/media/E1bxikWWEAEGYHU.png
2022-12-01 16:50:29 +00:00
import functools
import itertools
import operator
import sys
import warnings
import numbers
import numpy as np
from . import multiarray
from .multiarray import (
_fastCopyAndTranspose as fastCopyAndTranspose, ALLOW_THREADS,
BUFSIZE, CLIP, MAXDIMS, MAY_SHARE_BOUNDS, MAY_SHARE_EXACT, RAISE,
WRAP, arange, array, asarray, asanyarray, ascontiguousarray,
asfortranarray, broadcast, can_cast, compare_chararrays,
concatenate, copyto, dot, dtype, empty,
empty_like, flatiter, frombuffer, from_dlpack, fromfile, fromiter,
fromstring, inner, lexsort, matmul, may_share_memory,
min_scalar_type, ndarray, nditer, nested_iters, promote_types,
putmask, result_type, set_numeric_ops, shares_memory, vdot, where,
zeros, normalize_axis_index)
from . import overrides
from . import umath
from . import shape_base
from .overrides import set_array_function_like_doc, set_module
from .umath import (multiply, invert, sin, PINF, NAN)
from . import numerictypes
from .numerictypes import longlong, intc, int_, float_, complex_, bool_
from ._exceptions import TooHardError, AxisError
from ._ufunc_config import errstate
bitwise_not = invert
ufunc = type(sin)
newaxis = None
array_function_dispatch = functools.partial(
overrides.array_function_dispatch, module='numpy')
__all__ = [
'newaxis', 'ndarray', 'flatiter', 'nditer', 'nested_iters', 'ufunc',
'arange', 'array', 'asarray', 'asanyarray', 'ascontiguousarray',
'asfortranarray', 'zeros', 'count_nonzero', 'empty', 'broadcast', 'dtype',
'fromstring', 'fromfile', 'frombuffer', 'from_dlpack', 'where',
'argwhere', 'copyto', 'concatenate', 'fastCopyAndTranspose', 'lexsort',
'set_numeric_ops', 'can_cast', 'promote_types', 'min_scalar_type',
'result_type', 'isfortran', 'empty_like', 'zeros_like', 'ones_like',
'correlate', 'convolve', 'inner', 'dot', 'outer', 'vdot', 'roll',
'rollaxis', 'moveaxis', 'cross', 'tensordot', 'little_endian',
'fromiter', 'array_equal', 'array_equiv', 'indices', 'fromfunction',
'isclose', 'isscalar', 'binary_repr', 'base_repr', 'ones',
'identity', 'allclose', 'compare_chararrays', 'putmask',
'flatnonzero', 'Inf', 'inf', 'infty', 'Infinity', 'nan', 'NaN',
'False_', 'True_', 'bitwise_not', 'CLIP', 'RAISE', 'WRAP', 'MAXDIMS',
'BUFSIZE', 'ALLOW_THREADS', 'ComplexWarning', 'full', 'full_like',
'matmul', 'shares_memory', 'may_share_memory', 'MAY_SHARE_BOUNDS',
'MAY_SHARE_EXACT', 'TooHardError', 'AxisError']
@set_module('numpy')
class ComplexWarning(RuntimeWarning):
"""
The warning raised when casting a complex dtype to a real dtype.
As implemented, casting a complex number to a real discards its imaginary
part, but this behavior may not be what the user actually wants.
"""
pass
def _zeros_like_dispatcher(a, dtype=None, order=None, subok=None, shape=None):
return (a,)
@array_function_dispatch(_zeros_like_dispatcher)
def zeros_like(a, dtype=None, order='K', subok=True, shape=None):
"""
Return an array of zeros with the same shape and type as a given array.
Parameters
----------
a : array_like
The shape and data-type of `a` define these same attributes of
the returned array.
dtype : data-type, optional
Overrides the data type of the result.
.. versionadded:: 1.6.0
order : {'C', 'F', 'A', or 'K'}, optional
Overrides the memory layout of the result. 'C' means C-order,
'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous,
'C' otherwise. 'K' means match the layout of `a` as closely
as possible.
.. versionadded:: 1.6.0
subok : bool, optional.
If True, then the newly created array will use the sub-class
type of `a`, otherwise it will be a base-class array. Defaults
to True.
shape : int or sequence of ints, optional.
Overrides the shape of the result. If order='K' and the number of
dimensions is unchanged, will try to keep order, otherwise,
order='C' is implied.
.. versionadded:: 1.17.0
Returns
-------
out : ndarray
Array of zeros with the same shape and type as `a`.
See Also
--------
empty_like : Return an empty array with shape and type of input.
ones_like : Return an array of ones with shape and type of input.
full_like : Return a new array with shape of input filled with value.
zeros : Return a new array setting values to zero.
Examples
--------
>>> x = np.arange(6)
>>> x = x.reshape((2, 3))
>>> x
array([[0, 1, 2],
[3, 4, 5]])
>>> np.zeros_like(x)
array([[0, 0, 0],
[0, 0, 0]])
>>> y = np.arange(3, dtype=float)
>>> y
array([0., 1., 2.])
>>> np.zeros_like(y)
array([0., 0., 0.])
"""
res = empty_like(a, dtype=dtype, order=order, subok=subok, shape=shape)
# needed instead of a 0 to get same result as zeros for string dtypes
z = zeros(1, dtype=res.dtype)
multiarray.copyto(res, z, casting='unsafe')
return res
def _ones_dispatcher(shape, dtype=None, order=None, *, like=None):
return(like,)
@set_array_function_like_doc
@set_module('numpy')
def ones(shape, dtype=None, order='C', *, like=None):
"""
Return a new array of given shape and type, filled with ones.
Parameters
----------
shape : int or sequence of ints
Shape of the new array, e.g., ``(2, 3)`` or ``2``.
dtype : data-type, optional
The desired data-type for the array, e.g., `numpy.int8`. Default is
`numpy.float64`.
order : {'C', 'F'}, optional, default: C
Whether to store multi-dimensional data in row-major
(C-style) or column-major (Fortran-style) order in
memory.
${ARRAY_FUNCTION_LIKE}
.. versionadded:: 1.20.0
Returns
-------
out : ndarray
Array of ones with the given shape, dtype, and order.
See Also
--------
ones_like : Return an array of ones with shape and type of input.
empty : Return a new uninitialized array.
zeros : Return a new array setting values to zero.
full : Return a new array of given shape filled with value.
Examples
--------
>>> np.ones(5)
array([1., 1., 1., 1., 1.])
>>> np.ones((5,), dtype=int)
array([1, 1, 1, 1, 1])
>>> np.ones((2, 1))
array([[1.],
[1.]])
>>> s = (2,2)
>>> np.ones(s)
array([[1., 1.],
[1., 1.]])
"""
if like is not None:
return _ones_with_like(shape, dtype=dtype, order=order, like=like)
a = empty(shape, dtype, order)
multiarray.copyto(a, 1, casting='unsafe')
return a
_ones_with_like = array_function_dispatch(
_ones_dispatcher
)(ones)
def _ones_like_dispatcher(a, dtype=None, order=None, subok=None, shape=None):
return (a,)
@array_function_dispatch(_ones_like_dispatcher)
def ones_like(a, dtype=None, order='K', subok=True, shape=None):
"""
Return an array of ones with the same shape and type as a given array.
Parameters
----------
a : array_like
The shape and data-type of `a` define these same attributes of
the returned array.
dtype : data-type, optional
Overrides the data type of the result.
.. versionadded:: 1.6.0
order : {'C', 'F', 'A', or 'K'}, optional
Overrides the memory layout of the result. 'C' means C-order,
'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous,
'C' otherwise. 'K' means match the layout of `a` as closely
as possible.
.. versionadded:: 1.6.0
subok : bool, optional.
If True, then the newly created array will use the sub-class
type of `a`, otherwise it will be a base-class array. Defaults
to True.
shape : int or sequence of ints, optional.
Overrides the shape of the result. If order='K' and the number of
dimensions is unchanged, will try to keep order, otherwise,
order='C' is implied.
.. versionadded:: 1.17.0
Returns
-------
out : ndarray
Array of ones with the same shape and type as `a`.
See Also
--------
empty_like : Return an empty array with shape and type of input.
zeros_like : Return an array of zeros with shape and type of input.
full_like : Return a new array with shape of input filled with value.
ones : Return a new array setting values to one.
Examples
--------
>>> x = np.arange(6)
>>> x = x.reshape((2, 3))
>>> x
array([[0, 1, 2],
[3, 4, 5]])
>>> np.ones_like(x)
array([[1, 1, 1],
[1, 1, 1]])
>>> y = np.arange(3, dtype=float)
>>> y
array([0., 1., 2.])
>>> np.ones_like(y)
array([1., 1., 1.])
"""
res = empty_like(a, dtype=dtype, order=order, subok=subok, shape=shape)
multiarray.copyto(res, 1, casting='unsafe')
return res
def _full_dispatcher(shape, fill_value, dtype=None, order=None, *, like=None):
return(like,)
@set_array_function_like_doc
@set_module('numpy')
def full(shape, fill_value, dtype=None, order='C', *, like=None):
"""
Return a new array of given shape and type, filled with `fill_value`.
Parameters
----------
shape : int or sequence of ints
Shape of the new array, e.g., ``(2, 3)`` or ``2``.
fill_value : scalar or array_like
Fill value.
dtype : data-type, optional
The desired data-type for the array The default, None, means
``np.array(fill_value).dtype``.
order : {'C', 'F'}, optional
Whether to store multidimensional data in C- or Fortran-contiguous
(row- or column-wise) order in memory.
${ARRAY_FUNCTION_LIKE}
.. versionadded:: 1.20.0
Returns
-------
out : ndarray
Array of `fill_value` with the given shape, dtype, and order.
See Also
--------
full_like : Return a new array with shape of input filled with value.
empty : Return a new uninitialized array.
ones : Return a new array setting values to one.
zeros : Return a new array setting values to zero.
Examples
--------
>>> np.full((2, 2), np.inf)
array([[inf, inf],
[inf, inf]])
>>> np.full((2, 2), 10)
array([[10, 10],
[10, 10]])
>>> np.full((2, 2), [1, 2])
array([[1, 2],
[1, 2]])
"""
if like is not None:
return _full_with_like(shape, fill_value, dtype=dtype, order=order, like=like)
if dtype is None:
fill_value = asarray(fill_value)
dtype = fill_value.dtype
a = empty(shape, dtype, order)
multiarray.copyto(a, fill_value, casting='unsafe')
return a
_full_with_like = array_function_dispatch(
_full_dispatcher
)(full)
def _full_like_dispatcher(a, fill_value, dtype=None, order=None, subok=None, shape=None):
return (a,)
@array_function_dispatch(_full_like_dispatcher)
def full_like(a, fill_value, dtype=None, order='K', subok=True, shape=None):
"""
Return a full array with the same shape and type as a given array.
Parameters
----------
a : array_like
The shape and data-type of `a` define these same attributes of
the returned array.
fill_value : array_like
Fill value.
dtype : data-type, optional
Overrides the data type of the result.
order : {'C', 'F', 'A', or 'K'}, optional
Overrides the memory layout of the result. 'C' means C-order,
'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous,
'C' otherwise. 'K' means match the layout of `a` as closely
as possible.
subok : bool, optional.
If True, then the newly created array will use the sub-class
type of `a`, otherwise it will be a base-class array. Defaults
to True.
shape : int or sequence of ints, optional.
Overrides the shape of the result. If order='K' and the number of
dimensions is unchanged, will try to keep order, otherwise,
order='C' is implied.
.. versionadded:: 1.17.0
Returns
-------
out : ndarray
Array of `fill_value` with the same shape and type as `a`.
See Also
--------
empty_like : Return an empty array with shape and type of input.
ones_like : Return an array of ones with shape and type of input.
zeros_like : Return an array of zeros with shape and type of input.
full : Return a new array of given shape filled with value.
Examples
--------
>>> x = np.arange(6, dtype=int)
>>> np.full_like(x, 1)
array([1, 1, 1, 1, 1, 1])
>>> np.full_like(x, 0.1)
array([0, 0, 0, 0, 0, 0])
>>> np.full_like(x, 0.1, dtype=np.double)
array([0.1, 0.1, 0.1, 0.1, 0.1, 0.1])
>>> np.full_like(x, np.nan, dtype=np.double)
array([nan, nan, nan, nan, nan, nan])
>>> y = np.arange(6, dtype=np.double)
>>> np.full_like(y, 0.1)
array([0.1, 0.1, 0.1, 0.1, 0.1, 0.1])
>>> y = np.zeros([2, 2, 3], dtype=int)
>>> np.full_like(y, [0, 0, 255])
array([[[ 0, 0, 255],
[ 0, 0, 255]],
[[ 0, 0, 255],
[ 0, 0, 255]]])
"""
res = empty_like(a, dtype=dtype, order=order, subok=subok, shape=shape)
multiarray.copyto(res, fill_value, casting='unsafe')
return res
def _count_nonzero_dispatcher(a, axis=None, *, keepdims=None):
return (a,)
@array_function_dispatch(_count_nonzero_dispatcher)
def count_nonzero(a, axis=None, *, keepdims=False):
"""
Counts the number of non-zero values in the array ``a``.
The word "non-zero" is in reference to the Python 2.x
built-in method ``__nonzero__()`` (renamed ``__bool__()``
in Python 3.x) of Python objects that tests an object's
"truthfulness". For example, any number is considered
truthful if it is nonzero, whereas any string is considered
truthful if it is not the empty string. Thus, this function
(recursively) counts how many elements in ``a`` (and in
sub-arrays thereof) have their ``__nonzero__()`` or ``__bool__()``
method evaluated to ``True``.
Parameters
----------
a : array_like
The array for which to count non-zeros.
axis : int or tuple, optional
Axis or tuple of axes along which to count non-zeros.
Default is None, meaning that non-zeros will be counted
along a flattened version of ``a``.
.. versionadded:: 1.12.0
keepdims : bool, optional
If this is set to True, the axes that are counted are left
in the result as dimensions with size one. With this option,
the result will broadcast correctly against the input array.
.. versionadded:: 1.19.0
Returns
-------
count : int or array of int
Number of non-zero values in the array along a given axis.
Otherwise, the total number of non-zero values in the array
is returned.
See Also
--------
nonzero : Return the coordinates of all the non-zero values.
Examples
--------
>>> np.count_nonzero(np.eye(4))
4
>>> a = np.array([[0, 1, 7, 0],
... [3, 0, 2, 19]])
>>> np.count_nonzero(a)
5
>>> np.count_nonzero(a, axis=0)
array([1, 1, 2, 1])
>>> np.count_nonzero(a, axis=1)
array([2, 3])
>>> np.count_nonzero(a, axis=1, keepdims=True)
array([[2],
[3]])
"""
if axis is None and not keepdims:
return multiarray.count_nonzero(a)
a = asanyarray(a)
# TODO: this works around .astype(bool) not working properly (gh-9847)
if np.issubdtype(a.dtype, np.character):
a_bool = a != a.dtype.type()
else:
a_bool = a.astype(np.bool_, copy=False)
return a_bool.sum(axis=axis, dtype=np.intp, keepdims=keepdims)
@set_module('numpy')
def isfortran(a):
"""
Check if the array is Fortran contiguous but *not* C contiguous.
This function is obsolete and, because of changes due to relaxed stride
checking, its return value for the same array may differ for versions
of NumPy >= 1.10.0 and previous versions. If you only want to check if an
array is Fortran contiguous use ``a.flags.f_contiguous`` instead.
Parameters
----------
a : ndarray
Input array.
Returns
-------
isfortran : bool
Returns True if the array is Fortran contiguous but *not* C contiguous.
Examples
--------
np.array allows to specify whether the array is written in C-contiguous
order (last index varies the fastest), or FORTRAN-contiguous order in
memory (first index varies the fastest).
>>> a = np.array([[1, 2, 3], [4, 5, 6]], order='C')
>>> a
array([[1, 2, 3],
[4, 5, 6]])
>>> np.isfortran(a)
False
>>> b = np.array([[1, 2, 3], [4, 5, 6]], order='F')
>>> b
array([[1, 2, 3],
[4, 5, 6]])
>>> np.isfortran(b)
True
The transpose of a C-ordered array is a FORTRAN-ordered array.
>>> a = np.array([[1, 2, 3], [4, 5, 6]], order='C')
>>> a
array([[1, 2, 3],
[4, 5, 6]])
>>> np.isfortran(a)
False
>>> b = a.T
>>> b
array([[1, 4],
[2, 5],
[3, 6]])
>>> np.isfortran(b)
True
C-ordered arrays evaluate as False even if they are also FORTRAN-ordered.
>>> np.isfortran(np.array([1, 2], order='F'))
False
"""
return a.flags.fnc
def _argwhere_dispatcher(a):
return (a,)
@array_function_dispatch(_argwhere_dispatcher)
def argwhere(a):
"""
Find the indices of array elements that are non-zero, grouped by element.
Parameters
----------
a : array_like
Input data.
Returns
-------
index_array : (N, a.ndim) ndarray
Indices of elements that are non-zero. Indices are grouped by element.
This array will have shape ``(N, a.ndim)`` where ``N`` is the number of
non-zero items.
See Also
--------
where, nonzero
Notes
-----
``np.argwhere(a)`` is almost the same as ``np.transpose(np.nonzero(a))``,
but produces a result of the correct shape for a 0D array.
The output of ``argwhere`` is not suitable for indexing arrays.
For this purpose use ``nonzero(a)`` instead.
Examples
--------
>>> x = np.arange(6).reshape(2,3)
>>> x
array([[0, 1, 2],
[3, 4, 5]])
>>> np.argwhere(x>1)
array([[0, 2],
[1, 0],
[1, 1],
[1, 2]])
"""
# nonzero does not behave well on 0d, so promote to 1d
if np.ndim(a) == 0:
a = shape_base.atleast_1d(a)
# then remove the added dimension
return argwhere(a)[:,:0]
return transpose(nonzero(a))
def _flatnonzero_dispatcher(a):
return (a,)
@array_function_dispatch(_flatnonzero_dispatcher)
def flatnonzero(a):
"""
Return indices that are non-zero in the flattened version of a.
This is equivalent to ``np.nonzero(np.ravel(a))[0]``.
Parameters
----------
a : array_like
Input data.
Returns
-------
res : ndarray
Output array, containing the indices of the elements of ``a.ravel()``
that are non-zero.
See Also
--------
nonzero : Return the indices of the non-zero elements of the input array.
ravel : Return a 1-D array containing the elements of the input array.
Examples
--------
>>> x = np.arange(-2, 3)
>>> x
array([-2, -1, 0, 1, 2])
>>> np.flatnonzero(x)
array([0, 1, 3, 4])
Use the indices of the non-zero elements as an index array to extract
these elements:
>>> x.ravel()[np.flatnonzero(x)]
array([-2, -1, 1, 2])
"""
return np.nonzero(np.ravel(a))[0]
def _correlate_dispatcher(a, v, mode=None):
return (a, v)
@array_function_dispatch(_correlate_dispatcher)
def correlate(a, v, mode='valid'):
r"""
Cross-correlation of two 1-dimensional sequences.
This function computes the correlation as generally defined in signal
processing texts:
.. math:: c_k = \sum_n a_{n+k} \cdot \overline{v_n}
with a and v sequences being zero-padded where necessary and
:math:`\overline x` denoting complex conjugation.
Parameters
----------
a, v : array_like
Input sequences.
mode : {'valid', 'same', 'full'}, optional
Refer to the `convolve` docstring. Note that the default
is 'valid', unlike `convolve`, which uses 'full'.
old_behavior : bool
`old_behavior` was removed in NumPy 1.10. If you need the old
behavior, use `multiarray.correlate`.
Returns
-------
out : ndarray
Discrete cross-correlation of `a` and `v`.
See Also
--------
convolve : Discrete, linear convolution of two one-dimensional sequences.
multiarray.correlate : Old, no conjugate, version of correlate.
scipy.signal.correlate : uses FFT which has superior performance on large arrays.
Notes
-----
The definition of correlation above is not unique and sometimes correlation
may be defined differently. Another common definition is:
.. math:: c'_k = \sum_n a_{n} \cdot \overline{v_{n+k}}
which is related to :math:`c_k` by :math:`c'_k = c_{-k}`.
`numpy.correlate` may perform slowly in large arrays (i.e. n = 1e5) because it does
not use the FFT to compute the convolution; in that case, `scipy.signal.correlate` might
be preferable.
Examples
--------
>>> np.correlate([1, 2, 3], [0, 1, 0.5])
array([3.5])
>>> np.correlate([1, 2, 3], [0, 1, 0.5], "same")
array([2. , 3.5, 3. ])
>>> np.correlate([1, 2, 3], [0, 1, 0.5], "full")
array([0.5, 2. , 3.5, 3. , 0. ])
Using complex sequences:
>>> np.correlate([1+1j, 2, 3-1j], [0, 1, 0.5j], 'full')
array([ 0.5-0.5j, 1.0+0.j , 1.5-1.5j, 3.0-1.j , 0.0+0.j ])
Note that you get the time reversed, complex conjugated result
(:math:`\overline{c_{-k}}`) when the two input sequences a and v change
places:
>>> np.correlate([0, 1, 0.5j], [1+1j, 2, 3-1j], 'full')
array([ 0.0+0.j , 3.0+1.j , 1.5+1.5j, 1.0+0.j , 0.5+0.5j])
"""
return multiarray.correlate2(a, v, mode)
def _convolve_dispatcher(a, v, mode=None):
return (a, v)
@array_function_dispatch(_convolve_dispatcher)
def convolve(a, v, mode='full'):
"""
Returns the discrete, linear convolution of two one-dimensional sequences.
The convolution operator is often seen in signal processing, where it
models the effect of a linear time-invariant system on a signal [1]_. In
probability theory, the sum of two independent random variables is
distributed according to the convolution of their individual
distributions.
If `v` is longer than `a`, the arrays are swapped before computation.
Parameters
----------
a : (N,) array_like
First one-dimensional input array.
v : (M,) array_like
Second one-dimensional input array.
mode : {'full', 'valid', 'same'}, optional
'full':
By default, mode is 'full'. This returns the convolution
at each point of overlap, with an output shape of (N+M-1,). At
the end-points of the convolution, the signals do not overlap
completely, and boundary effects may be seen.
'same':
Mode 'same' returns output of length ``max(M, N)``. Boundary
effects are still visible.
'valid':
Mode 'valid' returns output of length
``max(M, N) - min(M, N) + 1``. The convolution product is only given
for points where the signals overlap completely. Values outside
the signal boundary have no effect.
Returns
-------
out : ndarray
Discrete, linear convolution of `a` and `v`.
See Also
--------
scipy.signal.fftconvolve : Convolve two arrays using the Fast Fourier
Transform.
scipy.linalg.toeplitz : Used to construct the convolution operator.
polymul : Polynomial multiplication. Same output as convolve, but also
accepts poly1d objects as input.
Notes
-----
The discrete convolution operation is defined as
.. math:: (a * v)_n = \\sum_{m = -\\infty}^{\\infty} a_m v_{n - m}
It can be shown that a convolution :math:`x(t) * y(t)` in time/space
is equivalent to the multiplication :math:`X(f) Y(f)` in the Fourier
domain, after appropriate padding (padding is necessary to prevent
circular convolution). Since multiplication is more efficient (faster)
than convolution, the function `scipy.signal.fftconvolve` exploits the
FFT to calculate the convolution of large data-sets.
References
----------
.. [1] Wikipedia, "Convolution",
https://en.wikipedia.org/wiki/Convolution
Examples
--------
Note how the convolution operator flips the second array
before "sliding" the two across one another:
>>> np.convolve([1, 2, 3], [0, 1, 0.5])
array([0. , 1. , 2.5, 4. , 1.5])
Only return the middle values of the convolution.
Contains boundary effects, where zeros are taken
into account:
>>> np.convolve([1,2,3],[0,1,0.5], 'same')
array([1. , 2.5, 4. ])
The two arrays are of the same length, so there
is only one position where they completely overlap:
>>> np.convolve([1,2,3],[0,1,0.5], 'valid')
array([2.5])
"""
a, v = array(a, copy=False, ndmin=1), array(v, copy=False, ndmin=1)
if (len(v) > len(a)):
a, v = v, a
if len(a) == 0:
raise ValueError('a cannot be empty')
if len(v) == 0:
raise ValueError('v cannot be empty')
return multiarray.correlate(a, v[::-1], mode)
def _outer_dispatcher(a, b, out=None):
return (a, b, out)
@array_function_dispatch(_outer_dispatcher)
def outer(a, b, out=None):
"""
Compute the outer product of two vectors.
Given two vectors, ``a = [a0, a1, ..., aM]`` and
``b = [b0, b1, ..., bN]``,
the outer product [1]_ is::
[[a0*b0 a0*b1 ... a0*bN ]
[a1*b0 .
[ ... .
[aM*b0 aM*bN ]]
Parameters
----------
a : (M,) array_like
First input vector. Input is flattened if
not already 1-dimensional.
b : (N,) array_like
Second input vector. Input is flattened if
not already 1-dimensional.
out : (M, N) ndarray, optional
A location where the result is stored
.. versionadded:: 1.9.0
Returns
-------
out : (M, N) ndarray
``out[i, j] = a[i] * b[j]``
See also
--------
inner
einsum : ``einsum('i,j->ij', a.ravel(), b.ravel())`` is the equivalent.
ufunc.outer : A generalization to dimensions other than 1D and other
operations. ``np.multiply.outer(a.ravel(), b.ravel())``
is the equivalent.
tensordot : ``np.tensordot(a.ravel(), b.ravel(), axes=((), ()))``
is the equivalent.
References
----------
.. [1] : G. H. Golub and C. F. Van Loan, *Matrix Computations*, 3rd
ed., Baltimore, MD, Johns Hopkins University Press, 1996,
pg. 8.
Examples
--------
Make a (*very* coarse) grid for computing a Mandelbrot set:
>>> rl = np.outer(np.ones((5,)), np.linspace(-2, 2, 5))
>>> rl
array([[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.]])
>>> im = np.outer(1j*np.linspace(2, -2, 5), np.ones((5,)))
>>> im
array([[0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j],
[0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j],
[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j],
[0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j]])
>>> grid = rl + im
>>> grid
array([[-2.+2.j, -1.+2.j, 0.+2.j, 1.+2.j, 2.+2.j],
[-2.+1.j, -1.+1.j, 0.+1.j, 1.+1.j, 2.+1.j],
[-2.+0.j, -1.+0.j, 0.+0.j, 1.+0.j, 2.+0.j],
[-2.-1.j, -1.-1.j, 0.-1.j, 1.-1.j, 2.-1.j],
[-2.-2.j, -1.-2.j, 0.-2.j, 1.-2.j, 2.-2.j]])
An example using a "vector" of letters:
>>> x = np.array(['a', 'b', 'c'], dtype=object)
>>> np.outer(x, [1, 2, 3])
array([['a', 'aa', 'aaa'],
['b', 'bb', 'bbb'],
['c', 'cc', 'ccc']], dtype=object)
"""
a = asarray(a)
b = asarray(b)
return multiply(a.ravel()[:, newaxis], b.ravel()[newaxis, :], out)
def _tensordot_dispatcher(a, b, axes=None):
return (a, b)
@array_function_dispatch(_tensordot_dispatcher)
def tensordot(a, b, axes=2):
"""
Compute tensor dot product along specified axes.
Given two tensors, `a` and `b`, and an array_like object containing
two array_like objects, ``(a_axes, b_axes)``, sum the products of
`a`'s and `b`'s elements (components) over the axes specified by
``a_axes`` and ``b_axes``. The third argument can be a single non-negative
integer_like scalar, ``N``; if it is such, then the last ``N`` dimensions
of `a` and the first ``N`` dimensions of `b` are summed over.
Parameters
----------
a, b : array_like
Tensors to "dot".
axes : int or (2,) array_like
* integer_like
If an int N, sum over the last N axes of `a` and the first N axes
of `b` in order. The sizes of the corresponding axes must match.
* (2,) array_like
Or, a list of axes to be summed over, first sequence applying to `a`,
second to `b`. Both elements array_like must be of the same length.
Returns
-------
output : ndarray
The tensor dot product of the input.
See Also
--------
dot, einsum
Notes
-----
Three common use cases are:
* ``axes = 0`` : tensor product :math:`a\\otimes b`
* ``axes = 1`` : tensor dot product :math:`a\\cdot b`
* ``axes = 2`` : (default) tensor double contraction :math:`a:b`
When `axes` is integer_like, the sequence for evaluation will be: first
the -Nth axis in `a` and 0th axis in `b`, and the -1th axis in `a` and
Nth axis in `b` last.
When there is more than one axis to sum over - and they are not the last
(first) axes of `a` (`b`) - the argument `axes` should consist of
two sequences of the same length, with the first axis to sum over given
first in both sequences, the second axis second, and so forth.
The shape of the result consists of the non-contracted axes of the
first tensor, followed by the non-contracted axes of the second.
Examples
--------
A "traditional" example:
>>> a = np.arange(60.).reshape(3,4,5)
>>> b = np.arange(24.).reshape(4,3,2)
>>> c = np.tensordot(a,b, axes=([1,0],[0,1]))
>>> c.shape
(5, 2)
>>> c
array([[4400., 4730.],
[4532., 4874.],
[4664., 5018.],
[4796., 5162.],
[4928., 5306.]])
>>> # A slower but equivalent way of computing the same...
>>> d = np.zeros((5,2))
>>> for i in range(5):
... for j in range(2):
... for k in range(3):
... for n in range(4):
... d[i,j] += a[k,n,i] * b[n,k,j]
>>> c == d
array([[ True, True],
[ True, True],
[ True, True],
[ True, True],
[ True, True]])
An extended example taking advantage of the overloading of + and \\*:
>>> a = np.array(range(1, 9))
>>> a.shape = (2, 2, 2)
>>> A = np.array(('a', 'b', 'c', 'd'), dtype=object)
>>> A.shape = (2, 2)
>>> a; A
array([[[1, 2],
[3, 4]],
[[5, 6],
[7, 8]]])
array([['a', 'b'],
['c', 'd']], dtype=object)
>>> np.tensordot(a, A) # third argument default is 2 for double-contraction
array(['abbcccdddd', 'aaaaabbbbbbcccccccdddddddd'], dtype=object)
>>> np.tensordot(a, A, 1)
array([[['acc', 'bdd'],
['aaacccc', 'bbbdddd']],
[['aaaaacccccc', 'bbbbbdddddd'],
['aaaaaaacccccccc', 'bbbbbbbdddddddd']]], dtype=object)
>>> np.tensordot(a, A, 0) # tensor product (result too long to incl.)
array([[[[['a', 'b'],
['c', 'd']],
...
>>> np.tensordot(a, A, (0, 1))
array([[['abbbbb', 'cddddd'],
['aabbbbbb', 'ccdddddd']],
[['aaabbbbbbb', 'cccddddddd'],
['aaaabbbbbbbb', 'ccccdddddddd']]], dtype=object)
>>> np.tensordot(a, A, (2, 1))
array([[['abb', 'cdd'],
['aaabbbb', 'cccdddd']],
[['aaaaabbbbbb', 'cccccdddddd'],
['aaaaaaabbbbbbbb', 'cccccccdddddddd']]], dtype=object)
>>> np.tensordot(a, A, ((0, 1), (0, 1)))
array(['abbbcccccddddddd', 'aabbbbccccccdddddddd'], dtype=object)
>>> np.tensordot(a, A, ((2, 1), (1, 0)))
array(['acccbbdddd', 'aaaaacccccccbbbbbbdddddddd'], dtype=object)
"""
try:
iter(axes)
except Exception:
axes_a = list(range(-axes, 0))
axes_b = list(range(0, axes))
else:
axes_a, axes_b = axes
try:
na = len(axes_a)
axes_a = list(axes_a)
except TypeError:
axes_a = [axes_a]
na = 1
try:
nb = len(axes_b)
axes_b = list(axes_b)
except TypeError:
axes_b = [axes_b]
nb = 1
a, b = asarray(a), asarray(b)
as_ = a.shape
nda = a.ndim
bs = b.shape
ndb = b.ndim
equal = True
if na != nb:
equal = False
else:
for k in range(na):
if as_[axes_a[k]] != bs[axes_b[k]]:
equal = False
break
if axes_a[k] < 0:
axes_a[k] += nda
if axes_b[k] < 0:
axes_b[k] += ndb
if not equal:
raise ValueError("shape-mismatch for sum")
# Move the axes to sum over to the end of "a"
# and to the front of "b"
notin = [k for k in range(nda) if k not in axes_a]
newaxes_a = notin + axes_a
N2 = 1
for axis in axes_a:
N2 *= as_[axis]
newshape_a = (int(multiply.reduce([as_[ax] for ax in notin])), N2)
olda = [as_[axis] for axis in notin]
notin = [k for k in range(ndb) if k not in axes_b]
newaxes_b = axes_b + notin
N2 = 1
for axis in axes_b:
N2 *= bs[axis]
newshape_b = (N2, int(multiply.reduce([bs[ax] for ax in notin])))
oldb = [bs[axis] for axis in notin]
at = a.transpose(newaxes_a).reshape(newshape_a)
bt = b.transpose(newaxes_b).reshape(newshape_b)
res = dot(at, bt)
return res.reshape(olda + oldb)
def _roll_dispatcher(a, shift, axis=None):
return (a,)
@array_function_dispatch(_roll_dispatcher)
def roll(a, shift, axis=None):
"""
Roll array elements along a given axis.
Elements that roll beyond the last position are re-introduced at
the first.
Parameters
----------
a : array_like
Input array.
shift : int or tuple of ints
The number of places by which elements are shifted. If a tuple,
then `axis` must be a tuple of the same size, and each of the
given axes is shifted by the corresponding number. If an int
while `axis` is a tuple of ints, then the same value is used for
all given axes.
axis : int or tuple of ints, optional
Axis or axes along which elements are shifted. By default, the
array is flattened before shifting, after which the original
shape is restored.
Returns
-------
res : ndarray
Output array, with the same shape as `a`.
See Also
--------
rollaxis : Roll the specified axis backwards, until it lies in a
given position.
Notes
-----
.. versionadded:: 1.12.0
Supports rolling over multiple dimensions simultaneously.
Examples
--------
>>> x = np.arange(10)
>>> np.roll(x, 2)
array([8, 9, 0, 1, 2, 3, 4, 5, 6, 7])
>>> np.roll(x, -2)
array([2, 3, 4, 5, 6, 7, 8, 9, 0, 1])
>>> x2 = np.reshape(x, (2, 5))
>>> x2
array([[0, 1, 2, 3, 4],
[5, 6, 7, 8, 9]])
>>> np.roll(x2, 1)
array([[9, 0, 1, 2, 3],
[4, 5, 6, 7, 8]])
>>> np.roll(x2, -1)
array([[1, 2, 3, 4, 5],
[6, 7, 8, 9, 0]])
>>> np.roll(x2, 1, axis=0)
array([[5, 6, 7, 8, 9],
[0, 1, 2, 3, 4]])
>>> np.roll(x2, -1, axis=0)
array([[5, 6, 7, 8, 9],
[0, 1, 2, 3, 4]])
>>> np.roll(x2, 1, axis=1)
array([[4, 0, 1, 2, 3],
[9, 5, 6, 7, 8]])
>>> np.roll(x2, -1, axis=1)
array([[1, 2, 3, 4, 0],
[6, 7, 8, 9, 5]])
>>> np.roll(x2, (1, 1), axis=(1, 0))
array([[9, 5, 6, 7, 8],
[4, 0, 1, 2, 3]])
>>> np.roll(x2, (2, 1), axis=(1, 0))
array([[8, 9, 5, 6, 7],
[3, 4, 0, 1, 2]])
"""
a = asanyarray(a)
if axis is None:
return roll(a.ravel(), shift, 0).reshape(a.shape)
else:
axis = normalize_axis_tuple(axis, a.ndim, allow_duplicate=True)
broadcasted = broadcast(shift, axis)
if broadcasted.ndim > 1:
raise ValueError(
"'shift' and 'axis' should be scalars or 1D sequences")
shifts = {ax: 0 for ax in range(a.ndim)}
for sh, ax in broadcasted:
shifts[ax] += sh
rolls = [((slice(None), slice(None)),)] * a.ndim
for ax, offset in shifts.items():
offset %= a.shape[ax] or 1 # If `a` is empty, nothing matters.
if offset:
# (original, result), (original, result)
rolls[ax] = ((slice(None, -offset), slice(offset, None)),
(slice(-offset, None), slice(None, offset)))
result = empty_like(a)
for indices in itertools.product(*rolls):
arr_index, res_index = zip(*indices)
result[res_index] = a[arr_index]
return result
def _rollaxis_dispatcher(a, axis, start=None):
return (a,)
@array_function_dispatch(_rollaxis_dispatcher)
def rollaxis(a, axis, start=0):
"""
Roll the specified axis backwards, until it lies in a given position.
This function continues to be supported for backward compatibility, but you
should prefer `moveaxis`. The `moveaxis` function was added in NumPy
1.11.
Parameters
----------
a : ndarray
Input array.
axis : int
The axis to be rolled. The positions of the other axes do not
change relative to one another.
start : int, optional
When ``start <= axis``, the axis is rolled back until it lies in
this position. When ``start > axis``, the axis is rolled until it
lies before this position. The default, 0, results in a "complete"
roll. The following table describes how negative values of ``start``
are interpreted:
.. table::
:align: left
+-------------------+----------------------+
| ``start`` | Normalized ``start`` |
+===================+======================+
| ``-(arr.ndim+1)`` | raise ``AxisError`` |
+-------------------+----------------------+
| ``-arr.ndim`` | 0 |
+-------------------+----------------------+
| |vdots| | |vdots| |
+-------------------+----------------------+
| ``-1`` | ``arr.ndim-1`` |
+-------------------+----------------------+
| ``0`` | ``0`` |
+-------------------+----------------------+
| |vdots| | |vdots| |
+-------------------+----------------------+
| ``arr.ndim`` | ``arr.ndim`` |
+-------------------+----------------------+
| ``arr.ndim + 1`` | raise ``AxisError`` |
+-------------------+----------------------+
.. |vdots| unicode:: U+22EE .. Vertical Ellipsis
Returns
-------
res : ndarray
For NumPy >= 1.10.0 a view of `a` is always returned. For earlier
NumPy versions a view of `a` is returned only if the order of the
axes is changed, otherwise the input array is returned.
See Also
--------
moveaxis : Move array axes to new positions.
roll : Roll the elements of an array by a number of positions along a
given axis.
Examples
--------
>>> a = np.ones((3,4,5,6))
>>> np.rollaxis(a, 3, 1).shape
(3, 6, 4, 5)
>>> np.rollaxis(a, 2).shape
(5, 3, 4, 6)
>>> np.rollaxis(a, 1, 4).shape
(3, 5, 6, 4)
"""
n = a.ndim
axis = normalize_axis_index(axis, n)
if start < 0:
start += n
msg = "'%s' arg requires %d <= %s < %d, but %d was passed in"
if not (0 <= start < n + 1):
raise AxisError(msg % ('start', -n, 'start', n + 1, start))
if axis < start:
# it's been removed
start -= 1
if axis == start:
return a[...]
axes = list(range(0, n))
axes.remove(axis)
axes.insert(start, axis)
return a.transpose(axes)
def normalize_axis_tuple(axis, ndim, argname=None, allow_duplicate=False):
"""
Normalizes an axis argument into a tuple of non-negative integer axes.
This handles shorthands such as ``1`` and converts them to ``(1,)``,
as well as performing the handling of negative indices covered by
`normalize_axis_index`.
By default, this forbids axes from being specified multiple times.
Used internally by multi-axis-checking logic.
.. versionadded:: 1.13.0
Parameters
----------
axis : int, iterable of int
The un-normalized index or indices of the axis.
ndim : int
The number of dimensions of the array that `axis` should be normalized
against.
argname : str, optional
A prefix to put before the error message, typically the name of the
argument.
allow_duplicate : bool, optional
If False, the default, disallow an axis from being specified twice.
Returns
-------
normalized_axes : tuple of int
The normalized axis index, such that `0 <= normalized_axis < ndim`
Raises
------
AxisError
If any axis provided is out of range
ValueError
If an axis is repeated
See also
--------
normalize_axis_index : normalizing a single scalar axis
"""
# Optimization to speed-up the most common cases.
if type(axis) not in (tuple, list):
try:
axis = [operator.index(axis)]
except TypeError:
pass
# Going via an iterator directly is slower than via list comprehension.
axis = tuple([normalize_axis_index(ax, ndim, argname) for ax in axis])
if not allow_duplicate and len(set(axis)) != len(axis):
if argname:
raise ValueError('repeated axis in `{}` argument'.format(argname))
else:
raise ValueError('repeated axis')
return axis
def _moveaxis_dispatcher(a, source, destination):
return (a,)
@array_function_dispatch(_moveaxis_dispatcher)
def moveaxis(a, source, destination):
"""
Move axes of an array to new positions.
Other axes remain in their original order.
.. versionadded:: 1.11.0
Parameters
----------
a : np.ndarray
The array whose axes should be reordered.
source : int or sequence of int
Original positions of the axes to move. These must be unique.
destination : int or sequence of int
Destination positions for each of the original axes. These must also be
unique.
Returns
-------
result : np.ndarray
Array with moved axes. This array is a view of the input array.
See Also
--------
transpose : Permute the dimensions of an array.
swapaxes : Interchange two axes of an array.
Examples
--------
>>> x = np.zeros((3, 4, 5))
>>> np.moveaxis(x, 0, -1).shape
(4, 5, 3)
>>> np.moveaxis(x, -1, 0).shape
(5, 3, 4)
These all achieve the same result:
>>> np.transpose(x).shape
(5, 4, 3)
>>> np.swapaxes(x, 0, -1).shape
(5, 4, 3)
>>> np.moveaxis(x, [0, 1], [-1, -2]).shape
(5, 4, 3)
>>> np.moveaxis(x, [0, 1, 2], [-1, -2, -3]).shape
(5, 4, 3)
"""
try:
# allow duck-array types if they define transpose
transpose = a.transpose
except AttributeError:
a = asarray(a)
transpose = a.transpose
source = normalize_axis_tuple(source, a.ndim, 'source')
destination = normalize_axis_tuple(destination, a.ndim, 'destination')
if len(source) != len(destination):
raise ValueError('`source` and `destination` arguments must have '
'the same number of elements')
order = [n for n in range(a.ndim) if n not in source]
for dest, src in sorted(zip(destination, source)):
order.insert(dest, src)
result = transpose(order)
return result
def _cross_dispatcher(a, b, axisa=None, axisb=None, axisc=None, axis=None):
return (a, b)
@array_function_dispatch(_cross_dispatcher)
def cross(a, b, axisa=-1, axisb=-1, axisc=-1, axis=None):
"""
Return the cross product of two (arrays of) vectors.
The cross product of `a` and `b` in :math:`R^3` is a vector perpendicular
to both `a` and `b`. If `a` and `b` are arrays of vectors, the vectors
are defined by the last axis of `a` and `b` by default, and these axes
can have dimensions 2 or 3. Where the dimension of either `a` or `b` is
2, the third component of the input vector is assumed to be zero and the
cross product calculated accordingly. In cases where both input vectors
have dimension 2, the z-component of the cross product is returned.
Parameters
----------
a : array_like
Components of the first vector(s).
b : array_like
Components of the second vector(s).
axisa : int, optional
Axis of `a` that defines the vector(s). By default, the last axis.
axisb : int, optional
Axis of `b` that defines the vector(s). By default, the last axis.
axisc : int, optional
Axis of `c` containing the cross product vector(s). Ignored if
both input vectors have dimension 2, as the return is scalar.
By default, the last axis.
axis : int, optional
If defined, the axis of `a`, `b` and `c` that defines the vector(s)
and cross product(s). Overrides `axisa`, `axisb` and `axisc`.
Returns
-------
c : ndarray
Vector cross product(s).
Raises
------
ValueError
When the dimension of the vector(s) in `a` and/or `b` does not
equal 2 or 3.
See Also
--------
inner : Inner product
outer : Outer product.
ix_ : Construct index arrays.
Notes
-----
.. versionadded:: 1.9.0
Supports full broadcasting of the inputs.
Examples
--------
Vector cross-product.
>>> x = [1, 2, 3]
>>> y = [4, 5, 6]
>>> np.cross(x, y)
array([-3, 6, -3])
One vector with dimension 2.
>>> x = [1, 2]
>>> y = [4, 5, 6]
>>> np.cross(x, y)
array([12, -6, -3])
Equivalently:
>>> x = [1, 2, 0]
>>> y = [4, 5, 6]
>>> np.cross(x, y)
array([12, -6, -3])
Both vectors with dimension 2.
>>> x = [1,2]
>>> y = [4,5]
>>> np.cross(x, y)
array(-3)
Multiple vector cross-products. Note that the direction of the cross
product vector is defined by the *right-hand rule*.
>>> x = np.array([[1,2,3], [4,5,6]])
>>> y = np.array([[4,5,6], [1,2,3]])
>>> np.cross(x, y)
array([[-3, 6, -3],
[ 3, -6, 3]])
The orientation of `c` can be changed using the `axisc` keyword.
>>> np.cross(x, y, axisc=0)
array([[-3, 3],
[ 6, -6],
[-3, 3]])
Change the vector definition of `x` and `y` using `axisa` and `axisb`.
>>> x = np.array([[1,2,3], [4,5,6], [7, 8, 9]])
>>> y = np.array([[7, 8, 9], [4,5,6], [1,2,3]])
>>> np.cross(x, y)
array([[ -6, 12, -6],
[ 0, 0, 0],
[ 6, -12, 6]])
>>> np.cross(x, y, axisa=0, axisb=0)
array([[-24, 48, -24],
[-30, 60, -30],
[-36, 72, -36]])
"""
if axis is not None:
axisa, axisb, axisc = (axis,) * 3
a = asarray(a)
b = asarray(b)
# Check axisa and axisb are within bounds
axisa = normalize_axis_index(axisa, a.ndim, msg_prefix='axisa')
axisb = normalize_axis_index(axisb, b.ndim, msg_prefix='axisb')
# Move working axis to the end of the shape
a = moveaxis(a, axisa, -1)
b = moveaxis(b, axisb, -1)
msg = ("incompatible dimensions for cross product\n"
"(dimension must be 2 or 3)")
if a.shape[-1] not in (2, 3) or b.shape[-1] not in (2, 3):
raise ValueError(msg)
# Create the output array
shape = broadcast(a[..., 0], b[..., 0]).shape
if a.shape[-1] == 3 or b.shape[-1] == 3:
shape += (3,)
# Check axisc is within bounds
axisc = normalize_axis_index(axisc, len(shape), msg_prefix='axisc')
dtype = promote_types(a.dtype, b.dtype)
cp = empty(shape, dtype)
# create local aliases for readability
a0 = a[..., 0]
a1 = a[..., 1]
if a.shape[-1] == 3:
a2 = a[..., 2]
b0 = b[..., 0]
b1 = b[..., 1]
if b.shape[-1] == 3:
b2 = b[..., 2]
if cp.ndim != 0 and cp.shape[-1] == 3:
cp0 = cp[..., 0]
cp1 = cp[..., 1]
cp2 = cp[..., 2]
if a.shape[-1] == 2:
if b.shape[-1] == 2:
# a0 * b1 - a1 * b0
multiply(a0, b1, out=cp)
cp -= a1 * b0
return cp
else:
assert b.shape[-1] == 3
# cp0 = a1 * b2 - 0 (a2 = 0)
# cp1 = 0 - a0 * b2 (a2 = 0)
# cp2 = a0 * b1 - a1 * b0
multiply(a1, b2, out=cp0)
multiply(a0, b2, out=cp1)
negative(cp1, out=cp1)
multiply(a0, b1, out=cp2)
cp2 -= a1 * b0
else:
assert a.shape[-1] == 3
if b.shape[-1] == 3:
# cp0 = a1 * b2 - a2 * b1
# cp1 = a2 * b0 - a0 * b2
# cp2 = a0 * b1 - a1 * b0
multiply(a1, b2, out=cp0)
tmp = array(a2 * b1)
cp0 -= tmp
multiply(a2, b0, out=cp1)
multiply(a0, b2, out=tmp)
cp1 -= tmp
multiply(a0, b1, out=cp2)
multiply(a1, b0, out=tmp)
cp2 -= tmp
else:
assert b.shape[-1] == 2
# cp0 = 0 - a2 * b1 (b2 = 0)
# cp1 = a2 * b0 - 0 (b2 = 0)
# cp2 = a0 * b1 - a1 * b0
multiply(a2, b1, out=cp0)
negative(cp0, out=cp0)
multiply(a2, b0, out=cp1)
multiply(a0, b1, out=cp2)
cp2 -= a1 * b0
return moveaxis(cp, -1, axisc)
little_endian = (sys.byteorder == 'little')
@set_module('numpy')
def indices(dimensions, dtype=int, sparse=False):
"""
Return an array representing the indices of a grid.
Compute an array where the subarrays contain index values 0, 1, ...
varying only along the corresponding axis.
Parameters
----------
dimensions : sequence of ints
The shape of the grid.
dtype : dtype, optional
Data type of the result.
sparse : boolean, optional
Return a sparse representation of the grid instead of a dense
representation. Default is False.
.. versionadded:: 1.17
Returns
-------
grid : one ndarray or tuple of ndarrays
If sparse is False:
Returns one array of grid indices,
``grid.shape = (len(dimensions),) + tuple(dimensions)``.
If sparse is True:
Returns a tuple of arrays, with
``grid[i].shape = (1, ..., 1, dimensions[i], 1, ..., 1)`` with
dimensions[i] in the ith place
See Also
--------
mgrid, ogrid, meshgrid
Notes
-----
The output shape in the dense case is obtained by prepending the number
of dimensions in front of the tuple of dimensions, i.e. if `dimensions`
is a tuple ``(r0, ..., rN-1)`` of length ``N``, the output shape is
``(N, r0, ..., rN-1)``.
The subarrays ``grid[k]`` contains the N-D array of indices along the
``k-th`` axis. Explicitly::
grid[k, i0, i1, ..., iN-1] = ik
Examples
--------
>>> grid = np.indices((2, 3))
>>> grid.shape
(2, 2, 3)
>>> grid[0] # row indices
array([[0, 0, 0],
[1, 1, 1]])
>>> grid[1] # column indices
array([[0, 1, 2],
[0, 1, 2]])
The indices can be used as an index into an array.
>>> x = np.arange(20).reshape(5, 4)
>>> row, col = np.indices((2, 3))
>>> x[row, col]
array([[0, 1, 2],
[4, 5, 6]])
Note that it would be more straightforward in the above example to
extract the required elements directly with ``x[:2, :3]``.
If sparse is set to true, the grid will be returned in a sparse
representation.
>>> i, j = np.indices((2, 3), sparse=True)
>>> i.shape
(2, 1)
>>> j.shape
(1, 3)
>>> i # row indices
array([[0],
[1]])
>>> j # column indices
array([[0, 1, 2]])
"""
dimensions = tuple(dimensions)
N = len(dimensions)
shape = (1,)*N
if sparse:
res = tuple()
else:
res = empty((N,)+dimensions, dtype=dtype)
for i, dim in enumerate(dimensions):
idx = arange(dim, dtype=dtype).reshape(
shape[:i] + (dim,) + shape[i+1:]
)
if sparse:
res = res + (idx,)
else:
res[i] = idx
return res
def _fromfunction_dispatcher(function, shape, *, dtype=None, like=None, **kwargs):
return (like,)
@set_array_function_like_doc
@set_module('numpy')
def fromfunction(function, shape, *, dtype=float, like=None, **kwargs):
"""
Construct an array by executing a function over each coordinate.
The resulting array therefore has a value ``fn(x, y, z)`` at
coordinate ``(x, y, z)``.
Parameters
----------
function : callable
The function is called with N parameters, where N is the rank of
`shape`. Each parameter represents the coordinates of the array
varying along a specific axis. For example, if `shape`
were ``(2, 2)``, then the parameters would be
``array([[0, 0], [1, 1]])`` and ``array([[0, 1], [0, 1]])``
shape : (N,) tuple of ints
Shape of the output array, which also determines the shape of
the coordinate arrays passed to `function`.
dtype : data-type, optional
Data-type of the coordinate arrays passed to `function`.
By default, `dtype` is float.
${ARRAY_FUNCTION_LIKE}
.. versionadded:: 1.20.0
Returns
-------
fromfunction : any
The result of the call to `function` is passed back directly.
Therefore the shape of `fromfunction` is completely determined by
`function`. If `function` returns a scalar value, the shape of
`fromfunction` would not match the `shape` parameter.
See Also
--------
indices, meshgrid
Notes
-----
Keywords other than `dtype` are passed to `function`.
Examples
--------
>>> np.fromfunction(lambda i, j: i, (2, 2), dtype=float)
array([[0., 0.],
[1., 1.]])
>>> np.fromfunction(lambda i, j: j, (2, 2), dtype=float)
array([[0., 1.],
[0., 1.]])
>>> np.fromfunction(lambda i, j: i == j, (3, 3), dtype=int)
array([[ True, False, False],
[False, True, False],
[False, False, True]])
>>> np.fromfunction(lambda i, j: i + j, (3, 3), dtype=int)
array([[0, 1, 2],
[1, 2, 3],
[2, 3, 4]])
"""
if like is not None:
return _fromfunction_with_like(function, shape, dtype=dtype, like=like, **kwargs)
args = indices(shape, dtype=dtype)
return function(*args, **kwargs)
_fromfunction_with_like = array_function_dispatch(
_fromfunction_dispatcher
)(fromfunction)
def _frombuffer(buf, dtype, shape, order):
return frombuffer(buf, dtype=dtype).reshape(shape, order=order)
@set_module('numpy')
def isscalar(element):
"""
Returns True if the type of `element` is a scalar type.
Parameters
----------
element : any
Input argument, can be of any type and shape.
Returns
-------
val : bool
True if `element` is a scalar type, False if it is not.
See Also
--------
ndim : Get the number of dimensions of an array
Notes
-----
If you need a stricter way to identify a *numerical* scalar, use
``isinstance(x, numbers.Number)``, as that returns ``False`` for most
non-numerical elements such as strings.
In most cases ``np.ndim(x) == 0`` should be used instead of this function,
as that will also return true for 0d arrays. This is how numpy overloads
functions in the style of the ``dx`` arguments to `gradient` and the ``bins``
argument to `histogram`. Some key differences:
+--------------------------------------+---------------+-------------------+
| x |``isscalar(x)``|``np.ndim(x) == 0``|
+======================================+===============+===================+
| PEP 3141 numeric objects (including | ``True`` | ``True`` |
| builtins) | | |
+--------------------------------------+---------------+-------------------+
| builtin string and buffer objects | ``True`` | ``True`` |
+--------------------------------------+---------------+-------------------+
| other builtin objects, like | ``False`` | ``True`` |
| `pathlib.Path`, `Exception`, | | |
| the result of `re.compile` | | |
+--------------------------------------+---------------+-------------------+
| third-party objects like | ``False`` | ``True`` |
| `matplotlib.figure.Figure` | | |
+--------------------------------------+---------------+-------------------+
| zero-dimensional numpy arrays | ``False`` | ``True`` |
+--------------------------------------+---------------+-------------------+
| other numpy arrays | ``False`` | ``False`` |
+--------------------------------------+---------------+-------------------+
| `list`, `tuple`, and other sequence | ``False`` | ``False`` |
| objects | | |
+--------------------------------------+---------------+-------------------+
Examples
--------
>>> np.isscalar(3.1)
True
>>> np.isscalar(np.array(3.1))
False
>>> np.isscalar([3.1])
False
>>> np.isscalar(False)
True
>>> np.isscalar('numpy')
True
NumPy supports PEP 3141 numbers:
>>> from fractions import Fraction
>>> np.isscalar(Fraction(5, 17))
True
>>> from numbers import Number
>>> np.isscalar(Number())
True
"""
return (isinstance(element, generic)
or type(element) in ScalarType
or isinstance(element, numbers.Number))
@set_module('numpy')
def binary_repr(num, width=None):
"""
Return the binary representation of the input number as a string.
For negative numbers, if width is not given, a minus sign is added to the
front. If width is given, the two's complement of the number is
returned, with respect to that width.
In a two's-complement system negative numbers are represented by the two's
complement of the absolute value. This is the most common method of
representing signed integers on computers [1]_. A N-bit two's-complement
system can represent every integer in the range
:math:`-2^{N-1}` to :math:`+2^{N-1}-1`.
Parameters
----------
num : int
Only an integer decimal number can be used.
width : int, optional
The length of the returned string if `num` is positive, or the length
of the two's complement if `num` is negative, provided that `width` is
at least a sufficient number of bits for `num` to be represented in the
designated form.
If the `width` value is insufficient, it will be ignored, and `num` will
be returned in binary (`num` > 0) or two's complement (`num` < 0) form
with its width equal to the minimum number of bits needed to represent
the number in the designated form. This behavior is deprecated and will
later raise an error.
.. deprecated:: 1.12.0
Returns
-------
bin : str
Binary representation of `num` or two's complement of `num`.
See Also
--------
base_repr: Return a string representation of a number in the given base
system.
bin: Python's built-in binary representation generator of an integer.
Notes
-----
`binary_repr` is equivalent to using `base_repr` with base 2, but about 25x
faster.
References
----------
.. [1] Wikipedia, "Two's complement",
https://en.wikipedia.org/wiki/Two's_complement
Examples
--------
>>> np.binary_repr(3)
'11'
>>> np.binary_repr(-3)
'-11'
>>> np.binary_repr(3, width=4)
'0011'
The two's complement is returned when the input number is negative and
width is specified:
>>> np.binary_repr(-3, width=3)
'101'
>>> np.binary_repr(-3, width=5)
'11101'
"""
def warn_if_insufficient(width, binwidth):
if width is not None and width < binwidth:
warnings.warn(
"Insufficient bit width provided. This behavior "
"will raise an error in the future.", DeprecationWarning,
stacklevel=3)
# Ensure that num is a Python integer to avoid overflow or unwanted
# casts to floating point.
num = operator.index(num)
if num == 0:
return '0' * (width or 1)
elif num > 0:
binary = bin(num)[2:]
binwidth = len(binary)
outwidth = (binwidth if width is None
else max(binwidth, width))
warn_if_insufficient(width, binwidth)
return binary.zfill(outwidth)
else:
if width is None:
return '-' + bin(-num)[2:]
else:
poswidth = len(bin(-num)[2:])
# See gh-8679: remove extra digit
# for numbers at boundaries.
if 2**(poswidth - 1) == -num:
poswidth -= 1
twocomp = 2**(poswidth + 1) + num
binary = bin(twocomp)[2:]
binwidth = len(binary)
outwidth = max(binwidth, width)
warn_if_insufficient(width, binwidth)
return '1' * (outwidth - binwidth) + binary
@set_module('numpy')
def base_repr(number, base=2, padding=0):
"""
Return a string representation of a number in the given base system.
Parameters
----------
number : int
The value to convert. Positive and negative values are handled.
base : int, optional
Convert `number` to the `base` number system. The valid range is 2-36,
the default value is 2.
padding : int, optional
Number of zeros padded on the left. Default is 0 (no padding).
Returns
-------
out : str
String representation of `number` in `base` system.
See Also
--------
binary_repr : Faster version of `base_repr` for base 2.
Examples
--------
>>> np.base_repr(5)
'101'
>>> np.base_repr(6, 5)
'11'
>>> np.base_repr(7, base=5, padding=3)
'00012'
>>> np.base_repr(10, base=16)
'A'
>>> np.base_repr(32, base=16)
'20'
"""
digits = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ'
if base > len(digits):
raise ValueError("Bases greater than 36 not handled in base_repr.")
elif base < 2:
raise ValueError("Bases less than 2 not handled in base_repr.")
num = abs(number)
res = []
while num:
res.append(digits[num % base])
num //= base
if padding:
res.append('0' * padding)
if number < 0:
res.append('-')
return ''.join(reversed(res or '0'))
# These are all essentially abbreviations
# These might wind up in a special abbreviations module
def _maketup(descr, val):
dt = dtype(descr)
# Place val in all scalar tuples:
fields = dt.fields
if fields is None:
return val
else:
res = [_maketup(fields[name][0], val) for name in dt.names]
return tuple(res)
def _identity_dispatcher(n, dtype=None, *, like=None):
return (like,)
@set_array_function_like_doc
@set_module('numpy')
def identity(n, dtype=None, *, like=None):
"""
Return the identity array.
The identity array is a square array with ones on
the main diagonal.
Parameters
----------
n : int
Number of rows (and columns) in `n` x `n` output.
dtype : data-type, optional
Data-type of the output. Defaults to ``float``.
${ARRAY_FUNCTION_LIKE}
.. versionadded:: 1.20.0
Returns
-------
out : ndarray
`n` x `n` array with its main diagonal set to one,
and all other elements 0.
Examples
--------
>>> np.identity(3)
array([[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])
"""
if like is not None:
return _identity_with_like(n, dtype=dtype, like=like)
from numpy import eye
return eye(n, dtype=dtype, like=like)
_identity_with_like = array_function_dispatch(
_identity_dispatcher
)(identity)
def _allclose_dispatcher(a, b, rtol=None, atol=None, equal_nan=None):
return (a, b)
@array_function_dispatch(_allclose_dispatcher)
def allclose(a, b, rtol=1.e-5, atol=1.e-8, equal_nan=False):
"""
Returns True if two arrays are element-wise equal within a tolerance.
The tolerance values are positive, typically very small numbers. The
relative difference (`rtol` * abs(`b`)) and the absolute difference
`atol` are added together to compare against the absolute difference
between `a` and `b`.
NaNs are treated as equal if they are in the same place and if
``equal_nan=True``. Infs are treated as equal if they are in the same
place and of the same sign in both arrays.
Parameters
----------
a, b : array_like
Input arrays to compare.
rtol : float
The relative tolerance parameter (see Notes).
atol : float
The absolute tolerance parameter (see Notes).
equal_nan : bool
Whether to compare NaN's as equal. If True, NaN's in `a` will be
considered equal to NaN's in `b` in the output array.
.. versionadded:: 1.10.0
Returns
-------
allclose : bool
Returns True if the two arrays are equal within the given
tolerance; False otherwise.
See Also
--------
isclose, all, any, equal
Notes
-----
If the following equation is element-wise True, then allclose returns
True.
absolute(`a` - `b`) <= (`atol` + `rtol` * absolute(`b`))
The above equation is not symmetric in `a` and `b`, so that
``allclose(a, b)`` might be different from ``allclose(b, a)`` in
some rare cases.
The comparison of `a` and `b` uses standard broadcasting, which
means that `a` and `b` need not have the same shape in order for
``allclose(a, b)`` to evaluate to True. The same is true for
`equal` but not `array_equal`.
`allclose` is not defined for non-numeric data types.
`bool` is considered a numeric data-type for this purpose.
Examples
--------
>>> np.allclose([1e10,1e-7], [1.00001e10,1e-8])
False
>>> np.allclose([1e10,1e-8], [1.00001e10,1e-9])
True
>>> np.allclose([1e10,1e-8], [1.0001e10,1e-9])
False
>>> np.allclose([1.0, np.nan], [1.0, np.nan])
False
>>> np.allclose([1.0, np.nan], [1.0, np.nan], equal_nan=True)
True
"""
res = all(isclose(a, b, rtol=rtol, atol=atol, equal_nan=equal_nan))
return bool(res)
def _isclose_dispatcher(a, b, rtol=None, atol=None, equal_nan=None):
return (a, b)
@array_function_dispatch(_isclose_dispatcher)
def isclose(a, b, rtol=1.e-5, atol=1.e-8, equal_nan=False):
"""
Returns a boolean array where two arrays are element-wise equal within a
tolerance.
The tolerance values are positive, typically very small numbers. The
relative difference (`rtol` * abs(`b`)) and the absolute difference
`atol` are added together to compare against the absolute difference
between `a` and `b`.
.. warning:: The default `atol` is not appropriate for comparing numbers
that are much smaller than one (see Notes).
Parameters
----------
a, b : array_like
Input arrays to compare.
rtol : float
The relative tolerance parameter (see Notes).
atol : float
The absolute tolerance parameter (see Notes).
equal_nan : bool
Whether to compare NaN's as equal. If True, NaN's in `a` will be
considered equal to NaN's in `b` in the output array.
Returns
-------
y : array_like
Returns a boolean array of where `a` and `b` are equal within the
given tolerance. If both `a` and `b` are scalars, returns a single
boolean value.
See Also
--------
allclose
math.isclose
Notes
-----
.. versionadded:: 1.7.0
For finite values, isclose uses the following equation to test whether
two floating point values are equivalent.
absolute(`a` - `b`) <= (`atol` + `rtol` * absolute(`b`))
Unlike the built-in `math.isclose`, the above equation is not symmetric
in `a` and `b` -- it assumes `b` is the reference value -- so that
`isclose(a, b)` might be different from `isclose(b, a)`. Furthermore,
the default value of atol is not zero, and is used to determine what
small values should be considered close to zero. The default value is
appropriate for expected values of order unity: if the expected values
are significantly smaller than one, it can result in false positives.
`atol` should be carefully selected for the use case at hand. A zero value
for `atol` will result in `False` if either `a` or `b` is zero.
`isclose` is not defined for non-numeric data types.
`bool` is considered a numeric data-type for this purpose.
Examples
--------
>>> np.isclose([1e10,1e-7], [1.00001e10,1e-8])
array([ True, False])
>>> np.isclose([1e10,1e-8], [1.00001e10,1e-9])
array([ True, True])
>>> np.isclose([1e10,1e-8], [1.0001e10,1e-9])
array([False, True])
>>> np.isclose([1.0, np.nan], [1.0, np.nan])
array([ True, False])
>>> np.isclose([1.0, np.nan], [1.0, np.nan], equal_nan=True)
array([ True, True])
>>> np.isclose([1e-8, 1e-7], [0.0, 0.0])
array([ True, False])
>>> np.isclose([1e-100, 1e-7], [0.0, 0.0], atol=0.0)
array([False, False])
>>> np.isclose([1e-10, 1e-10], [1e-20, 0.0])
array([ True, True])
>>> np.isclose([1e-10, 1e-10], [1e-20, 0.999999e-10], atol=0.0)
array([False, True])
"""
def within_tol(x, y, atol, rtol):
with errstate(invalid='ignore'):
return less_equal(abs(x-y), atol + rtol * abs(y))
x = asanyarray(a)
y = asanyarray(b)
# Make sure y is an inexact type to avoid bad behavior on abs(MIN_INT).
# This will cause casting of x later. Also, make sure to allow subclasses
# (e.g., for numpy.ma).
# NOTE: We explicitly allow timedelta, which used to work. This could
# possibly be deprecated. See also gh-18286.
# timedelta works if `atol` is an integer or also a timedelta.
# Although, the default tolerances are unlikely to be useful
if y.dtype.kind != "m":
dt = multiarray.result_type(y, 1.)
y = asanyarray(y, dtype=dt)
xfin = isfinite(x)
yfin = isfinite(y)
if all(xfin) and all(yfin):
return within_tol(x, y, atol, rtol)
else:
finite = xfin & yfin
cond = zeros_like(finite, subok=True)
# Because we're using boolean indexing, x & y must be the same shape.
# Ideally, we'd just do x, y = broadcast_arrays(x, y). It's in
# lib.stride_tricks, though, so we can't import it here.
x = x * ones_like(cond)
y = y * ones_like(cond)
# Avoid subtraction with infinite/nan values...
cond[finite] = within_tol(x[finite], y[finite], atol, rtol)
# Check for equality of infinite values...
cond[~finite] = (x[~finite] == y[~finite])
if equal_nan:
# Make NaN == NaN
both_nan = isnan(x) & isnan(y)
# Needed to treat masked arrays correctly. = True would not work.
cond[both_nan] = both_nan[both_nan]
return cond[()] # Flatten 0d arrays to scalars
def _array_equal_dispatcher(a1, a2, equal_nan=None):
return (a1, a2)
@array_function_dispatch(_array_equal_dispatcher)
def array_equal(a1, a2, equal_nan=False):
"""
True if two arrays have the same shape and elements, False otherwise.
Parameters
----------
a1, a2 : array_like
Input arrays.
equal_nan : bool
Whether to compare NaN's as equal. If the dtype of a1 and a2 is
complex, values will be considered equal if either the real or the
imaginary component of a given value is ``nan``.
.. versionadded:: 1.19.0
Returns
-------
b : bool
Returns True if the arrays are equal.
See Also
--------
allclose: Returns True if two arrays are element-wise equal within a
tolerance.
array_equiv: Returns True if input arrays are shape consistent and all
elements equal.
Examples
--------
>>> np.array_equal([1, 2], [1, 2])
True
>>> np.array_equal(np.array([1, 2]), np.array([1, 2]))
True
>>> np.array_equal([1, 2], [1, 2, 3])
False
>>> np.array_equal([1, 2], [1, 4])
False
>>> a = np.array([1, np.nan])
>>> np.array_equal(a, a)
False
>>> np.array_equal(a, a, equal_nan=True)
True
When ``equal_nan`` is True, complex values with nan components are
considered equal if either the real *or* the imaginary components are nan.
>>> a = np.array([1 + 1j])
>>> b = a.copy()
>>> a.real = np.nan
>>> b.imag = np.nan
>>> np.array_equal(a, b, equal_nan=True)
True
"""
try:
a1, a2 = asarray(a1), asarray(a2)
except Exception:
return False
if a1.shape != a2.shape:
return False
if not equal_nan:
return bool(asarray(a1 == a2).all())
# Handling NaN values if equal_nan is True
a1nan, a2nan = isnan(a1), isnan(a2)
# NaN's occur at different locations
if not (a1nan == a2nan).all():
return False
# Shapes of a1, a2 and masks are guaranteed to be consistent by this point
return bool(asarray(a1[~a1nan] == a2[~a1nan]).all())
def _array_equiv_dispatcher(a1, a2):
return (a1, a2)
@array_function_dispatch(_array_equiv_dispatcher)
def array_equiv(a1, a2):
"""
Returns True if input arrays are shape consistent and all elements equal.
Shape consistent means they are either the same shape, or one input array
can be broadcasted to create the same shape as the other one.
Parameters
----------
a1, a2 : array_like
Input arrays.
Returns
-------
out : bool
True if equivalent, False otherwise.
Examples
--------
>>> np.array_equiv([1, 2], [1, 2])
True
>>> np.array_equiv([1, 2], [1, 3])
False
Showing the shape equivalence:
>>> np.array_equiv([1, 2], [[1, 2], [1, 2]])
True
>>> np.array_equiv([1, 2], [[1, 2, 1, 2], [1, 2, 1, 2]])
False
>>> np.array_equiv([1, 2], [[1, 2], [1, 3]])
False
"""
try:
a1, a2 = asarray(a1), asarray(a2)
except Exception:
return False
try:
multiarray.broadcast(a1, a2)
except Exception:
return False
return bool(asarray(a1 == a2).all())
Inf = inf = infty = Infinity = PINF
nan = NaN = NAN
False_ = bool_(False)
True_ = bool_(True)
def extend_all(module):
existing = set(__all__)
mall = getattr(module, '__all__')
for a in mall:
if a not in existing:
__all__.append(a)
from .umath import *
from .numerictypes import *
from . import fromnumeric
from .fromnumeric import *
from . import arrayprint
from .arrayprint import *
from . import _asarray
from ._asarray import *
from . import _ufunc_config
from ._ufunc_config import *
extend_all(fromnumeric)
extend_all(umath)
extend_all(numerictypes)
extend_all(arrayprint)
extend_all(_asarray)
extend_all(_ufunc_config)