index_tricks.py

from __future__ import division, absolute_import, print_function

import functools
import sys
import math

import numpy.core.numeric as _nx
from numpy.core.numeric import (
    asarray, ScalarType, array, alltrue, cumprod, arange, ndim
    )
from numpy.core.numerictypes import find_common_type, issubdtype

import numpy.matrixlib as matrixlib
from .function_base import diff
from numpy.core.multiarray import ravel_multi_index, unravel_index
from numpy.core.overrides import set_module
from numpy.core import overrides, linspace
from numpy.lib.stride_tricks import as_strided


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


__all__ = [
    'ravel_multi_index', 'unravel_index', 'mgrid', 'ogrid', 'r_', 'c_',
    's_', 'index_exp', 'ix_', 'ndenumerate', 'ndindex', 'fill_diagonal',
    'diag_indices', 'diag_indices_from'
    ]


def _ix__dispatcher(*args):
    return args


@array_function_dispatch(_ix__dispatcher)
def ix_(*args):
    """
    Construct an open mesh from multiple sequences.

    This function takes N 1-D sequences and returns N outputs with N
    dimensions each, such that the shape is 1 in all but one dimension
    and the dimension with the non-unit shape value cycles through all
    N dimensions.

    Using `ix_` one can quickly construct index arrays that will index
    the cross product. ``a[np.ix_([1,3],[2,5])]`` returns the array
    ``[[a[1,2] a[1,5]], [a[3,2] a[3,5]]]``.

    Parameters
    ----------
    args : 1-D sequences
        Each sequence should be of integer or boolean type.
        Boolean sequences will be interpreted as boolean masks for the
        corresponding dimension (equivalent to passing in
        ``np.nonzero(boolean_sequence)``).

    Returns
    -------
    out : tuple of ndarrays
        N arrays with N dimensions each, with N the number of input
        sequences. Together these arrays form an open mesh.

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

    Examples
    --------
    >>> a = np.arange(10).reshape(2, 5)
    >>> a
    array([[0, 1, 2, 3, 4],
           [5, 6, 7, 8, 9]])
    >>> ixgrid = np.ix_([0, 1], [2, 4])
    >>> ixgrid
    (array([[0],
           [1]]), array([[2, 4]]))
    >>> ixgrid[0].shape, ixgrid[1].shape
    ((2, 1), (1, 2))
    >>> a[ixgrid]
    array([[2, 4],
           [7, 9]])

    >>> ixgrid = np.ix_([True, True], [2, 4])
    >>> a[ixgrid]
    array([[2, 4],
           [7, 9]])
    >>> ixgrid = np.ix_([True, True], [False, False, True, False, True])
    >>> a[ixgrid]
    array([[2, 4],
           [7, 9]])

    """
    out = []
    nd = len(args)
    for k, new in enumerate(args):
        new = asarray(new)
        if new.ndim != 1:
            raise ValueError("Cross index must be 1 dimensional")
        if new.size == 0:
            # Explicitly type empty arrays to avoid float default
            new = new.astype(_nx.intp)
        if issubdtype(new.dtype, _nx.bool_):
            new, = new.nonzero()
        new = new.reshape((1,)*k + (new.size,) + (1,)*(nd-k-1))
        out.append(new)
    return tuple(out)

class nd_grid(object):
    """
    Construct a multi-dimensional "meshgrid".

    ``grid = nd_grid()`` creates an instance which will return a mesh-grid
    when indexed.  The dimension and number of the output arrays are equal
    to the number of indexing dimensions.  If the step length is not a
    complex number, then the stop is not inclusive.

    However, if the step length is a **complex number** (e.g. 5j), then the
    integer part of its magnitude is interpreted as specifying the
    number of points to create between the start and stop values, where
    the stop value **is inclusive**.

    If instantiated with an argument of ``sparse=True``, the mesh-grid is
    open (or not fleshed out) so that only one-dimension of each returned
    argument is greater than 1.

    Parameters
    ----------
    sparse : bool, optional
        Whether the grid is sparse or not. Default is False.

    Notes
    -----
    Two instances of `nd_grid` are made available in the NumPy namespace,
    `mgrid` and `ogrid`, approximately defined as::

        mgrid = nd_grid(sparse=False)
        ogrid = nd_grid(sparse=True)

    Users should use these pre-defined instances instead of using `nd_grid`
    directly.
    """

    def __init__(self, sparse=False):
        self.sparse = sparse

    def __getitem__(self, key):
        try:
            size = []
            typ = int
            for k in range(len(key)):
                step = key[k].step
                start = key[k].start
                if start is None:
                    start = 0
                if step is None:
                    step = 1
                if isinstance(step, complex):
                    size.append(int(abs(step)))
                    typ = float
                else:
                    size.append(
                        int(math.ceil((key[k].stop - start)/(step*1.0))))
                if (isinstance(step, float) or
                        isinstance(start, float) or
                        isinstance(key[k].stop, float)):
                    typ = float
            if self.sparse:
                nn = [_nx.arange(_x, dtype=_t)
                        for _x, _t in zip(size, (typ,)*len(size))]
            else:
                nn = _nx.indices(size, typ)
            for k in range(len(size)):
                step = key[k].step
                start = key[k].start
                if start is None:
                    start = 0
                if step is None:
                    step = 1
                if isinstance(step, complex):
                    step = int(abs(step))
                    if step != 1:
                        step = (key[k].stop - start)/float(step-1)
                nn[k] = (nn[k]*step+start)
            if self.sparse:
                slobj = [_nx.newaxis]*len(size)
                for k in range(len(size)):
                    slobj[k] = slice(None, None)
                    nn[k] = nn[k][tuple(slobj)]
                    slobj[k] = _nx.newaxis
            return nn
        except (IndexError, TypeError):
            step = key.step
            stop = key.stop
            start = key.start
            if start is None:
                start = 0
            if isinstance(step, complex):
                step = abs(step)
                length = int(step)
                if step != 1:
                    step = (key.stop-start)/float(step-1)
                stop = key.stop + step
                return _nx.arange(0, length, 1, float)*step + start
            else:
                return _nx.arange(start, stop, step)


class MGridClass(nd_grid):
    """
    `nd_grid` instance which returns a dense multi-dimensional "meshgrid".

    An instance of `numpy.lib.index_tricks.nd_grid` which returns an dense
    (or fleshed out) mesh-grid when indexed, so that each returned argument
    has the same shape.  The dimensions and number of the output arrays are
    equal to the number of indexing dimensions.  If the step length is not a
    complex number, then the stop is not inclusive.

    However, if the step length is a **complex number** (e.g. 5j), then
    the integer part of its magnitude is interpreted as specifying the
    number of points to create between the start and stop values, where
    the stop value **is inclusive**.

    Returns
    ----------
    mesh-grid `ndarrays` all of the same dimensions

    See Also
    --------
    numpy.lib.index_tricks.nd_grid : class of `ogrid` and `mgrid` objects
    ogrid : like mgrid but returns open (not fleshed out) mesh grids
    r_ : array concatenator

    Examples
    --------
    >>> np.mgrid[0:5,0:5]
    array([[[0, 0, 0, 0, 0],
            [1, 1, 1, 1, 1],
            [2, 2, 2, 2, 2],
            [3, 3, 3, 3, 3],
            [4, 4, 4, 4, 4]],
           [[0, 1, 2, 3, 4],
            [0, 1, 2, 3, 4],
            [0, 1, 2, 3, 4],
            [0, 1, 2, 3, 4],
            [0, 1, 2, 3, 4]]])
    >>> np.mgrid[-1:1:5j]
    array([-1. , -0.5,  0. ,  0.5,  1. ])

    """
    def __init__(self):
        super(MGridClass, self).__init__(sparse=False)

mgrid = MGridClass()

class OGridClass(nd_grid):
    """
    `nd_grid` instance which returns an open multi-dimensional "meshgrid".

    An instance of `numpy.lib.index_tricks.nd_grid` which returns an open
    (i.e. not fleshed out) mesh-grid when indexed, so that only one dimension
    of each returned array is greater than 1.  The dimension and number of the
    output arrays are equal to the number of indexing dimensions.  If the step
    length is not a complex number, then the stop is not inclusive.

    However, if the step length is a **complex number** (e.g. 5j), then
    the integer part of its magnitude is interpreted as specifying the
    number of points to create between the start and stop values, where
    the stop value **is inclusive**.

    Returns
    ----------
    mesh-grid `ndarrays` with only one dimension :math:`\\neq 1`

    See Also
    --------
    np.lib.index_tricks.nd_grid : class of `ogrid` and `mgrid` objects
    mgrid : like `ogrid` but returns dense (or fleshed out) mesh grids
    r_ : array concatenator

    Examples
    --------
    >>> from numpy import ogrid
    >>> ogrid[-1:1:5j]
    array([-1. , -0.5,  0. ,  0.5,  1. ])
    >>> ogrid[0:5,0:5]
    [array([[0],
            [1],
            [2],
            [3],
            [4]]), array([[0, 1, 2, 3, 4]])]

    """
    def __init__(self):
        super(OGridClass, self).__init__(sparse=True)

ogrid = OGridClass()


class AxisConcatenator(object):
    """
    Translates slice objects to concatenation along an axis.

    For detailed documentation on usage, see `r_`.
    """
    # allow ma.mr_ to override this
    concatenate = staticmethod(_nx.concatenate)
    makemat = staticmethod(matrixlib.matrix)

    def __init__(self, axis=0, matrix=False, ndmin=1, trans1d=-1):
        self.axis = axis
        self.matrix = matrix
        self.trans1d = trans1d
        self.ndmin = ndmin

    def __getitem__(self, key):
        # handle matrix builder syntax
        if isinstance(key, str):
            frame = sys._getframe().f_back
            mymat = matrixlib.bmat(key, frame.f_globals, frame.f_locals)
            return mymat

        if not isinstance(key, tuple):
            key = (key,)

        # copy attributes, since they can be overridden in the first argument
        trans1d = self.trans1d
        ndmin = self.ndmin
        matrix = self.matrix
        axis = self.axis

        objs = []
        scalars = []
        arraytypes = []
        scalartypes = []

        for k, item in enumerate(key):
            scalar = False
            if isinstance(item, slice):
                step = item.step
                start = item.start
                stop = item.stop
                if start is None:
                    start = 0
                if step is None:
                    step = 1
                if isinstance(step, complex):
                    size = int(abs(step))
                    newobj = linspace(start, stop, num=size)
                else:
                    newobj = _nx.arange(start, stop, step)
                if ndmin > 1:
                    newobj = array(newobj, copy=False, ndmin=ndmin)
                    if trans1d != -1:
                        newobj = newobj.swapaxes(-1, trans1d)
            elif isinstance(item, str):
                if k != 0:
                    raise ValueError("special directives must be the "
                            "first entry.")
                if item in ('r', 'c'):
                    matrix = True
                    col = (item == 'c')
                    continue
                if ',' in item:
                    vec = item.split(',')
                    try:
                        axis, ndmin = [int(x) for x in vec[:2]]
                        if len(vec) == 3:
                            trans1d = int(vec[2])
                        continue
                    except Exception:
                        raise ValueError("unknown special directive")
                try:
                    axis = int(item)
                    continue
                except (ValueError, TypeError):
                    raise ValueError("unknown special directive")
            elif type(item) in ScalarType:
                newobj = array(item, ndmin=ndmin)
                scalars.append(len(objs))
                scalar = True
                scalartypes.append(newobj.dtype)
            else:
                item_ndim = ndim(item)
                newobj = array(item, copy=False, subok=True, ndmin=ndmin)
                if trans1d != -1 and item_ndim < ndmin:
                    k2 = ndmin - item_ndim
                    k1 = trans1d
                    if k1 < 0:
                        k1 += k2 + 1
                    defaxes = list(range(ndmin))
                    axes = defaxes[:k1] + defaxes[k2:] + defaxes[k1:k2]
                    newobj = newobj.transpose(axes)
            objs.append(newobj)
            if not scalar and isinstance(newobj, _nx.ndarray):
                arraytypes.append(newobj.dtype)

        # Ensure that scalars won't up-cast unless warranted
        final_dtype = find_common_type(arraytypes, scalartypes)
        if final_dtype is not None:
            for k in scalars:
                objs[k] = objs[k].astype(final_dtype)

        res = self.concatenate(tuple(objs), axis=axis)

        if matrix:
            oldndim = res.ndim
            res = self.makemat(res)
            if oldndim == 1 and col:
                res = res.T
        return res

    def __len__(self):
        return 0

# separate classes are used here instead of just making r_ = concatentor(0),
# etc. because otherwise we couldn't get the doc string to come out right
# in help(r_)

class RClass(AxisConcatenator):
    """
    Translates slice objects to concatenation along the first axis.

    This is a simple way to build up arrays quickly. There are two use cases.

    1. If the index expression contains comma separated arrays, then stack
       them along their first axis.
    2. If the index expression contains slice notation or scalars then create
       a 1-D array with a range indicated by the slice notation.

    If slice notation is used, the syntax ``start:stop:step`` is equivalent
    to ``np.arange(start, stop, step)`` inside of the brackets. However, if
    ``step`` is an imaginary number (i.e. 100j) then its integer portion is
    interpreted as a number-of-points desired and the start and stop are
    inclusive. In other words ``start:stop:stepj`` is interpreted as
    ``np.linspace(start, stop, step, endpoint=1)`` inside of the brackets.
    After expansion of slice notation, all comma separated sequences are
    concatenated together.

    Optional character strings placed as the first element of the index
    expression can be used to change the output. The strings 'r' or 'c' result
    in matrix output. If the result is 1-D and 'r' is specified a 1 x N (row)
    matrix is produced. If the result is 1-D and 'c' is specified, then a N x 1
    (column) matrix is produced. If the result is 2-D then both provide the
    same matrix result.

    A string integer specifies which axis to stack multiple comma separated
    arrays along. A string of two comma-separated integers allows indication
    of the minimum number of dimensions to force each entry into as the
    second integer (the axis to concatenate along is still the first integer).

    A string with three comma-separated integers allows specification of the
    axis to concatenate along, the minimum number of dimensions to force the
    entries to, and which axis should contain the start of the arrays which
    are less than the specified number of dimensions. In other words the third
    integer allows you to specify where the 1's should be placed in the shape
    of the arrays that have their shapes upgraded. By default, they are placed
    in the front of the shape tuple. The third argument allows you to specify
    where the start of the array should be instead. Thus, a third argument of
    '0' would place the 1's at the end of the array shape. Negative integers
    specify where in the new shape tuple the last dimension of upgraded arrays
    should be placed, so the default is '-1'.

    Parameters
    ----------
    Not a function, so takes no parameters


    Returns
    -------
    A concatenated ndarray or matrix.

    See Also
    --------
    concatenate : Join a sequence of arrays along an existing axis.
    c_ : Translates slice objects to concatenation along the second axis.

    Examples
    --------
    >>> np.r_[np.array([1,2,3]), 0, 0, np.array([4,5,6])]
    array([1, 2, 3, 0, 0, 4, 5, 6])
    >>> np.r_[-1:1:6j, [0]*3, 5, 6]
    array([-1. , -0.6, -0.2,  0.2,  0.6,  1. ,  0. ,  0. ,  0. ,  5. ,  6. ])

    String integers specify the axis to concatenate along or the minimum
    number of dimensions to force entries into.

    >>> a = np.array([[0, 1, 2], [3, 4, 5]])
    >>> np.r_['-1', a, a] # concatenate along last axis
    array([[0, 1, 2, 0, 1, 2],
           [3, 4, 5, 3, 4, 5]])
    >>> np.r_['0,2', [1,2,3], [4,5,6]] # concatenate along first axis, dim>=2
    array([[1, 2, 3],
           [4, 5, 6]])

    >>> np.r_['0,2,0', [1,2,3], [4,5,6]]
    array([[1],
           [2],
           [3],
           [4],
           [5],
           [6]])
    >>> np.r_['1,2,0', [1,2,3], [4,5,6]]
    array([[1, 4],
           [2, 5],
           [3, 6]])

    Using 'r' or 'c' as a first string argument creates a matrix.

    >>> np.r_['r',[1,2,3], [4,5,6]]
    matrix([[1, 2, 3, 4, 5, 6]])

    """

    def __init__(self):
        AxisConcatenator.__init__(self, 0)

r_ = RClass()

class CClass(AxisConcatenator):
    """
    Translates slice objects to concatenation along the second axis.

    This is short-hand for ``np.r_['-1,2,0', index expression]``, which is
    useful because of its common occurrence. In particular, arrays will be
    stacked along their last axis after being upgraded to at least 2-D with
    1's post-pended to the shape (column vectors made out of 1-D arrays).
    
    See Also
    --------
    column_stack : Stack 1-D arrays as columns into a 2-D array.
    r_ : For more detailed documentation.

    Examples
    --------
    >>> np.c_[np.array([1,2,3]), np.array([4,5,6])]
    array([[1, 4],
           [2, 5],
           [3, 6]])
    >>> np.c_[np.array([[1,2,3]]), 0, 0, np.array([[4,5,6]])]
    array([[1, 2, 3, 0, 0, 4, 5, 6]])

    """

    def __init__(self):
        AxisConcatenator.__init__(self, -1, ndmin=2, trans1d=0)


c_ = CClass()


@set_module('numpy')
class ndenumerate(object):
    """
    Multidimensional index iterator.

    Return an iterator yielding pairs of array coordinates and values.

    Parameters
    ----------
    arr : ndarray
      Input array.

    See Also
    --------
    ndindex, flatiter

    Examples
    --------
    >>> a = np.array([[1, 2], [3, 4]])
    >>> for index, x in np.ndenumerate(a):
    ...     print(index, x)
    (0, 0) 1
    (0, 1) 2
    (1, 0) 3
    (1, 1) 4

    """

    def __init__(self, arr):
        self.iter = asarray(arr).flat

    def __next__(self):
        """
        Standard iterator method, returns the index tuple and array value.

        Returns
        -------
        coords : tuple of ints
            The indices of the current iteration.
        val : scalar
            The array element of the current iteration.

        """
        return self.iter.coords, next(self.iter)

    def __iter__(self):
        return self

    next = __next__


@set_module('numpy')
class ndindex(object):
    """
    An N-dimensional iterator object to index arrays.

    Given the shape of an array, an `ndindex` instance iterates over
    the N-dimensional index of the array. At each iteration a tuple
    of indices is returned, the last dimension is iterated over first.

    Parameters
    ----------
    `*args` : ints
      The size of each dimension of the array.

    See Also
    --------
    ndenumerate, flatiter

    Examples
    --------
    >>> for index in np.ndindex(3, 2, 1):
    ...     print(index)
    (0, 0, 0)
    (0, 1, 0)
    (1, 0, 0)
    (1, 1, 0)
    (2, 0, 0)
    (2, 1, 0)

    """

    def __init__(self, *shape):
        if len(shape) == 1 and isinstance(shape[0], tuple):
            shape = shape[0]
        x = as_strided(_nx.zeros(1), shape=shape,
                       strides=_nx.zeros_like(shape))
        self._it = _nx.nditer(x, flags=['multi_index', 'zerosize_ok'],
                              order='C')

    def __iter__(self):
        return self

    def ndincr(self):
        """
        Increment the multi-dimensional index by one.

        This method is for backward compatibility only: do not use.
        """
        next(self)

    def __next__(self):
        """
        Standard iterator method, updates the index and returns the index
        tuple.

        Returns
        -------
        val : tuple of ints
            Returns a tuple containing the indices of the current
            iteration.

        """
        next(self._it)
        return self._it.multi_index

    next = __next__


# You can do all this with slice() plus a few special objects,
# but there's a lot to remember. This version is simpler because
# it uses the standard array indexing syntax.
#
# Written by Konrad Hinsen <hinsen@cnrs-orleans.fr>
# last revision: 1999-7-23
#
# Cosmetic changes by T. Oliphant 2001
#
#

class IndexExpression(object):
    """
    A nicer way to build up index tuples for arrays.

    .. note::
       Use one of the two predefined instances `index_exp` or `s_`
       rather than directly using `IndexExpression`.

    For any index combination, including slicing and axis insertion,
    ``a[indices]`` is the same as ``a[np.index_exp[indices]]`` for any
    array `a`. However, ``np.index_exp[indices]`` can be used anywhere
    in Python code and returns a tuple of slice objects that can be
    used in the construction of complex index expressions.

    Parameters
    ----------
    maketuple : bool
        If True, always returns a tuple.

    See Also
    --------
    index_exp : Predefined instance that always returns a tuple:
       `index_exp = IndexExpression(maketuple=True)`.
    s_ : Predefined instance without tuple conversion:
       `s_ = IndexExpression(maketuple=False)`.

    Notes
    -----
    You can do all this with `slice()` plus a few special objects,
    but there's a lot to remember and this version is simpler because
    it uses the standard array indexing syntax.

    Examples
    --------
    >>> np.s_[2::2]
    slice(2, None, 2)
    >>> np.index_exp[2::2]
    (slice(2, None, 2),)

    >>> np.array([0, 1, 2, 3, 4])[np.s_[2::2]]
    array([2, 4])

    """

    def __init__(self, maketuple):
        self.maketuple = maketuple

    def __getitem__(self, item):
        if self.maketuple and not isinstance(item, tuple):
            return (item,)
        else:
            return item

index_exp = IndexExpression(maketuple=True)
s_ = IndexExpression(maketuple=False)

# End contribution from Konrad.


# The following functions complement those in twodim_base, but are
# applicable to N-dimensions.


def _fill_diagonal_dispatcher(a, val, wrap=None):
    return (a,)


@array_function_dispatch(_fill_diagonal_dispatcher)
def fill_diagonal(a, val, wrap=False):
    """Fill the main diagonal of the given array of any dimensionality.

    For an array `a` with ``a.ndim >= 2``, the diagonal is the list of
    locations with indices ``a[i, ..., i]`` all identical. This function
    modifies the input array in-place, it does not return a value.

    Parameters
    ----------
    a : array, at least 2-D.
      Array whose diagonal is to be filled, it gets modified in-place.

    val : scalar
      Value to be written on the diagonal, its type must be compatible with
      that of the array a.

    wrap : bool
      For tall matrices in NumPy version up to 1.6.2, the
      diagonal "wrapped" after N columns. You can have this behavior
      with this option. This affects only tall matrices.

    See also
    --------
    diag_indices, diag_indices_from

    Notes
    -----
    .. versionadded:: 1.4.0

    This functionality can be obtained via `diag_indices`, but internally
    this version uses a much faster implementation that never constructs the
    indices and uses simple slicing.

    Examples
    --------
    >>> a = np.zeros((3, 3), int)
    >>> np.fill_diagonal(a, 5)
    >>> a
    array([[5, 0, 0],
           [0, 5, 0],
           [0, 0, 5]])

    The same function can operate on a 4-D array:

    >>> a = np.zeros((3, 3, 3, 3), int)
    >>> np.fill_diagonal(a, 4)

    We only show a few blocks for clarity:

    >>> a[0, 0]
    array([[4, 0, 0],
           [0, 0, 0],
           [0, 0, 0]])
    >>> a[1, 1]
    array([[0, 0, 0],
           [0, 4, 0],
           [0, 0, 0]])
    >>> a[2, 2]
    array([[0, 0, 0],
           [0, 0, 0],
           [0, 0, 4]])

    The wrap option affects only tall matrices:

    >>> # tall matrices no wrap
    >>> a = np.zeros((5, 3),int)
    >>> fill_diagonal(a, 4)
    >>> a
    array([[4, 0, 0],
           [0, 4, 0],
           [0, 0, 4],
           [0, 0, 0],
           [0, 0, 0]])

    >>> # tall matrices wrap
    >>> a = np.zeros((5, 3),int)
    >>> fill_diagonal(a, 4, wrap=True)
    >>> a
    array([[4, 0, 0],
           [0, 4, 0],
           [0, 0, 4],
           [0, 0, 0],
           [4, 0, 0]])

    >>> # wide matrices
    >>> a = np.zeros((3, 5),int)
    >>> fill_diagonal(a, 4, wrap=True)
    >>> a
    array([[4, 0, 0, 0, 0],
           [0, 4, 0, 0, 0],
           [0, 0, 4, 0, 0]])

    """
    if a.ndim < 2:
        raise ValueError("array must be at least 2-d")
    end = None
    if a.ndim == 2:
        # Explicit, fast formula for the common case.  For 2-d arrays, we
        # accept rectangular ones.
        step = a.shape[1] + 1
        #This is needed to don't have tall matrix have the diagonal wrap.
        if not wrap:
            end = a.shape[1] * a.shape[1]
    else:
        # For more than d=2, the strided formula is only valid for arrays with
        # all dimensions equal, so we check first.
        if not alltrue(diff(a.shape) == 0):
            raise ValueError("All dimensions of input must be of equal length")
        step = 1 + (cumprod(a.shape[:-1])).sum()

    # Write the value out into the diagonal.
    a.flat[:end:step] = val


@set_module('numpy')
def diag_indices(n, ndim=2):
    """
    Return the indices to access the main diagonal of an array.

    This returns a tuple of indices that can be used to access the main
    diagonal of an array `a` with ``a.ndim >= 2`` dimensions and shape
    (n, n, ..., n). For ``a.ndim = 2`` this is the usual diagonal, for
    ``a.ndim > 2`` this is the set of indices to access ``a[i, i, ..., i]``
    for ``i = [0..n-1]``.

    Parameters
    ----------
    n : int
      The size, along each dimension, of the arrays for which the returned
      indices can be used.

    ndim : int, optional
      The number of dimensions.

    See also
    --------
    diag_indices_from

    Notes
    -----
    .. versionadded:: 1.4.0

    Examples
    --------
    Create a set of indices to access the diagonal of a (4, 4) array:

    >>> di = np.diag_indices(4)
    >>> di
    (array([0, 1, 2, 3]), array([0, 1, 2, 3]))
    >>> a = np.arange(16).reshape(4, 4)
    >>> a
    array([[ 0,  1,  2,  3],
           [ 4,  5,  6,  7],
           [ 8,  9, 10, 11],
           [12, 13, 14, 15]])
    >>> a[di] = 100
    >>> a
    array([[100,   1,   2,   3],
           [  4, 100,   6,   7],
           [  8,   9, 100,  11],
           [ 12,  13,  14, 100]])

    Now, we create indices to manipulate a 3-D array:

    >>> d3 = np.diag_indices(2, 3)
    >>> d3
    (array([0, 1]), array([0, 1]), array([0, 1]))

    And use it to set the diagonal of an array of zeros to 1:

    >>> a = np.zeros((2, 2, 2), dtype=int)
    >>> a[d3] = 1
    >>> a
    array([[[1, 0],
            [0, 0]],
           [[0, 0],
            [0, 1]]])

    """
    idx = arange(n)
    return (idx,) * ndim


def _diag_indices_from(arr):
    return (arr,)


@array_function_dispatch(_diag_indices_from)
def diag_indices_from(arr):
    """
    Return the indices to access the main diagonal of an n-dimensional array.

    See `diag_indices` for full details.

    Parameters
    ----------
    arr : array, at least 2-D

    See Also
    --------
    diag_indices

    Notes
    -----
    .. versionadded:: 1.4.0

    """

    if not arr.ndim >= 2:
        raise ValueError("input array must be at least 2-d")
    # For more than d=2, the strided formula is only valid for arrays with
    # all dimensions equal, so we check first.
    if not alltrue(diff(arr.shape) == 0):
        raise ValueError("All dimensions of input must be of equal length")

    return diag_indices(arr.shape[0], arr.ndim)