shape_base.py

from __future__ import division, absolute_import, print_function

import functools
import warnings

import numpy.core.numeric as _nx
from numpy.core.numeric import (
    asarray, zeros, outer, concatenate, array, asanyarray
    )
from numpy.core.fromnumeric import product, reshape, transpose
from numpy.core.multiarray import normalize_axis_index
from numpy.core import overrides
from numpy.core import vstack, atleast_3d
from numpy.core.shape_base import (
    _arrays_for_stack_dispatcher, _warn_for_nonsequence)
from numpy.lib.index_tricks import ndindex
from numpy.matrixlib.defmatrix import matrix  # this raises all the right alarm bells


__all__ = [
    'column_stack', 'row_stack', 'dstack', 'array_split', 'split',
    'hsplit', 'vsplit', 'dsplit', 'apply_over_axes', 'expand_dims',
    'apply_along_axis', 'kron', 'tile', 'get_array_wrap', 'take_along_axis',
    'put_along_axis'
    ]


array_function_dispatch = functools.partial(
    overrides.array_function_dispatch, module='numpy')


def _make_along_axis_idx(arr_shape, indices, axis):
	# compute dimensions to iterate over
    if not _nx.issubdtype(indices.dtype, _nx.integer):
        raise IndexError('`indices` must be an integer array')
    if len(arr_shape) != indices.ndim:
        raise ValueError(
            "`indices` and `arr` must have the same number of dimensions")
    shape_ones = (1,) * indices.ndim
    dest_dims = list(range(axis)) + [None] + list(range(axis+1, indices.ndim))

    # build a fancy index, consisting of orthogonal aranges, with the
    # requested index inserted at the right location
    fancy_index = []
    for dim, n in zip(dest_dims, arr_shape):
        if dim is None:
            fancy_index.append(indices)
        else:
            ind_shape = shape_ones[:dim] + (-1,) + shape_ones[dim+1:]
            fancy_index.append(_nx.arange(n).reshape(ind_shape))

    return tuple(fancy_index)


def _take_along_axis_dispatcher(arr, indices, axis):
    return (arr, indices)


@array_function_dispatch(_take_along_axis_dispatcher)
def take_along_axis(arr, indices, axis):
    """
    Take values from the input array by matching 1d index and data slices.

    This iterates over matching 1d slices oriented along the specified axis in
    the index and data arrays, and uses the former to look up values in the
    latter. These slices can be different lengths.

    Functions returning an index along an axis, like `argsort` and
    `argpartition`, produce suitable indices for this function.

    .. versionadded:: 1.15.0

    Parameters
    ----------
    arr: ndarray (Ni..., M, Nk...)
        Source array
    indices: ndarray (Ni..., J, Nk...)
        Indices to take along each 1d slice of `arr`. This must match the
        dimension of arr, but dimensions Ni and Nj only need to broadcast
        against `arr`.
    axis: int
        The axis to take 1d slices along. If axis is None, the input array is
        treated as if it had first been flattened to 1d, for consistency with
        `sort` and `argsort`.

    Returns
    -------
    out: ndarray (Ni..., J, Nk...)
        The indexed result.

    Notes
    -----
    This is equivalent to (but faster than) the following use of `ndindex` and
    `s_`, which sets each of ``ii`` and ``kk`` to a tuple of indices::

        Ni, M, Nk = a.shape[:axis], a.shape[axis], a.shape[axis+1:]
        J = indices.shape[axis]  # Need not equal M
        out = np.empty(Nk + (J,) + Nk)

        for ii in ndindex(Ni):
            for kk in ndindex(Nk):
                a_1d       = a      [ii + s_[:,] + kk]
                indices_1d = indices[ii + s_[:,] + kk]
                out_1d     = out    [ii + s_[:,] + kk]
                for j in range(J):
                    out_1d[j] = a_1d[indices_1d[j]]

    Equivalently, eliminating the inner loop, the last two lines would be::

                out_1d[:] = a_1d[indices_1d]

    See Also
    --------
    take : Take along an axis, using the same indices for every 1d slice
    put_along_axis :
        Put values into the destination array by matching 1d index and data slices

    Examples
    --------

    For this sample array

    >>> a = np.array([[10, 30, 20], [60, 40, 50]])

    We can sort either by using sort directly, or argsort and this function

    >>> np.sort(a, axis=1)
    array([[10, 20, 30],
           [40, 50, 60]])
    >>> ai = np.argsort(a, axis=1); ai
    array([[0, 2, 1],
           [1, 2, 0]], dtype=int64)
    >>> np.take_along_axis(a, ai, axis=1)
    array([[10, 20, 30],
           [40, 50, 60]])

    The same works for max and min, if you expand the dimensions:

    >>> np.expand_dims(np.max(a, axis=1), axis=1)
    array([[30],
           [60]])
    >>> ai = np.expand_dims(np.argmax(a, axis=1), axis=1)
    >>> ai
    array([[1],
           [0], dtype=int64)
    >>> np.take_along_axis(a, ai, axis=1)
    array([[30],
           [60]])

    If we want to get the max and min at the same time, we can stack the
    indices first

    >>> ai_min = np.expand_dims(np.argmin(a, axis=1), axis=1)
    >>> ai_max = np.expand_dims(np.argmax(a, axis=1), axis=1)
    >>> ai = np.concatenate([ai_min, ai_max], axis=axis)
    >> ai
    array([[0, 1],
           [1, 0]], dtype=int64)
    >>> np.take_along_axis(a, ai, axis=1)
    array([[10, 30],
           [40, 60]])
    """
    # normalize inputs
    if axis is None:
        arr = arr.flat
        arr_shape = (len(arr),)  # flatiter has no .shape
        axis = 0
    else:
        axis = normalize_axis_index(axis, arr.ndim)
        arr_shape = arr.shape

    # use the fancy index
    return arr[_make_along_axis_idx(arr_shape, indices, axis)]


def _put_along_axis_dispatcher(arr, indices, values, axis):
    return (arr, indices, values)


@array_function_dispatch(_put_along_axis_dispatcher)
def put_along_axis(arr, indices, values, axis):
    """
    Put values into the destination array by matching 1d index and data slices.

    This iterates over matching 1d slices oriented along the specified axis in
    the index and data arrays, and uses the former to place values into the
    latter. These slices can be different lengths.

    Functions returning an index along an axis, like `argsort` and
    `argpartition`, produce suitable indices for this function.

    .. versionadded:: 1.15.0

    Parameters
    ----------
    arr: ndarray (Ni..., M, Nk...)
        Destination array.
    indices: ndarray (Ni..., J, Nk...)
        Indices to change along each 1d slice of `arr`. This must match the
        dimension of arr, but dimensions in Ni and Nj may be 1 to broadcast
        against `arr`.
    values: array_like (Ni..., J, Nk...)
        values to insert at those indices. Its shape and dimension are
        broadcast to match that of `indices`.
    axis: int
        The axis to take 1d slices along. If axis is None, the destination
        array is treated as if a flattened 1d view had been created of it.

    Notes
    -----
    This is equivalent to (but faster than) the following use of `ndindex` and
    `s_`, which sets each of ``ii`` and ``kk`` to a tuple of indices::

        Ni, M, Nk = a.shape[:axis], a.shape[axis], a.shape[axis+1:]
        J = indices.shape[axis]  # Need not equal M

        for ii in ndindex(Ni):
            for kk in ndindex(Nk):
                a_1d       = a      [ii + s_[:,] + kk]
                indices_1d = indices[ii + s_[:,] + kk]
                values_1d  = values [ii + s_[:,] + kk]
                for j in range(J):
                    a_1d[indices_1d[j]] = values_1d[j]

    Equivalently, eliminating the inner loop, the last two lines would be::

                a_1d[indices_1d] = values_1d

    See Also
    --------
    take_along_axis :
        Take values from the input array by matching 1d index and data slices

    Examples
    --------

    For this sample array

    >>> a = np.array([[10, 30, 20], [60, 40, 50]])

    We can replace the maximum values with:

    >>> ai = np.expand_dims(np.argmax(a, axis=1), axis=1)
    >>> ai
    array([[1],
           [0]], dtype=int64)
    >>> np.put_along_axis(a, ai, 99, axis=1)
    >>> a
    array([[10, 99, 20],
           [99, 40, 50]])

    """
    # normalize inputs
    if axis is None:
        arr = arr.flat
        axis = 0
        arr_shape = (len(arr),)  # flatiter has no .shape
    else:
        axis = normalize_axis_index(axis, arr.ndim)
        arr_shape = arr.shape

    # use the fancy index
    arr[_make_along_axis_idx(arr_shape, indices, axis)] = values


def _apply_along_axis_dispatcher(func1d, axis, arr, *args, **kwargs):
    return (arr,)


@array_function_dispatch(_apply_along_axis_dispatcher)
def apply_along_axis(func1d, axis, arr, *args, **kwargs):
    """
    Apply a function to 1-D slices along the given axis.

    Execute `func1d(a, *args)` where `func1d` operates on 1-D arrays and `a`
    is a 1-D slice of `arr` along `axis`.

    This is equivalent to (but faster than) the following use of `ndindex` and
    `s_`, which sets each of ``ii``, ``jj``, and ``kk`` to a tuple of indices::

        Ni, Nk = a.shape[:axis], a.shape[axis+1:]
        for ii in ndindex(Ni):
            for kk in ndindex(Nk):
                f = func1d(arr[ii + s_[:,] + kk])
                Nj = f.shape
                for jj in ndindex(Nj):
                    out[ii + jj + kk] = f[jj]

    Equivalently, eliminating the inner loop, this can be expressed as::

        Ni, Nk = a.shape[:axis], a.shape[axis+1:]
        for ii in ndindex(Ni):
            for kk in ndindex(Nk):
                out[ii + s_[...,] + kk] = func1d(arr[ii + s_[:,] + kk])

    Parameters
    ----------
    func1d : function (M,) -> (Nj...)
        This function should accept 1-D arrays. It is applied to 1-D
        slices of `arr` along the specified axis.
    axis : integer
        Axis along which `arr` is sliced.
    arr : ndarray (Ni..., M, Nk...)
        Input array.
    args : any
        Additional arguments to `func1d`.
    kwargs : any
        Additional named arguments to `func1d`.

        .. versionadded:: 1.9.0


    Returns
    -------
    out : ndarray  (Ni..., Nj..., Nk...)
        The output array. The shape of `out` is identical to the shape of
        `arr`, except along the `axis` dimension. This axis is removed, and
        replaced with new dimensions equal to the shape of the return value
        of `func1d`. So if `func1d` returns a scalar `out` will have one
        fewer dimensions than `arr`.

    See Also
    --------
    apply_over_axes : Apply a function repeatedly over multiple axes.

    Examples
    --------
    >>> def my_func(a):
    ...     \"\"\"Average first and last element of a 1-D array\"\"\"
    ...     return (a[0] + a[-1]) * 0.5
    >>> b = np.array([[1,2,3], [4,5,6], [7,8,9]])
    >>> np.apply_along_axis(my_func, 0, b)
    array([ 4.,  5.,  6.])
    >>> np.apply_along_axis(my_func, 1, b)
    array([ 2.,  5.,  8.])

    For a function that returns a 1D array, the number of dimensions in
    `outarr` is the same as `arr`.

    >>> b = np.array([[8,1,7], [4,3,9], [5,2,6]])
    >>> np.apply_along_axis(sorted, 1, b)
    array([[1, 7, 8],
           [3, 4, 9],
           [2, 5, 6]])

    For a function that returns a higher dimensional array, those dimensions
    are inserted in place of the `axis` dimension.

    >>> b = np.array([[1,2,3], [4,5,6], [7,8,9]])
    >>> np.apply_along_axis(np.diag, -1, b)
    array([[[1, 0, 0],
            [0, 2, 0],
            [0, 0, 3]],
           [[4, 0, 0],
            [0, 5, 0],
            [0, 0, 6]],
           [[7, 0, 0],
            [0, 8, 0],
            [0, 0, 9]]])
    """
    # handle negative axes
    arr = asanyarray(arr)
    nd = arr.ndim
    axis = normalize_axis_index(axis, nd)

    # arr, with the iteration axis at the end
    in_dims = list(range(nd))
    inarr_view = transpose(arr, in_dims[:axis] + in_dims[axis+1:] + [axis])

    # compute indices for the iteration axes, and append a trailing ellipsis to
    # prevent 0d arrays decaying to scalars, which fixes gh-8642
    inds = ndindex(inarr_view.shape[:-1])
    inds = (ind + (Ellipsis,) for ind in inds)

    # invoke the function on the first item
    try:
        ind0 = next(inds)
    except StopIteration:
        raise ValueError('Cannot apply_along_axis when any iteration dimensions are 0')
    res = asanyarray(func1d(inarr_view[ind0], *args, **kwargs))

    # build a buffer for storing evaluations of func1d.
    # remove the requested axis, and add the new ones on the end.
    # laid out so that each write is contiguous.
    # for a tuple index inds, buff[inds] = func1d(inarr_view[inds])
    buff = zeros(inarr_view.shape[:-1] + res.shape, res.dtype)

    # permutation of axes such that out = buff.transpose(buff_permute)
    buff_dims = list(range(buff.ndim))
    buff_permute = (
        buff_dims[0 : axis] +
        buff_dims[buff.ndim-res.ndim : buff.ndim] +
        buff_dims[axis : buff.ndim-res.ndim]
    )

    # matrices have a nasty __array_prepare__ and __array_wrap__
    if not isinstance(res, matrix):
        buff = res.__array_prepare__(buff)

    # save the first result, then compute and save all remaining results
    buff[ind0] = res
    for ind in inds:
        buff[ind] = asanyarray(func1d(inarr_view[ind], *args, **kwargs))

    if not isinstance(res, matrix):
        # wrap the array, to preserve subclasses
        buff = res.__array_wrap__(buff)

        # finally, rotate the inserted axes back to where they belong
        return transpose(buff, buff_permute)

    else:
        # matrices have to be transposed first, because they collapse dimensions!
        out_arr = transpose(buff, buff_permute)
        return res.__array_wrap__(out_arr)


def _apply_over_axes_dispatcher(func, a, axes):
    return (a,)


@array_function_dispatch(_apply_over_axes_dispatcher)
def apply_over_axes(func, a, axes):
    """
    Apply a function repeatedly over multiple axes.

    `func` is called as `res = func(a, axis)`, where `axis` is the first
    element of `axes`.  The result `res` of the function call must have
    either the same dimensions as `a` or one less dimension.  If `res`
    has one less dimension than `a`, a dimension is inserted before
    `axis`.  The call to `func` is then repeated for each axis in `axes`,
    with `res` as the first argument.

    Parameters
    ----------
    func : function
        This function must take two arguments, `func(a, axis)`.
    a : array_like
        Input array.
    axes : array_like
        Axes over which `func` is applied; the elements must be integers.

    Returns
    -------
    apply_over_axis : ndarray
        The output array.  The number of dimensions is the same as `a`,
        but the shape can be different.  This depends on whether `func`
        changes the shape of its output with respect to its input.

    See Also
    --------
    apply_along_axis :
        Apply a function to 1-D slices of an array along the given axis.

    Notes
    ------
    This function is equivalent to tuple axis arguments to reorderable ufuncs
    with keepdims=True. Tuple axis arguments to ufuncs have been available since
    version 1.7.0.

    Examples
    --------
    >>> a = np.arange(24).reshape(2,3,4)
    >>> a
    array([[[ 0,  1,  2,  3],
            [ 4,  5,  6,  7],
            [ 8,  9, 10, 11]],
           [[12, 13, 14, 15],
            [16, 17, 18, 19],
            [20, 21, 22, 23]]])

    Sum over axes 0 and 2. The result has same number of dimensions
    as the original array:

    >>> np.apply_over_axes(np.sum, a, [0,2])
    array([[[ 60],
            [ 92],
            [124]]])

    Tuple axis arguments to ufuncs are equivalent:

    >>> np.sum(a, axis=(0,2), keepdims=True)
    array([[[ 60],
            [ 92],
            [124]]])

    """
    val = asarray(a)
    N = a.ndim
    if array(axes).ndim == 0:
        axes = (axes,)
    for axis in axes:
        if axis < 0:
            axis = N + axis
        args = (val, axis)
        res = func(*args)
        if res.ndim == val.ndim:
            val = res
        else:
            res = expand_dims(res, axis)
            if res.ndim == val.ndim:
                val = res
            else:
                raise ValueError("function is not returning "
                                 "an array of the correct shape")
    return val


def _expand_dims_dispatcher(a, axis):
    return (a,)


@array_function_dispatch(_expand_dims_dispatcher)
def expand_dims(a, axis):
    """
    Expand the shape of an array.

    Insert a new axis that will appear at the `axis` position in the expanded
    array shape.

    .. note:: Previous to NumPy 1.13.0, neither ``axis < -a.ndim - 1`` nor
       ``axis > a.ndim`` raised errors or put the new axis where documented.
       Those axis values are now deprecated and will raise an AxisError in the
       future.

    Parameters
    ----------
    a : array_like
        Input array.
    axis : int
        Position in the expanded axes where the new axis is placed.

    Returns
    -------
    res : ndarray
        Output array. The number of dimensions is one greater than that of
        the input array.

    See Also
    --------
    squeeze : The inverse operation, removing singleton dimensions
    reshape : Insert, remove, and combine dimensions, and resize existing ones
    doc.indexing, atleast_1d, atleast_2d, atleast_3d

    Examples
    --------
    >>> x = np.array([1,2])
    >>> x.shape
    (2,)

    The following is equivalent to ``x[np.newaxis,:]`` or ``x[np.newaxis]``:

    >>> y = np.expand_dims(x, axis=0)
    >>> y
    array([[1, 2]])
    >>> y.shape
    (1, 2)

    >>> y = np.expand_dims(x, axis=1)  # Equivalent to x[:,np.newaxis]
    >>> y
    array([[1],
           [2]])
    >>> y.shape
    (2, 1)

    Note that some examples may use ``None`` instead of ``np.newaxis``.  These
    are the same objects:

    >>> np.newaxis is None
    True

    """
    if isinstance(a, matrix):
        a = asarray(a)
    else:
        a = asanyarray(a)

    shape = a.shape
    if axis > a.ndim or axis < -a.ndim - 1:
        # 2017-05-17, 1.13.0
        warnings.warn("Both axis > a.ndim and axis < -a.ndim - 1 are "
                      "deprecated and will raise an AxisError in the future.",
                      DeprecationWarning, stacklevel=2)
    # When the deprecation period expires, delete this if block,
    if axis < 0:
        axis = axis + a.ndim + 1
    # and uncomment the following line.
    # axis = normalize_axis_index(axis, a.ndim + 1)
    return a.reshape(shape[:axis] + (1,) + shape[axis:])


row_stack = vstack


def _column_stack_dispatcher(tup):
    return _arrays_for_stack_dispatcher(tup)


@array_function_dispatch(_column_stack_dispatcher)
def column_stack(tup):
    """
    Stack 1-D arrays as columns into a 2-D array.

    Take a sequence of 1-D arrays and stack them as columns
    to make a single 2-D array. 2-D arrays are stacked as-is,
    just like with `hstack`.  1-D arrays are turned into 2-D columns
    first.

    Parameters
    ----------
    tup : sequence of 1-D or 2-D arrays.
        Arrays to stack. All of them must have the same first dimension.

    Returns
    -------
    stacked : 2-D array
        The array formed by stacking the given arrays.

    See Also
    --------
    stack, hstack, vstack, concatenate

    Examples
    --------
    >>> a = np.array((1,2,3))
    >>> b = np.array((2,3,4))
    >>> np.column_stack((a,b))
    array([[1, 2],
           [2, 3],
           [3, 4]])

    """
    _warn_for_nonsequence(tup)
    arrays = []
    for v in tup:
        arr = array(v, copy=False, subok=True)
        if arr.ndim < 2:
            arr = array(arr, copy=False, subok=True, ndmin=2).T
        arrays.append(arr)
    return _nx.concatenate(arrays, 1)


def _dstack_dispatcher(tup):
    return _arrays_for_stack_dispatcher(tup)


@array_function_dispatch(_dstack_dispatcher)
def dstack(tup):
    """
    Stack arrays in sequence depth wise (along third axis).

    This is equivalent to concatenation along the third axis after 2-D arrays
    of shape `(M,N)` have been reshaped to `(M,N,1)` and 1-D arrays of shape
    `(N,)` have been reshaped to `(1,N,1)`. Rebuilds arrays divided by
    `dsplit`.

    This function makes most sense for arrays with up to 3 dimensions. For
    instance, for pixel-data with a height (first axis), width (second axis),
    and r/g/b channels (third axis). The functions `concatenate`, `stack` and
    `block` provide more general stacking and concatenation operations.

    Parameters
    ----------
    tup : sequence of arrays
        The arrays must have the same shape along all but the third axis.
        1-D or 2-D arrays must have the same shape.

    Returns
    -------
    stacked : ndarray
        The array formed by stacking the given arrays, will be at least 3-D.

    See Also
    --------
    stack : Join a sequence of arrays along a new axis.
    vstack : Stack along first axis.
    hstack : Stack along second axis.
    concatenate : Join a sequence of arrays along an existing axis.
    dsplit : Split array along third axis.

    Examples
    --------
    >>> a = np.array((1,2,3))
    >>> b = np.array((2,3,4))
    >>> np.dstack((a,b))
    array([[[1, 2],
            [2, 3],
            [3, 4]]])

    >>> a = np.array([[1],[2],[3]])
    >>> b = np.array([[2],[3],[4]])
    >>> np.dstack((a,b))
    array([[[1, 2]],
           [[2, 3]],
           [[3, 4]]])

    """
    _warn_for_nonsequence(tup)
    return _nx.concatenate([atleast_3d(_m) for _m in tup], 2)


def _replace_zero_by_x_arrays(sub_arys):
    for i in range(len(sub_arys)):
        if _nx.ndim(sub_arys[i]) == 0:
            sub_arys[i] = _nx.empty(0, dtype=sub_arys[i].dtype)
        elif _nx.sometrue(_nx.equal(_nx.shape(sub_arys[i]), 0)):
            sub_arys[i] = _nx.empty(0, dtype=sub_arys[i].dtype)
    return sub_arys


def _array_split_dispatcher(ary, indices_or_sections, axis=None):
    return (ary, indices_or_sections)


@array_function_dispatch(_array_split_dispatcher)
def array_split(ary, indices_or_sections, axis=0):
    """
    Split an array into multiple sub-arrays.

    Please refer to the ``split`` documentation.  The only difference
    between these functions is that ``array_split`` allows
    `indices_or_sections` to be an integer that does *not* equally
    divide the axis. For an array of length l that should be split
    into n sections, it returns l % n sub-arrays of size l//n + 1
    and the rest of size l//n.

    See Also
    --------
    split : Split array into multiple sub-arrays of equal size.

    Examples
    --------
    >>> x = np.arange(8.0)
    >>> np.array_split(x, 3)
        [array([ 0.,  1.,  2.]), array([ 3.,  4.,  5.]), array([ 6.,  7.])]

    >>> x = np.arange(7.0)
    >>> np.array_split(x, 3)
        [array([ 0.,  1.,  2.]), array([ 3.,  4.]), array([ 5.,  6.])]

    """
    try:
        Ntotal = ary.shape[axis]
    except AttributeError:
        Ntotal = len(ary)
    try:
        # handle array case.
        Nsections = len(indices_or_sections) + 1
        div_points = [0] + list(indices_or_sections) + [Ntotal]
    except TypeError:
        # indices_or_sections is a scalar, not an array.
        Nsections = int(indices_or_sections)
        if Nsections <= 0:
            raise ValueError('number sections must be larger than 0.')
        Neach_section, extras = divmod(Ntotal, Nsections)
        section_sizes = ([0] +
                         extras * [Neach_section+1] +
                         (Nsections-extras) * [Neach_section])
        div_points = _nx.array(section_sizes, dtype=_nx.intp).cumsum()

    sub_arys = []
    sary = _nx.swapaxes(ary, axis, 0)
    for i in range(Nsections):
        st = div_points[i]
        end = div_points[i + 1]
        sub_arys.append(_nx.swapaxes(sary[st:end], axis, 0))

    return sub_arys


def _split_dispatcher(ary, indices_or_sections, axis=None):
    return (ary, indices_or_sections)


@array_function_dispatch(_split_dispatcher)
def split(ary, indices_or_sections, axis=0):
    """
    Split an array into multiple sub-arrays.

    Parameters
    ----------
    ary : ndarray
        Array to be divided into sub-arrays.
    indices_or_sections : int or 1-D array
        If `indices_or_sections` is an integer, N, the array will be divided
        into N equal arrays along `axis`.  If such a split is not possible,
        an error is raised.

        If `indices_or_sections` is a 1-D array of sorted integers, the entries
        indicate where along `axis` the array is split.  For example,
        ``[2, 3]`` would, for ``axis=0``, result in

          - ary[:2]
          - ary[2:3]
          - ary[3:]

        If an index exceeds the dimension of the array along `axis`,
        an empty sub-array is returned correspondingly.
    axis : int, optional
        The axis along which to split, default is 0.

    Returns
    -------
    sub-arrays : list of ndarrays
        A list of sub-arrays.

    Raises
    ------
    ValueError
        If `indices_or_sections` is given as an integer, but
        a split does not result in equal division.

    See Also
    --------
    array_split : Split an array into multiple sub-arrays of equal or
                  near-equal size.  Does not raise an exception if
                  an equal division cannot be made.
    hsplit : Split array into multiple sub-arrays horizontally (column-wise).
    vsplit : Split array into multiple sub-arrays vertically (row wise).
    dsplit : Split array into multiple sub-arrays along the 3rd axis (depth).
    concatenate : Join a sequence of arrays along an existing axis.
    stack : Join a sequence of arrays along a new axis.
    hstack : Stack arrays in sequence horizontally (column wise).
    vstack : Stack arrays in sequence vertically (row wise).
    dstack : Stack arrays in sequence depth wise (along third dimension).

    Examples
    --------
    >>> x = np.arange(9.0)
    >>> np.split(x, 3)
    [array([ 0.,  1.,  2.]), array([ 3.,  4.,  5.]), array([ 6.,  7.,  8.])]

    >>> x = np.arange(8.0)
    >>> np.split(x, [3, 5, 6, 10])
    [array([ 0.,  1.,  2.]),
     array([ 3.,  4.]),
     array([ 5.]),
     array([ 6.,  7.]),
     array([], dtype=float64)]

    """
    try:
        len(indices_or_sections)
    except TypeError:
        sections = indices_or_sections
        N = ary.shape[axis]
        if N % sections:
            raise ValueError(
                'array split does not result in an equal division')
    res = array_split(ary, indices_or_sections, axis)
    return res


def _hvdsplit_dispatcher(ary, indices_or_sections):
    return (ary, indices_or_sections)


@array_function_dispatch(_hvdsplit_dispatcher)
def hsplit(ary, indices_or_sections):
    """
    Split an array into multiple sub-arrays horizontally (column-wise).

    Please refer to the `split` documentation.  `hsplit` is equivalent
    to `split` with ``axis=1``, the array is always split along the second
    axis regardless of the array dimension.

    See Also
    --------
    split : Split an array into multiple sub-arrays of equal size.

    Examples
    --------
    >>> x = np.arange(16.0).reshape(4, 4)
    >>> x
    array([[  0.,   1.,   2.,   3.],
           [  4.,   5.,   6.,   7.],
           [  8.,   9.,  10.,  11.],
           [ 12.,  13.,  14.,  15.]])
    >>> np.hsplit(x, 2)
    [array([[  0.,   1.],
           [  4.,   5.],
           [  8.,   9.],
           [ 12.,  13.]]),
     array([[  2.,   3.],
           [  6.,   7.],
           [ 10.,  11.],
           [ 14.,  15.]])]
    >>> np.hsplit(x, np.array([3, 6]))
    [array([[  0.,   1.,   2.],
           [  4.,   5.,   6.],
           [  8.,   9.,  10.],
           [ 12.,  13.,  14.]]),
     array([[  3.],
           [  7.],
           [ 11.],
           [ 15.]]),
     array([], dtype=float64)]

    With a higher dimensional array the split is still along the second axis.

    >>> x = np.arange(8.0).reshape(2, 2, 2)
    >>> x
    array([[[ 0.,  1.],
            [ 2.,  3.]],
           [[ 4.,  5.],
            [ 6.,  7.]]])
    >>> np.hsplit(x, 2)
    [array([[[ 0.,  1.]],
           [[ 4.,  5.]]]),
     array([[[ 2.,  3.]],
           [[ 6.,  7.]]])]

    """
    if _nx.ndim(ary) == 0:
        raise ValueError('hsplit only works on arrays of 1 or more dimensions')
    if ary.ndim > 1:
        return split(ary, indices_or_sections, 1)
    else:
        return split(ary, indices_or_sections, 0)


@array_function_dispatch(_hvdsplit_dispatcher)
def vsplit(ary, indices_or_sections):
    """
    Split an array into multiple sub-arrays vertically (row-wise).

    Please refer to the ``split`` documentation.  ``vsplit`` is equivalent
    to ``split`` with `axis=0` (default), the array is always split along the
    first axis regardless of the array dimension.

    See Also
    --------
    split : Split an array into multiple sub-arrays of equal size.

    Examples
    --------
    >>> x = np.arange(16.0).reshape(4, 4)
    >>> x
    array([[  0.,   1.,   2.,   3.],
           [  4.,   5.,   6.,   7.],
           [  8.,   9.,  10.,  11.],
           [ 12.,  13.,  14.,  15.]])
    >>> np.vsplit(x, 2)
    [array([[ 0.,  1.,  2.,  3.],
           [ 4.,  5.,  6.,  7.]]),
     array([[  8.,   9.,  10.,  11.],
           [ 12.,  13.,  14.,  15.]])]
    >>> np.vsplit(x, np.array([3, 6]))
    [array([[  0.,   1.,   2.,   3.],
           [  4.,   5.,   6.,   7.],
           [  8.,   9.,  10.,  11.]]),
     array([[ 12.,  13.,  14.,  15.]]),
     array([], dtype=float64)]

    With a higher dimensional array the split is still along the first axis.

    >>> x = np.arange(8.0).reshape(2, 2, 2)
    >>> x
    array([[[ 0.,  1.],
            [ 2.,  3.]],
           [[ 4.,  5.],
            [ 6.,  7.]]])
    >>> np.vsplit(x, 2)
    [array([[[ 0.,  1.],
            [ 2.,  3.]]]),
     array([[[ 4.,  5.],
            [ 6.,  7.]]])]

    """
    if _nx.ndim(ary) < 2:
        raise ValueError('vsplit only works on arrays of 2 or more dimensions')
    return split(ary, indices_or_sections, 0)


@array_function_dispatch(_hvdsplit_dispatcher)
def dsplit(ary, indices_or_sections):
    """
    Split array into multiple sub-arrays along the 3rd axis (depth).

    Please refer to the `split` documentation.  `dsplit` is equivalent
    to `split` with ``axis=2``, the array is always split along the third
    axis provided the array dimension is greater than or equal to 3.

    See Also
    --------
    split : Split an array into multiple sub-arrays of equal size.

    Examples
    --------
    >>> x = np.arange(16.0).reshape(2, 2, 4)
    >>> x
    array([[[  0.,   1.,   2.,   3.],
            [  4.,   5.,   6.,   7.]],
           [[  8.,   9.,  10.,  11.],
            [ 12.,  13.,  14.,  15.]]])
    >>> np.dsplit(x, 2)
    [array([[[  0.,   1.],
            [  4.,   5.]],
           [[  8.,   9.],
            [ 12.,  13.]]]),
     array([[[  2.,   3.],
            [  6.,   7.]],
           [[ 10.,  11.],
            [ 14.,  15.]]])]
    >>> np.dsplit(x, np.array([3, 6]))
    [array([[[  0.,   1.,   2.],
            [  4.,   5.,   6.]],
           [[  8.,   9.,  10.],
            [ 12.,  13.,  14.]]]),
     array([[[  3.],
            [  7.]],
           [[ 11.],
            [ 15.]]]),
     array([], dtype=float64)]

    """
    if _nx.ndim(ary) < 3:
        raise ValueError('dsplit only works on arrays of 3 or more dimensions')
    return split(ary, indices_or_sections, 2)

def get_array_prepare(*args):
    """Find the wrapper for the array with the highest priority.

    In case of ties, leftmost wins. If no wrapper is found, return None
    """
    wrappers = sorted((getattr(x, '__array_priority__', 0), -i,
                 x.__array_prepare__) for i, x in enumerate(args)
                                   if hasattr(x, '__array_prepare__'))
    if wrappers:
        return wrappers[-1][-1]
    return None

def get_array_wrap(*args):
    """Find the wrapper for the array with the highest priority.

    In case of ties, leftmost wins. If no wrapper is found, return None
    """
    wrappers = sorted((getattr(x, '__array_priority__', 0), -i,
                 x.__array_wrap__) for i, x in enumerate(args)
                                   if hasattr(x, '__array_wrap__'))
    if wrappers:
        return wrappers[-1][-1]
    return None


def _kron_dispatcher(a, b):
    return (a, b)


@array_function_dispatch(_kron_dispatcher)
def kron(a, b):
    """
    Kronecker product of two arrays.

    Computes the Kronecker product, a composite array made of blocks of the
    second array scaled by the first.

    Parameters
    ----------
    a, b : array_like

    Returns
    -------
    out : ndarray

    See Also
    --------
    outer : The outer product

    Notes
    -----
    The function assumes that the number of dimensions of `a` and `b`
    are the same, if necessary prepending the smallest with ones.
    If `a.shape = (r0,r1,..,rN)` and `b.shape = (s0,s1,...,sN)`,
    the Kronecker product has shape `(r0*s0, r1*s1, ..., rN*SN)`.
    The elements are products of elements from `a` and `b`, organized
    explicitly by::

        kron(a,b)[k0,k1,...,kN] = a[i0,i1,...,iN] * b[j0,j1,...,jN]

    where::

        kt = it * st + jt,  t = 0,...,N

    In the common 2-D case (N=1), the block structure can be visualized::

        [[ a[0,0]*b,   a[0,1]*b,  ... , a[0,-1]*b  ],
         [  ...                              ...   ],
         [ a[-1,0]*b,  a[-1,1]*b, ... , a[-1,-1]*b ]]


    Examples
    --------
    >>> np.kron([1,10,100], [5,6,7])
    array([  5,   6,   7,  50,  60,  70, 500, 600, 700])
    >>> np.kron([5,6,7], [1,10,100])
    array([  5,  50, 500,   6,  60, 600,   7,  70, 700])

    >>> np.kron(np.eye(2), np.ones((2,2)))
    array([[ 1.,  1.,  0.,  0.],
           [ 1.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  1.],
           [ 0.,  0.,  1.,  1.]])

    >>> a = np.arange(100).reshape((2,5,2,5))
    >>> b = np.arange(24).reshape((2,3,4))
    >>> c = np.kron(a,b)
    >>> c.shape
    (2, 10, 6, 20)
    >>> I = (1,3,0,2)
    >>> J = (0,2,1)
    >>> J1 = (0,) + J             # extend to ndim=4
    >>> S1 = (1,) + b.shape
    >>> K = tuple(np.array(I) * np.array(S1) + np.array(J1))
    >>> c[K] == a[I]*b[J]
    True

    """
    b = asanyarray(b)
    a = array(a, copy=False, subok=True, ndmin=b.ndim)
    ndb, nda = b.ndim, a.ndim
    if (nda == 0 or ndb == 0):
        return _nx.multiply(a, b)
    as_ = a.shape
    bs = b.shape
    if not a.flags.contiguous:
        a = reshape(a, as_)
    if not b.flags.contiguous:
        b = reshape(b, bs)
    nd = ndb
    if (ndb != nda):
        if (ndb > nda):
            as_ = (1,)*(ndb-nda) + as_
        else:
            bs = (1,)*(nda-ndb) + bs
            nd = nda
    result = outer(a, b).reshape(as_+bs)
    axis = nd-1
    for _ in range(nd):
        result = concatenate(result, axis=axis)
    wrapper = get_array_prepare(a, b)
    if wrapper is not None:
        result = wrapper(result)
    wrapper = get_array_wrap(a, b)
    if wrapper is not None:
        result = wrapper(result)
    return result


def _tile_dispatcher(A, reps):
    return (A, reps)


@array_function_dispatch(_tile_dispatcher)
def tile(A, reps):
    """
    Construct an array by repeating A the number of times given by reps.

    If `reps` has length ``d``, the result will have dimension of
    ``max(d, A.ndim)``.

    If ``A.ndim < d``, `A` is promoted to be d-dimensional by prepending new
    axes. So a shape (3,) array is promoted to (1, 3) for 2-D replication,
    or shape (1, 1, 3) for 3-D replication. If this is not the desired
    behavior, promote `A` to d-dimensions manually before calling this
    function.

    If ``A.ndim > d``, `reps` is promoted to `A`.ndim by pre-pending 1's to it.
    Thus for an `A` of shape (2, 3, 4, 5), a `reps` of (2, 2) is treated as
    (1, 1, 2, 2).

    Note : Although tile may be used for broadcasting, it is strongly
    recommended to use numpy's broadcasting operations and functions.

    Parameters
    ----------
    A : array_like
        The input array.
    reps : array_like
        The number of repetitions of `A` along each axis.

    Returns
    -------
    c : ndarray
        The tiled output array.

    See Also
    --------
    repeat : Repeat elements of an array.
    broadcast_to : Broadcast an array to a new shape

    Examples
    --------
    >>> a = np.array([0, 1, 2])
    >>> np.tile(a, 2)
    array([0, 1, 2, 0, 1, 2])
    >>> np.tile(a, (2, 2))
    array([[0, 1, 2, 0, 1, 2],
           [0, 1, 2, 0, 1, 2]])
    >>> np.tile(a, (2, 1, 2))
    array([[[0, 1, 2, 0, 1, 2]],
           [[0, 1, 2, 0, 1, 2]]])

    >>> b = np.array([[1, 2], [3, 4]])
    >>> np.tile(b, 2)
    array([[1, 2, 1, 2],
           [3, 4, 3, 4]])
    >>> np.tile(b, (2, 1))
    array([[1, 2],
           [3, 4],
           [1, 2],
           [3, 4]])

    >>> c = np.array([1,2,3,4])
    >>> np.tile(c,(4,1))
    array([[1, 2, 3, 4],
           [1, 2, 3, 4],
           [1, 2, 3, 4],
           [1, 2, 3, 4]])
    """
    try:
        tup = tuple(reps)
    except TypeError:
        tup = (reps,)
    d = len(tup)
    if all(x == 1 for x in tup) and isinstance(A, _nx.ndarray):
        # Fixes the problem that the function does not make a copy if A is a
        # numpy array and the repetitions are 1 in all dimensions
        return _nx.array(A, copy=True, subok=True, ndmin=d)
    else:
        # Note that no copy of zero-sized arrays is made. However since they
        # have no data there is no risk of an inadvertent overwrite.
        c = _nx.array(A, copy=False, subok=True, ndmin=d)
    if (d < c.ndim):
        tup = (1,)*(c.ndim-d) + tup
    shape_out = tuple(s*t for s, t in zip(c.shape, tup))
    n = c.size
    if n > 0:
        for dim_in, nrep in zip(c.shape, tup):
            if nrep != 1:
                c = c.reshape(-1, n).repeat(nrep, 0)
            n //= dim_in
    return c.reshape(shape_out)