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편집 파일: numeric.py

from __future__ import division, absolute_import, print_function

import collections
import itertools
import operator
import sys
import warnings

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, broadcast, can_cast, compare_chararrays,
    concatenate, copyto, count_nonzero, dot, dtype, empty,
    empty_like, flatiter, frombuffer, fromfile, fromiter, fromstring,
    inner, int_asbuffer, 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)
if sys.version_info[0] < 3:
    from .multiarray import newbuffer, getbuffer

from . import umath
from .umath import (invert, sin, UFUNC_BUFSIZE_DEFAULT, ERR_IGNORE,
                    ERR_WARN, ERR_RAISE, ERR_CALL, ERR_PRINT, ERR_LOG,
                    ERR_DEFAULT, PINF, NAN)
from . import numerictypes
from .numerictypes import longlong, intc, int_, float_, complex_, bool_
from ._internal import TooHardError, AxisError

bitwise_not = invert
ufunc = type(sin)
newaxis = None

if sys.version_info[0] >= 3:
    import pickle
    basestring = str
    import builtins
else:
    import cPickle as pickle
    import __builtin__ as builtins

loads = pickle.loads

__all__ = [
    'newaxis', 'ndarray', 'flatiter', 'nditer', 'nested_iters', 'ufunc',
    'arange', 'array', 'zeros', 'count_nonzero', 'empty', 'broadcast',
    'dtype', 'fromstring', 'fromfile', 'frombuffer', 'int_asbuffer',
    'where', 'argwhere', 'copyto', 'concatenate', 'fastCopyAndTranspose',
    'lexsort', 'set_numeric_ops', 'can_cast', 'promote_types',
    'min_scalar_type', 'result_type', 'asarray', 'asanyarray',
    'ascontiguousarray', 'asfortranarray', 'isfortran', 'empty_like',
    'zeros_like', 'ones_like', 'correlate', 'convolve', 'inner', 'dot',
    'outer', 'vdot', 'roll',
    'rollaxis', 'moveaxis', 'cross', 'tensordot', 'array2string',
    'get_printoptions', 'set_printoptions', 'array_repr', 'array_str',
    'set_string_function', 'little_endian', 'require', 'fromiter',
    'array_equal', 'array_equiv', 'indices', 'fromfunction', 'isclose', 'load',
    'loads', 'isscalar', 'binary_repr', 'base_repr', 'ones', 'identity',
    'allclose', 'compare_chararrays', 'putmask', 'seterr', 'geterr',
    'setbufsize', 'getbufsize', 'seterrcall', 'geterrcall', 'errstate',
    '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'
    ]

if sys.version_info[0] < 3:
    __all__.extend(['getbuffer', 'newbuffer'])

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(a, dtype=None, order='K', subok=True):
    """
    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.

Returns
    -------
    out : ndarray
        Array of zeros with the same shape and type as `a`.

See Also
    --------
    ones_like : Return an array of ones with shape and type of input.
    empty_like : Return an empty array with shape and type of input.
    zeros : Return a new array setting values to zero.
    ones : Return a new array setting values to one.
    empty : Return a new uninitialized array.

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=np.float)
    >>> y
    array([ 0.,  1.,  2.])
    >>> np.zeros_like(y)
    array([ 0.,  0.,  0.])

"""
    res = empty_like(a, dtype=dtype, order=order, subok=subok)
    # needed instead of a 0 to get same result as zeros for for string dtypes
    z = zeros(1, dtype=res.dtype)
    multiarray.copyto(res, z, casting='unsafe')
    return res

def ones(shape, dtype=None, order='C'):
    """
    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
        Whether to store multidimensional data in C- or Fortran-contiguous
        (row- or column-wise) order in memory.

Returns
    -------
    out : ndarray
        Array of ones with the given shape, dtype, and order.

See Also
    --------
    zeros, ones_like

Examples
    --------
    >>> np.ones(5)
    array([ 1.,  1.,  1.,  1.,  1.])

>>> np.ones((5,), dtype=np.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.]])

"""
    a = empty(shape, dtype, order)
    multiarray.copyto(a, 1, casting='unsafe')
    return a

def ones_like(a, dtype=None, order='K', subok=True):
    """
    Return an array of ones with the same shape and type as a given array.

Returns
    -------
    out : ndarray
        Array of ones with the same shape and type as `a`.

See Also
    --------
    zeros_like : Return an array of zeros with shape and type of input.
    empty_like : Return an empty array with shape and type of input.
    zeros : Return a new array setting values to zero.
    ones : Return a new array setting values to one.
    empty : Return a new uninitialized array.

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=np.float)
    >>> y
    array([ 0.,  1.,  2.])
    >>> np.ones_like(y)
    array([ 1.,  1.,  1.])

"""
    res = empty_like(a, dtype=dtype, order=order, subok=subok)
    multiarray.copyto(res, 1, casting='unsafe')
    return res

def full(shape, fill_value, dtype=None, order='C'):
    """
    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
        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.

Returns
    -------
    out : ndarray
        Array of `fill_value` with the given shape, dtype, and order.

See Also
    --------
    zeros_like : Return an array of zeros with shape and type of input.
    ones_like : Return an array of ones with shape and type of input.
    empty_like : Return an empty array with shape and type of input.
    full_like : Fill an array with shape and type of input.
    zeros : Return a new array setting values to zero.
    ones : Return a new array setting values to one.
    empty : Return a new uninitialized array.

Examples
    --------
    >>> np.full((2, 2), np.inf)
    array([[ inf,  inf],
           [ inf,  inf]])
    >>> np.full((2, 2), 10)
    array([[10, 10],
           [10, 10]])

"""
    if dtype is None:
        dtype = array(fill_value).dtype
    a = empty(shape, dtype, order)
    multiarray.copyto(a, fill_value, casting='unsafe')
    return a

def full_like(a, fill_value, dtype=None, order='K', subok=True):
    """
    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 : scalar
        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.

Returns
    -------
    out : ndarray
        Array of `fill_value` with the same shape and type as `a`.

See Also
    --------
    zeros_like : Return an array of zeros with shape and type of input.
    ones_like : Return an array of ones with shape and type of input.
    empty_like : Return an empty array with shape and type of input.
    zeros : Return a new array setting values to zero.
    ones : Return a new array setting values to one.
    empty : Return a new uninitialized array.
    full : Fill a new array.

Examples
    --------
    >>> x = np.arange(6, dtype=np.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])

"""
    res = empty_like(a, dtype=dtype, order=order, subok=subok)
    multiarray.copyto(res, fill_value, casting='unsafe')
    return res

def extend_all(module):
    adict = {}
    for a in __all__:
        adict[a] = 1
    try:
        mall = getattr(module, '__all__')
    except AttributeError:
        mall = [k for k in module.__dict__.keys() if not k.startswith('_')]
    for a in mall:
        if a not in adict:
            __all__.append(a)

def count_nonzero(a, axis=None):
    """
    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

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
    >>> np.count_nonzero([[0,1,7,0,0],[3,0,0,2,19]])
    5
    >>> np.count_nonzero([[0,1,7,0,0],[3,0,0,2,19]], axis=0)
    array([1, 1, 1, 1, 1])
    >>> np.count_nonzero([[0,1,7,0,0],[3,0,0,2,19]], axis=1)
    array([2, 3])

"""
    if axis is None or axis == ():
        return multiarray.count_nonzero(a)

a = asanyarray(a)

if a.dtype == bool:
        return a.sum(axis=axis, dtype=np.intp)

if issubdtype(a.dtype, np.number):
        return (a != 0).sum(axis=axis, dtype=np.intp)

if (issubdtype(a.dtype, np.string_) or
            issubdtype(a.dtype, np.unicode_)):
        nullstr = a.dtype.type('')
        return (a != nullstr).sum(axis=axis, dtype=np.intp)

axis = asarray(normalize_axis_tuple(axis, a.ndim))
    counts = np.apply_along_axis(multiarray.count_nonzero, axis[0], a)

if axis.size == 1:
        return counts.astype(np.intp, copy=False)
    else:
        # for subsequent axis numbers, that number decreases
        # by one in this new 'counts' array if it was larger
        # than the first axis upon which 'count_nonzero' was
        # applied but remains unchanged if that number was
        # smaller than that first axis
        #
        # this trick enables us to perform counts on object-like
        # elements across multiple axes very quickly because integer
        # addition is very well optimized
        return counts.sum(axis=tuple(axis[1:] - (
            axis[1:] > axis[0])), dtype=np.intp)

def asarray(a, dtype=None, order=None):
    """Convert the input to an array.

Parameters
    ----------
    a : array_like
        Input data, in any form that can be converted to an array.  This
        includes lists, lists of tuples, tuples, tuples of tuples, tuples
        of lists and ndarrays.
    dtype : data-type, optional
        By default, the data-type is inferred from the input data.
    order : {'C', 'F'}, optional
        Whether to use row-major (C-style) or
        column-major (Fortran-style) memory representation.
        Defaults to 'C'.

Returns
    -------
    out : ndarray
        Array interpretation of `a`.  No copy is performed if the input
        is already an ndarray with matching dtype and order.  If `a` is a
        subclass of ndarray, a base class ndarray is returned.

See Also
    --------
    asanyarray : Similar function which passes through subclasses.
    ascontiguousarray : Convert input to a contiguous array.
    asfarray : Convert input to a floating point ndarray.
    asfortranarray : Convert input to an ndarray with column-major
                     memory order.
    asarray_chkfinite : Similar function which checks input for NaNs and Infs.
    fromiter : Create an array from an iterator.
    fromfunction : Construct an array by executing a function on grid
                   positions.

Examples
    --------
    Convert a list into an array:

>>> a = [1, 2]
    >>> np.asarray(a)
    array([1, 2])

Existing arrays are not copied:

>>> a = np.array([1, 2])
    >>> np.asarray(a) is a
    True

If `dtype` is set, array is copied only if dtype does not match:

>>> a = np.array([1, 2], dtype=np.float32)
    >>> np.asarray(a, dtype=np.float32) is a
    True
    >>> np.asarray(a, dtype=np.float64) is a
    False

Contrary to `asanyarray`, ndarray subclasses are not passed through:

>>> issubclass(np.matrix, np.ndarray)
    True
    >>> a = np.matrix([[1, 2]])
    >>> np.asarray(a) is a
    False
    >>> np.asanyarray(a) is a
    True

"""
    return array(a, dtype, copy=False, order=order)

def asanyarray(a, dtype=None, order=None):
    """Convert the input to an ndarray, but pass ndarray subclasses through.

Parameters
    ----------
    a : array_like
        Input data, in any form that can be converted to an array.  This
        includes scalars, lists, lists of tuples, tuples, tuples of tuples,
        tuples of lists, and ndarrays.
    dtype : data-type, optional
        By default, the data-type is inferred from the input data.
    order : {'C', 'F'}, optional
        Whether to use row-major (C-style) or column-major
        (Fortran-style) memory representation.  Defaults to 'C'.

Returns
    -------
    out : ndarray or an ndarray subclass
        Array interpretation of `a`.  If `a` is an ndarray or a subclass
        of ndarray, it is returned as-is and no copy is performed.

See Also
    --------
    asarray : Similar function which always returns ndarrays.
    ascontiguousarray : Convert input to a contiguous array.
    asfarray : Convert input to a floating point ndarray.
    asfortranarray : Convert input to an ndarray with column-major
                     memory order.
    asarray_chkfinite : Similar function which checks input for NaNs and
                        Infs.
    fromiter : Create an array from an iterator.
    fromfunction : Construct an array by executing a function on grid
                   positions.

Examples
    --------
    Convert a list into an array:

>>> a = [1, 2]
    >>> np.asanyarray(a)
    array([1, 2])

Instances of `ndarray` subclasses are passed through as-is:

>>> a = np.matrix([1, 2])
    >>> np.asanyarray(a) is a
    True

"""
    return array(a, dtype, copy=False, order=order, subok=True)

def ascontiguousarray(a, dtype=None):
    """
    Return a contiguous array in memory (C order).

Parameters
    ----------
    a : array_like
        Input array.
    dtype : str or dtype object, optional
        Data-type of returned array.

Returns
    -------
    out : ndarray
        Contiguous array of same shape and content as `a`, with type `dtype`
        if specified.

See Also
    --------
    asfortranarray : Convert input to an ndarray with column-major
                     memory order.
    require : Return an ndarray that satisfies requirements.
    ndarray.flags : Information about the memory layout of the array.

Examples
    --------
    >>> x = np.arange(6).reshape(2,3)
    >>> np.ascontiguousarray(x, dtype=np.float32)
    array([[ 0.,  1.,  2.],
           [ 3.,  4.,  5.]], dtype=float32)
    >>> x.flags['C_CONTIGUOUS']
    True

"""
    return array(a, dtype, copy=False, order='C', ndmin=1)

def asfortranarray(a, dtype=None):
    """
    Return an array laid out in Fortran order in memory.

Parameters
    ----------
    a : array_like
        Input array.
    dtype : str or dtype object, optional
        By default, the data-type is inferred from the input data.

Returns
    -------
    out : ndarray
        The input `a` in Fortran, or column-major, order.

See Also
    --------
    ascontiguousarray : Convert input to a contiguous (C order) array.
    asanyarray : Convert input to an ndarray with either row or
        column-major memory order.
    require : Return an ndarray that satisfies requirements.
    ndarray.flags : Information about the memory layout of the array.

Examples
    --------
    >>> x = np.arange(6).reshape(2,3)
    >>> y = np.asfortranarray(x)
    >>> x.flags['F_CONTIGUOUS']
    False
    >>> y.flags['F_CONTIGUOUS']
    True

"""
    return array(a, dtype, copy=False, order='F', ndmin=1)

def require(a, dtype=None, requirements=None):
    """
    Return an ndarray of the provided type that satisfies requirements.

This function is useful to be sure that an array with the correct flags
    is returned for passing to compiled code (perhaps through ctypes).

Parameters
    ----------
    a : array_like
       The object to be converted to a type-and-requirement-satisfying array.
    dtype : data-type
       The required data-type. If None preserve the current dtype. If your
       application requires the data to be in native byteorder, include
       a byteorder specification as a part of the dtype specification.
    requirements : str or list of str
       The requirements list can be any of the following

* 'F_CONTIGUOUS' ('F') - ensure a Fortran-contiguous array
       * 'C_CONTIGUOUS' ('C') - ensure a C-contiguous array
       * 'ALIGNED' ('A')      - ensure a data-type aligned array
       * 'WRITEABLE' ('W')    - ensure a writable array
       * 'OWNDATA' ('O')      - ensure an array that owns its own data
       * 'ENSUREARRAY', ('E') - ensure a base array, instead of a subclass

See Also
    --------
    asarray : Convert input to an ndarray.
    asanyarray : Convert to an ndarray, but pass through ndarray subclasses.
    ascontiguousarray : Convert input to a contiguous array.
    asfortranarray : Convert input to an ndarray with column-major
                     memory order.
    ndarray.flags : Information about the memory layout of the array.

Notes
    -----
    The returned array will be guaranteed to have the listed requirements
    by making a copy if needed.

Examples
    --------
    >>> x = np.arange(6).reshape(2,3)
    >>> x.flags
      C_CONTIGUOUS : True
      F_CONTIGUOUS : False
      OWNDATA : False
      WRITEABLE : True
      ALIGNED : True
      UPDATEIFCOPY : False

>>> y = np.require(x, dtype=np.float32, requirements=['A', 'O', 'W', 'F'])
    >>> y.flags
      C_CONTIGUOUS : False
      F_CONTIGUOUS : True
      OWNDATA : True
      WRITEABLE : True
      ALIGNED : True
      UPDATEIFCOPY : False

"""
    possible_flags = {'C':'C', 'C_CONTIGUOUS':'C', 'CONTIGUOUS':'C',
                      'F':'F', 'F_CONTIGUOUS':'F', 'FORTRAN':'F',
                      'A':'A', 'ALIGNED':'A',
                      'W':'W', 'WRITEABLE':'W',
                      'O':'O', 'OWNDATA':'O',
                      'E':'E', 'ENSUREARRAY':'E'}
    if not requirements:
        return asanyarray(a, dtype=dtype)
    else:
        requirements = set(possible_flags[x.upper()] for x in requirements)

if 'E' in requirements:
        requirements.remove('E')
        subok = False
    else:
        subok = True

order = 'A'
    if requirements >= set(['C', 'F']):
        raise ValueError('Cannot specify both "C" and "F" order')
    elif 'F' in requirements:
        order = 'F'
        requirements.remove('F')
    elif 'C' in requirements:
        order = 'C'
        requirements.remove('C')

arr = array(a, dtype=dtype, order=order, copy=False, subok=subok)

for prop in requirements:
        if not arr.flags[prop]:
            arr = arr.copy(order)
            break
    return arr

def isfortran(a):
    """
    Returns True 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.

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='FORTRAN')
    >>> 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='FORTRAN'))
    False

"""
    return a.flags.fnc

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 : ndarray
        Indices of elements that are non-zero. Indices are grouped by element.

See Also
    --------
    where, nonzero

Notes
    -----
    ``np.argwhere(a)`` is the same as ``np.transpose(np.nonzero(a))``.

The output of ``argwhere`` is not suitable for indexing arrays.
    For this purpose use ``where(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]])

"""
    return transpose(nonzero(a))

def flatnonzero(a):
    """
    Return indices that are non-zero in the flattened version of a.

This is equivalent to a.ravel().nonzero()[0].

Parameters
    ----------
    a : ndarray
        Input array.

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 a.ravel().nonzero()[0]

_mode_from_name_dict = {'v': 0,
                        's': 1,
                        'f': 2}

def _mode_from_name(mode):
    if isinstance(mode, basestring):
        return _mode_from_name_dict[mode.lower()[0]]
    return mode

def correlate(a, v, mode='valid'):
    """
    Cross-correlation of two 1-dimensional sequences.

This function computes the correlation as generally defined in signal
    processing texts::

c_{av}[k] = sum_n a[n+k] * conj(v[n])

with a and v sequences being zero-padded where necessary and conj being
    the conjugate.

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.

Notes
    -----
    The definition of correlation above is not unique and sometimes correlation
    may be defined differently. Another common definition is::

c'_{av}[k] = sum_n a[n] conj(v[n+k])

which is related to ``c_{av}[k]`` by ``c'_{av}[k] = c_{av}[-k]``.

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
    when the two input sequences change places, i.e.,
    ``c_{va}[k] = c^{*}_{av}[-k]``:

>>> 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])

"""
    mode = _mode_from_name(mode)
    return multiarray.correlate2(a, v, mode)

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", http://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')
    mode = _mode_from_name(mode)
    return multiarray.correlate(a, v[::-1], mode)

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
    --------
    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.

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]], dtype=bool)

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:
        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 = (-1, 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, -1)
    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(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])

>>> 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, 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]])

"""
    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(a, axis, start=0):
    """
    Roll the specified axis backwards, until it lies in a given position.

Parameters
    ----------
    a : ndarray
        Input array.
    axis : int
        The axis to roll backwards.  The positions of the other axes do not
        change relative to one another.
    start : int, optional
        The axis is rolled until it lies before this position.  The default,
        0, results in a "complete" roll.

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
    """
    try:
        axis = [operator.index(axis)]
    except TypeError:
        axis = tuple(axis)
    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(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

# fix hack in scipy which imports this function
def _move_axis_to_0(a, axis):
    return rollaxis(a, axis, 0)

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)
    -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 = rollaxis(a, axisa, a.ndim)
    b = rollaxis(b, axisb, b.ndim)
    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

# This works because we are moving the last axis
    return rollaxis(cp, -1, axisc)

# Use numarray's printing function
from .arrayprint import array2string, get_printoptions, set_printoptions

_typelessdata = [int_, float_, complex_]
if issubclass(intc, int):
    _typelessdata.append(intc)

if issubclass(longlong, int):
    _typelessdata.append(longlong)

def array_repr(arr, max_line_width=None, precision=None, suppress_small=None):
    """
    Return the string representation of an array.

Parameters
    ----------
    arr : ndarray
        Input array.
    max_line_width : int, optional
        The maximum number of columns the string should span. Newline
        characters split the string appropriately after array elements.
    precision : int, optional
        Floating point precision. Default is the current printing precision
        (usually 8), which can be altered using `set_printoptions`.
    suppress_small : bool, optional
        Represent very small numbers as zero, default is False. Very small
        is defined by `precision`, if the precision is 8 then
        numbers smaller than 5e-9 are represented as zero.

Returns
    -------
    string : str
      The string representation of an array.

See Also
    --------
    array_str, array2string, set_printoptions

Examples
    --------
    >>> np.array_repr(np.array([1,2]))
    'array([1, 2])'
    >>> np.array_repr(np.ma.array([0.]))
    'MaskedArray([ 0.])'
    >>> np.array_repr(np.array([], np.int32))
    'array([], dtype=int32)'

>>> x = np.array([1e-6, 4e-7, 2, 3])
    >>> np.array_repr(x, precision=6, suppress_small=True)
    'array([ 0.000001,  0.      ,  2.      ,  3.      ])'

"""
    if type(arr) is not ndarray:
        class_name = type(arr).__name__
    else:
        class_name = "array"

if arr.size > 0 or arr.shape == (0,):
        lst = array2string(arr, max_line_width, precision, suppress_small,
                           ', ', class_name + "(")
    else:  # show zero-length shape unless it is (0,)
        lst = "[], shape=%s" % (repr(arr.shape),)

skipdtype = (arr.dtype.type in _typelessdata) and arr.size > 0

if skipdtype:
        return "%s(%s)" % (class_name, lst)
    else:
        typename = arr.dtype.name
        # Quote typename in the output if it is "complex".
        if typename and not (typename[0].isalpha() and typename.isalnum()):
            typename = "'%s'" % typename

lf = ' '
        if issubclass(arr.dtype.type, flexible):
            if arr.dtype.names:
                typename = "%s" % str(arr.dtype)
            else:
                typename = "'%s'" % str(arr.dtype)
            lf = '\n'+' '*len(class_name + "(")
        return "%s(%s,%sdtype=%s)" % (class_name, lst, lf, typename)

def array_str(a, max_line_width=None, precision=None, suppress_small=None):
    """
    Return a string representation of the data in an array.

The data in the array is returned as a single string.  This function is
    similar to `array_repr`, the difference being that `array_repr` also
    returns information on the kind of array and its data type.

Parameters
    ----------
    a : ndarray
        Input array.
    max_line_width : int, optional
        Inserts newlines if text is longer than `max_line_width`.  The
        default is, indirectly, 75.
    precision : int, optional
        Floating point precision.  Default is the current printing precision
        (usually 8), which can be altered using `set_printoptions`.
    suppress_small : bool, optional
        Represent numbers "very close" to zero as zero; default is False.
        Very close is defined by precision: if the precision is 8, e.g.,
        numbers smaller (in absolute value) than 5e-9 are represented as
        zero.

See Also
    --------
    array2string, array_repr, set_printoptions

Examples
    --------
    >>> np.array_str(np.arange(3))
    '[0 1 2]'

"""
    return array2string(a, max_line_width, precision, suppress_small, ' ', "", str)

def set_string_function(f, repr=True):
    """
    Set a Python function to be used when pretty printing arrays.

Parameters
    ----------
    f : function or None
        Function to be used to pretty print arrays. The function should expect
        a single array argument and return a string of the representation of
        the array. If None, the function is reset to the default NumPy function
        to print arrays.
    repr : bool, optional
        If True (default), the function for pretty printing (``__repr__``)
        is set, if False the function that returns the default string
        representation (``__str__``) is set.

See Also
    --------
    set_printoptions, get_printoptions

Examples
    --------
    >>> def pprint(arr):
    ...     return 'HA! - What are you going to do now?'
    ...
    >>> np.set_string_function(pprint)
    >>> a = np.arange(10)
    >>> a
    HA! - What are you going to do now?
    >>> print(a)
    [0 1 2 3 4 5 6 7 8 9]

We can reset the function to the default:

>>> np.set_string_function(None)
    >>> a
    array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

`repr` affects either pretty printing or normal string representation.
    Note that ``__repr__`` is still affected by setting ``__str__``
    because the width of each array element in the returned string becomes
    equal to the length of the result of ``__str__()``.

>>> x = np.arange(4)
    >>> np.set_string_function(lambda x:'random', repr=False)
    >>> x.__str__()
    'random'
    >>> x.__repr__()
    'array([     0,      1,      2,      3])'

"""
    if f is None:
        if repr:
            return multiarray.set_string_function(array_repr, 1)
        else:
            return multiarray.set_string_function(array_str, 0)
    else:
        return multiarray.set_string_function(f, repr)

set_string_function(array_str, 0)
set_string_function(array_repr, 1)

little_endian = (sys.byteorder == 'little')

def indices(dimensions, dtype=int):
    """
    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.

Returns
    -------
    grid : ndarray
        The array of grid indices,
        ``grid.shape = (len(dimensions),) + tuple(dimensions)``.

See Also
    --------
    mgrid, meshgrid

Notes
    -----
    The output shape 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]``.

"""
    dimensions = tuple(dimensions)
    N = len(dimensions)
    shape = (1,)*N
    res = empty((N,)+dimensions, dtype=dtype)
    for i, dim in enumerate(dimensions):
        res[i] = arange(dim, dtype=dtype).reshape(
            shape[:i] + (dim,) + shape[i+1:]
        )
    return res

def fromfunction(function, shape, **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.

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 match the `shape` parameter.

See Also
    --------
    indices, meshgrid

Notes
    -----
    Keywords other than `dtype` are passed to `function`.

Examples
    --------
    >>> np.fromfunction(lambda i, j: i == j, (3, 3), dtype=int)
    array([[ True, False, False],
           [False,  True, False],
           [False, False,  True]], dtype=bool)

>>> np.fromfunction(lambda i, j: i + j, (3, 3), dtype=int)
    array([[0, 1, 2],
           [1, 2, 3],
           [2, 3, 4]])

"""
    dtype = kwargs.pop('dtype', float)
    args = indices(shape, dtype=dtype)
    return function(*args, **kwargs)

def isscalar(num):
    """
    Returns True if the type of `num` is a scalar type.

Parameters
    ----------
    num : any
        Input argument, can be of any type and shape.

Returns
    -------
    val : bool
        True if `num` is a scalar type, False if it is not.

Examples
    --------
    >>> np.isscalar(3.1)
    True
    >>> np.isscalar([3.1])
    False
    >>> np.isscalar(False)
    True

"""
    if isinstance(num, generic):
        return True
    else:
        return type(num) in ScalarType

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",
        http://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, binwdith):
        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)

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

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'))

def load(file):
    """
    Wrapper around cPickle.load which accepts either a file-like object or
    a filename.

Note that the NumPy binary format is not based on pickle/cPickle anymore.
    For details on the preferred way of loading and saving files, see `load`
    and `save`.