twodim_base.py

""" Basic functions for manipulating 2d arrays

"""
import functools
import operator

from numpy.core.numeric import (
    asanyarray, arange, zeros, greater_equal, multiply, ones,
    asarray, where, int8, int16, int32, int64, intp, empty, promote_types,
    diagonal, nonzero, indices
    )
from numpy.core.overrides import set_array_function_like_doc, set_module
from numpy.core import overrides
from numpy.core import iinfo
from numpy.lib.stride_tricks import broadcast_to


__all__ = [
    'diag', 'diagflat', 'eye', 'fliplr', 'flipud', 'tri', 'triu',
    'tril', 'vander', 'histogram2d', 'mask_indices', 'tril_indices',
    'tril_indices_from', 'triu_indices', 'triu_indices_from', ]


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


i1 = iinfo(int8)
i2 = iinfo(int16)
i4 = iinfo(int32)


def _min_int(low, high):
    """ get small int that fits the range """
    if high <= i1.max and low >= i1.min:
        return int8
    if high <= i2.max and low >= i2.min:
        return int16
    if high <= i4.max and low >= i4.min:
        return int32
    return int64


def _flip_dispatcher(m):
    return (m,)


@array_function_dispatch(_flip_dispatcher)
def fliplr(m):
    """
    Reverse the order of elements along axis 1 (left/right).

    For a 2-D array, this flips the entries in each row in the left/right
    direction. Columns are preserved, but appear in a different order than
    before.

    Parameters
    ----------
    m : array_like
        Input array, must be at least 2-D.

    Returns
    -------
    f : ndarray
        A view of `m` with the columns reversed.  Since a view
        is returned, this operation is :math:`\\mathcal O(1)`.

    See Also
    --------
    flipud : Flip array in the up/down direction.
    flip : Flip array in one or more dimensions.
    rot90 : Rotate array counterclockwise.

    Notes
    -----
    Equivalent to ``m[:,::-1]`` or ``np.flip(m, axis=1)``.
    Requires the array to be at least 2-D.

    Examples
    --------
    >>> A = np.diag([1.,2.,3.])
    >>> A
    array([[1.,  0.,  0.],
           [0.,  2.,  0.],
           [0.,  0.,  3.]])
    >>> np.fliplr(A)
    array([[0.,  0.,  1.],
           [0.,  2.,  0.],
           [3.,  0.,  0.]])

    >>> A = np.random.randn(2,3,5)
    >>> np.all(np.fliplr(A) == A[:,::-1,...])
    True

    """
    m = asanyarray(m)
    if m.ndim < 2:
        raise ValueError("Input must be >= 2-d.")
    return m[:, ::-1]


@array_function_dispatch(_flip_dispatcher)
def flipud(m):
    """
    Reverse the order of elements along axis 0 (up/down).

    For a 2-D array, this flips the entries in each column in the up/down
    direction. Rows are preserved, but appear in a different order than before.

    Parameters
    ----------
    m : array_like
        Input array.

    Returns
    -------
    out : array_like
        A view of `m` with the rows reversed.  Since a view is
        returned, this operation is :math:`\\mathcal O(1)`.

    See Also
    --------
    fliplr : Flip array in the left/right direction.
    flip : Flip array in one or more dimensions.
    rot90 : Rotate array counterclockwise.

    Notes
    -----
    Equivalent to ``m[::-1, ...]`` or ``np.flip(m, axis=0)``.
    Requires the array to be at least 1-D.

    Examples
    --------
    >>> A = np.diag([1.0, 2, 3])
    >>> A
    array([[1.,  0.,  0.],
           [0.,  2.,  0.],
           [0.,  0.,  3.]])
    >>> np.flipud(A)
    array([[0.,  0.,  3.],
           [0.,  2.,  0.],
           [1.,  0.,  0.]])

    >>> A = np.random.randn(2,3,5)
    >>> np.all(np.flipud(A) == A[::-1,...])
    True

    >>> np.flipud([1,2])
    array([2, 1])

    """
    m = asanyarray(m)
    if m.ndim < 1:
        raise ValueError("Input must be >= 1-d.")
    return m[::-1, ...]


@set_array_function_like_doc
@set_module('numpy')
def eye(N, M=None, k=0, dtype=float, order='C', *, like=None):
    """
    Return a 2-D array with ones on the diagonal and zeros elsewhere.

    Parameters
    ----------
    N : int
      Number of rows in the output.
    M : int, optional
      Number of columns in the output. If None, defaults to `N`.
    k : int, optional
      Index of the diagonal: 0 (the default) refers to the main diagonal,
      a positive value refers to an upper diagonal, and a negative value
      to a lower diagonal.
    dtype : data-type, optional
      Data-type of the returned array.
    order : {'C', 'F'}, optional
        Whether the output should be stored in row-major (C-style) or
        column-major (Fortran-style) order in memory.

        .. versionadded:: 1.14.0
    ${ARRAY_FUNCTION_LIKE}

        .. versionadded:: 1.20.0

    Returns
    -------
    I : ndarray of shape (N,M)
      An array where all elements are equal to zero, except for the `k`-th
      diagonal, whose values are equal to one.

    See Also
    --------
    identity : (almost) equivalent function
    diag : diagonal 2-D array from a 1-D array specified by the user.

    Examples
    --------
    >>> np.eye(2, dtype=int)
    array([[1, 0],
           [0, 1]])
    >>> np.eye(3, k=1)
    array([[0.,  1.,  0.],
           [0.,  0.,  1.],
           [0.,  0.,  0.]])

    """
    if like is not None:
        return _eye_with_like(like, N, M=M, k=k, dtype=dtype, order=order)
    if M is None:
        M = N
    m = zeros((N, M), dtype=dtype, order=order)
    if k >= M:
        return m
    # Ensure M and k are integers, so we don't get any surprise casting
    # results in the expressions `M-k` and `M+1` used below.  This avoids
    # a problem with inputs with type (for example) np.uint64.
    M = operator.index(M)
    k = operator.index(k)
    if k >= 0:
        i = k
    else:
        i = (-k) * M
    m[:M-k].flat[i::M+1] = 1
    return m


_eye_with_like = array_function_dispatch()(eye)


def _diag_dispatcher(v, k=None):
    return (v,)


@array_function_dispatch(_diag_dispatcher)
def diag(v, k=0):
    """
    Extract a diagonal or construct a diagonal array.

    See the more detailed documentation for ``numpy.diagonal`` if you use this
    function to extract a diagonal and wish to write to the resulting array;
    whether it returns a copy or a view depends on what version of numpy you
    are using.

    Parameters
    ----------
    v : array_like
        If `v` is a 2-D array, return a copy of its `k`-th diagonal.
        If `v` is a 1-D array, return a 2-D array with `v` on the `k`-th
        diagonal.
    k : int, optional
        Diagonal in question. The default is 0. Use `k>0` for diagonals
        above the main diagonal, and `k<0` for diagonals below the main
        diagonal.

    Returns
    -------
    out : ndarray
        The extracted diagonal or constructed diagonal array.

    See Also
    --------
    diagonal : Return specified diagonals.
    diagflat : Create a 2-D array with the flattened input as a diagonal.
    trace : Sum along diagonals.
    triu : Upper triangle of an array.
    tril : Lower triangle of an array.

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

    >>> np.diag(x)
    array([0, 4, 8])
    >>> np.diag(x, k=1)
    array([1, 5])
    >>> np.diag(x, k=-1)
    array([3, 7])

    >>> np.diag(np.diag(x))
    array([[0, 0, 0],
           [0, 4, 0],
           [0, 0, 8]])

    """
    v = asanyarray(v)
    s = v.shape
    if len(s) == 1:
        n = s[0]+abs(k)
        res = zeros((n, n), v.dtype)
        if k >= 0:
            i = k
        else:
            i = (-k) * n
        res[:n-k].flat[i::n+1] = v
        return res
    elif len(s) == 2:
        return diagonal(v, k)
    else:
        raise ValueError("Input must be 1- or 2-d.")


@array_function_dispatch(_diag_dispatcher)
def diagflat(v, k=0):
    """
    Create a two-dimensional array with the flattened input as a diagonal.

    Parameters
    ----------
    v : array_like
        Input data, which is flattened and set as the `k`-th
        diagonal of the output.
    k : int, optional
        Diagonal to set; 0, the default, corresponds to the "main" diagonal,
        a positive (negative) `k` giving the number of the diagonal above
        (below) the main.

    Returns
    -------
    out : ndarray
        The 2-D output array.

    See Also
    --------
    diag : MATLAB work-alike for 1-D and 2-D arrays.
    diagonal : Return specified diagonals.
    trace : Sum along diagonals.

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

    >>> np.diagflat([1,2], 1)
    array([[0, 1, 0],
           [0, 0, 2],
           [0, 0, 0]])

    """
    try:
        wrap = v.__array_wrap__
    except AttributeError:
        wrap = None
    v = asarray(v).ravel()
    s = len(v)
    n = s + abs(k)
    res = zeros((n, n), v.dtype)
    if (k >= 0):
        i = arange(0, n-k, dtype=intp)
        fi = i+k+i*n
    else:
        i = arange(0, n+k, dtype=intp)
        fi = i+(i-k)*n
    res.flat[fi] = v
    if not wrap:
        return res
    return wrap(res)


@set_array_function_like_doc
@set_module('numpy')
def tri(N, M=None, k=0, dtype=float, *, like=None):
    """
    An array with ones at and below the given diagonal and zeros elsewhere.

    Parameters
    ----------
    N : int
        Number of rows in the array.
    M : int, optional
        Number of columns in the array.
        By default, `M` is taken equal to `N`.
    k : int, optional
        The sub-diagonal at and below which the array is filled.
        `k` = 0 is the main diagonal, while `k` < 0 is below it,
        and `k` > 0 is above.  The default is 0.
    dtype : dtype, optional
        Data type of the returned array.  The default is float.
    ${ARRAY_FUNCTION_LIKE}

        .. versionadded:: 1.20.0

    Returns
    -------
    tri : ndarray of shape (N, M)
        Array with its lower triangle filled with ones and zero elsewhere;
        in other words ``T[i,j] == 1`` for ``j <= i + k``, 0 otherwise.

    Examples
    --------
    >>> np.tri(3, 5, 2, dtype=int)
    array([[1, 1, 1, 0, 0],
           [1, 1, 1, 1, 0],
           [1, 1, 1, 1, 1]])

    >>> np.tri(3, 5, -1)
    array([[0.,  0.,  0.,  0.,  0.],
           [1.,  0.,  0.,  0.,  0.],
           [1.,  1.,  0.,  0.,  0.]])

    """
    if like is not None:
        return _tri_with_like(like, N, M=M, k=k, dtype=dtype)

    if M is None:
        M = N

    m = greater_equal.outer(arange(N, dtype=_min_int(0, N)),
                            arange(-k, M-k, dtype=_min_int(-k, M - k)))

    # Avoid making a copy if the requested type is already bool
    m = m.astype(dtype, copy=False)

    return m


_tri_with_like = array_function_dispatch()(tri)


def _trilu_dispatcher(m, k=None):
    return (m,)


@array_function_dispatch(_trilu_dispatcher)
def tril(m, k=0):
    """
    Lower triangle of an array.

    Return a copy of an array with elements above the `k`-th diagonal zeroed.
    For arrays with ``ndim`` exceeding 2, `tril` will apply to the final two
    axes.

    Parameters
    ----------
    m : array_like, shape (..., M, N)
        Input array.
    k : int, optional
        Diagonal above which to zero elements.  `k = 0` (the default) is the
        main diagonal, `k < 0` is below it and `k > 0` is above.

    Returns
    -------
    tril : ndarray, shape (..., M, N)
        Lower triangle of `m`, of same shape and data-type as `m`.

    See Also
    --------
    triu : same thing, only for the upper triangle

    Examples
    --------
    >>> np.tril([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
    array([[ 0,  0,  0],
           [ 4,  0,  0],
           [ 7,  8,  0],
           [10, 11, 12]])

    >>> np.tril(np.arange(3*4*5).reshape(3, 4, 5))
    array([[[ 0,  0,  0,  0,  0],
            [ 5,  6,  0,  0,  0],
            [10, 11, 12,  0,  0],
            [15, 16, 17, 18,  0]],
           [[20,  0,  0,  0,  0],
            [25, 26,  0,  0,  0],
            [30, 31, 32,  0,  0],
            [35, 36, 37, 38,  0]],
           [[40,  0,  0,  0,  0],
            [45, 46,  0,  0,  0],
            [50, 51, 52,  0,  0],
            [55, 56, 57, 58,  0]]])

    """
    m = asanyarray(m)
    mask = tri(*m.shape[-2:], k=k, dtype=bool)

    return where(mask, m, zeros(1, m.dtype))


@array_function_dispatch(_trilu_dispatcher)
def triu(m, k=0):
    """
    Upper triangle of an array.

    Return a copy of an array with the elements below the `k`-th diagonal
    zeroed. For arrays with ``ndim`` exceeding 2, `triu` will apply to the
    final two axes.

    Please refer to the documentation for `tril` for further details.

    See Also
    --------
    tril : lower triangle of an array

    Examples
    --------
    >>> np.triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
    array([[ 1,  2,  3],
           [ 4,  5,  6],
           [ 0,  8,  9],
           [ 0,  0, 12]])

    >>> np.triu(np.arange(3*4*5).reshape(3, 4, 5))
    array([[[ 0,  1,  2,  3,  4],
            [ 0,  6,  7,  8,  9],
            [ 0,  0, 12, 13, 14],
            [ 0,  0,  0, 18, 19]],
           [[20, 21, 22, 23, 24],
            [ 0, 26, 27, 28, 29],
            [ 0,  0, 32, 33, 34],
            [ 0,  0,  0, 38, 39]],
           [[40, 41, 42, 43, 44],
            [ 0, 46, 47, 48, 49],
            [ 0,  0, 52, 53, 54],
            [ 0,  0,  0, 58, 59]]])

    """
    m = asanyarray(m)
    mask = tri(*m.shape[-2:], k=k-1, dtype=bool)

    return where(mask, zeros(1, m.dtype), m)


def _vander_dispatcher(x, N=None, increasing=None):
    return (x,)


# Originally borrowed from John Hunter and matplotlib
@array_function_dispatch(_vander_dispatcher)
def vander(x, N=None, increasing=False):
    """
    Generate a Vandermonde matrix.

    The columns of the output matrix are powers of the input vector. The
    order of the powers is determined by the `increasing` boolean argument.
    Specifically, when `increasing` is False, the `i`-th output column is
    the input vector raised element-wise to the power of ``N - i - 1``. Such
    a matrix with a geometric progression in each row is named for Alexandre-
    Theophile Vandermonde.

    Parameters
    ----------
    x : array_like
        1-D input array.
    N : int, optional
        Number of columns in the output.  If `N` is not specified, a square
        array is returned (``N = len(x)``).
    increasing : bool, optional
        Order of the powers of the columns.  If True, the powers increase
        from left to right, if False (the default) they are reversed.

        .. versionadded:: 1.9.0

    Returns
    -------
    out : ndarray
        Vandermonde matrix.  If `increasing` is False, the first column is
        ``x^(N-1)``, the second ``x^(N-2)`` and so forth. If `increasing` is
        True, the columns are ``x^0, x^1, ..., x^(N-1)``.

    See Also
    --------
    polynomial.polynomial.polyvander

    Examples
    --------
    >>> x = np.array([1, 2, 3, 5])
    >>> N = 3
    >>> np.vander(x, N)
    array([[ 1,  1,  1],
           [ 4,  2,  1],
           [ 9,  3,  1],
           [25,  5,  1]])

    >>> np.column_stack([x**(N-1-i) for i in range(N)])
    array([[ 1,  1,  1],
           [ 4,  2,  1],
           [ 9,  3,  1],
           [25,  5,  1]])

    >>> x = np.array([1, 2, 3, 5])
    >>> np.vander(x)
    array([[  1,   1,   1,   1],
           [  8,   4,   2,   1],
           [ 27,   9,   3,   1],
           [125,  25,   5,   1]])
    >>> np.vander(x, increasing=True)
    array([[  1,   1,   1,   1],
           [  1,   2,   4,   8],
           [  1,   3,   9,  27],
           [  1,   5,  25, 125]])

    The determinant of a square Vandermonde matrix is the product
    of the differences between the values of the input vector:

    >>> np.linalg.det(np.vander(x))
    48.000000000000043 # may vary
    >>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1)
    48

    """
    x = asarray(x)
    if x.ndim != 1:
        raise ValueError("x must be a one-dimensional array or sequence.")
    if N is None:
        N = len(x)

    v = empty((len(x), N), dtype=promote_types(x.dtype, int))
    tmp = v[:, ::-1] if not increasing else v

    if N > 0:
        tmp[:, 0] = 1
    if N > 1:
        tmp[:, 1:] = x[:, None]
        multiply.accumulate(tmp[:, 1:], out=tmp[:, 1:], axis=1)

    return v


def _histogram2d_dispatcher(x, y, bins=None, range=None, density=None,
                            weights=None):
    yield x
    yield y

    # This terrible logic is adapted from the checks in histogram2d
    try:
        N = len(bins)
    except TypeError:
        N = 1
    if N == 2:
        yield from bins  # bins=[x, y]
    else:
        yield bins

    yield weights


@array_function_dispatch(_histogram2d_dispatcher)
def histogram2d(x, y, bins=10, range=None, density=None, weights=None):
    """
    Compute the bi-dimensional histogram of two data samples.

    Parameters
    ----------
    x : array_like, shape (N,)
        An array containing the x coordinates of the points to be
        histogrammed.
    y : array_like, shape (N,)
        An array containing the y coordinates of the points to be
        histogrammed.
    bins : int or array_like or [int, int] or [array, array], optional
        The bin specification:

          * If int, the number of bins for the two dimensions (nx=ny=bins).
          * If array_like, the bin edges for the two dimensions
            (x_edges=y_edges=bins).
          * If [int, int], the number of bins in each dimension
            (nx, ny = bins).
          * If [array, array], the bin edges in each dimension
            (x_edges, y_edges = bins).
          * A combination [int, array] or [array, int], where int
            is the number of bins and array is the bin edges.

    range : array_like, shape(2,2), optional
        The leftmost and rightmost edges of the bins along each dimension
        (if not specified explicitly in the `bins` parameters):
        ``[[xmin, xmax], [ymin, ymax]]``. All values outside of this range
        will be considered outliers and not tallied in the histogram.
    density : bool, optional
        If False, the default, returns the number of samples in each bin.
        If True, returns the probability *density* function at the bin,
        ``bin_count / sample_count / bin_area``.
    weights : array_like, shape(N,), optional
        An array of values ``w_i`` weighing each sample ``(x_i, y_i)``.
        Weights are normalized to 1 if `density` is True. If `density` is
        False, the values of the returned histogram are equal to the sum of
        the weights belonging to the samples falling into each bin.

    Returns
    -------
    H : ndarray, shape(nx, ny)
        The bi-dimensional histogram of samples `x` and `y`. Values in `x`
        are histogrammed along the first dimension and values in `y` are
        histogrammed along the second dimension.
    xedges : ndarray, shape(nx+1,)
        The bin edges along the first dimension.
    yedges : ndarray, shape(ny+1,)
        The bin edges along the second dimension.

    See Also
    --------
    histogram : 1D histogram
    histogramdd : Multidimensional histogram

    Notes
    -----
    When `density` is True, then the returned histogram is the sample
    density, defined such that the sum over bins of the product
    ``bin_value * bin_area`` is 1.

    Please note that the histogram does not follow the Cartesian convention
    where `x` values are on the abscissa and `y` values on the ordinate
    axis.  Rather, `x` is histogrammed along the first dimension of the
    array (vertical), and `y` along the second dimension of the array
    (horizontal).  This ensures compatibility with `histogramdd`.

    Examples
    --------
    >>> from matplotlib.image import NonUniformImage
    >>> import matplotlib.pyplot as plt

    Construct a 2-D histogram with variable bin width. First define the bin
    edges:

    >>> xedges = [0, 1, 3, 5]
    >>> yedges = [0, 2, 3, 4, 6]

    Next we create a histogram H with random bin content:

    >>> x = np.random.normal(2, 1, 100)
    >>> y = np.random.normal(1, 1, 100)
    >>> H, xedges, yedges = np.histogram2d(x, y, bins=(xedges, yedges))
    >>> # Histogram does not follow Cartesian convention (see Notes),
    >>> # therefore transpose H for visualization purposes.
    >>> H = H.T

    :func:`imshow <matplotlib.pyplot.imshow>` can only display square bins:

    >>> fig = plt.figure(figsize=(7, 3))
    >>> ax = fig.add_subplot(131, title='imshow: square bins')
    >>> plt.imshow(H, interpolation='nearest', origin='lower',
    ...         extent=[xedges[0], xedges[-1], yedges[0], yedges[-1]])
    <matplotlib.image.AxesImage object at 0x...>

    :func:`pcolormesh <matplotlib.pyplot.pcolormesh>` can display actual edges:

    >>> ax = fig.add_subplot(132, title='pcolormesh: actual edges',
    ...         aspect='equal')
    >>> X, Y = np.meshgrid(xedges, yedges)
    >>> ax.pcolormesh(X, Y, H)
    <matplotlib.collections.QuadMesh object at 0x...>

    :class:`NonUniformImage <matplotlib.image.NonUniformImage>` can be used to
    display actual bin edges with interpolation:

    >>> ax = fig.add_subplot(133, title='NonUniformImage: interpolated',
    ...         aspect='equal', xlim=xedges[[0, -1]], ylim=yedges[[0, -1]])
    >>> im = NonUniformImage(ax, interpolation='bilinear')
    >>> xcenters = (xedges[:-1] + xedges[1:]) / 2
    >>> ycenters = (yedges[:-1] + yedges[1:]) / 2
    >>> im.set_data(xcenters, ycenters, H)
    >>> ax.add_image(im)
    >>> plt.show()

    It is also possible to construct a 2-D histogram without specifying bin
    edges:

    >>> # Generate non-symmetric test data
    >>> n = 10000
    >>> x = np.linspace(1, 100, n)
    >>> y = 2*np.log(x) + np.random.rand(n) - 0.5
    >>> # Compute 2d histogram. Note the order of x/y and xedges/yedges
    >>> H, yedges, xedges = np.histogram2d(y, x, bins=20)

    Now we can plot the histogram using
    :func:`pcolormesh <matplotlib.pyplot.pcolormesh>`, and a
    :func:`hexbin <matplotlib.pyplot.hexbin>` for comparison.

    >>> # Plot histogram using pcolormesh
    >>> fig, (ax1, ax2) = plt.subplots(ncols=2, sharey=True)
    >>> ax1.pcolormesh(xedges, yedges, H, cmap='rainbow')
    >>> ax1.plot(x, 2*np.log(x), 'k-')
    >>> ax1.set_xlim(x.min(), x.max())
    >>> ax1.set_ylim(y.min(), y.max())
    >>> ax1.set_xlabel('x')
    >>> ax1.set_ylabel('y')
    >>> ax1.set_title('histogram2d')
    >>> ax1.grid()

    >>> # Create hexbin plot for comparison
    >>> ax2.hexbin(x, y, gridsize=20, cmap='rainbow')
    >>> ax2.plot(x, 2*np.log(x), 'k-')
    >>> ax2.set_title('hexbin')
    >>> ax2.set_xlim(x.min(), x.max())
    >>> ax2.set_xlabel('x')
    >>> ax2.grid()

    >>> plt.show()
    """
    from numpy import histogramdd

    if len(x) != len(y):
        raise ValueError('x and y must have the same length.')

    try:
        N = len(bins)
    except TypeError:
        N = 1

    if N != 1 and N != 2:
        xedges = yedges = asarray(bins)
        bins = [xedges, yedges]
    hist, edges = histogramdd([x, y], bins, range, density, weights)
    return hist, edges[0], edges[1]


@set_module('numpy')
def mask_indices(n, mask_func, k=0):
    """
    Return the indices to access (n, n) arrays, given a masking function.

    Assume `mask_func` is a function that, for a square array a of size
    ``(n, n)`` with a possible offset argument `k`, when called as
    ``mask_func(a, k)`` returns a new array with zeros in certain locations
    (functions like `triu` or `tril` do precisely this). Then this function
    returns the indices where the non-zero values would be located.

    Parameters
    ----------
    n : int
        The returned indices will be valid to access arrays of shape (n, n).
    mask_func : callable
        A function whose call signature is similar to that of `triu`, `tril`.
        That is, ``mask_func(x, k)`` returns a boolean array, shaped like `x`.
        `k` is an optional argument to the function.
    k : scalar
        An optional argument which is passed through to `mask_func`. Functions
        like `triu`, `tril` take a second argument that is interpreted as an
        offset.

    Returns
    -------
    indices : tuple of arrays.
        The `n` arrays of indices corresponding to the locations where
        ``mask_func(np.ones((n, n)), k)`` is True.

    See Also
    --------
    triu, tril, triu_indices, tril_indices

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

    Examples
    --------
    These are the indices that would allow you to access the upper triangular
    part of any 3x3 array:

    >>> iu = np.mask_indices(3, np.triu)

    For example, if `a` is a 3x3 array:

    >>> a = np.arange(9).reshape(3, 3)
    >>> a
    array([[0, 1, 2],
           [3, 4, 5],
           [6, 7, 8]])
    >>> a[iu]
    array([0, 1, 2, 4, 5, 8])

    An offset can be passed also to the masking function.  This gets us the
    indices starting on the first diagonal right of the main one:

    >>> iu1 = np.mask_indices(3, np.triu, 1)

    with which we now extract only three elements:

    >>> a[iu1]
    array([1, 2, 5])

    """
    m = ones((n, n), int)
    a = mask_func(m, k)
    return nonzero(a != 0)


@set_module('numpy')
def tril_indices(n, k=0, m=None):
    """
    Return the indices for the lower-triangle of an (n, m) array.

    Parameters
    ----------
    n : int
        The row dimension of the arrays for which the returned
        indices will be valid.
    k : int, optional
        Diagonal offset (see `tril` for details).
    m : int, optional
        .. versionadded:: 1.9.0

        The column dimension of the arrays for which the returned
        arrays will be valid.
        By default `m` is taken equal to `n`.


    Returns
    -------
    inds : tuple of arrays
        The indices for the triangle. The returned tuple contains two arrays,
        each with the indices along one dimension of the array.

    See also
    --------
    triu_indices : similar function, for upper-triangular.
    mask_indices : generic function accepting an arbitrary mask function.
    tril, triu

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

    Examples
    --------
    Compute two different sets of indices to access 4x4 arrays, one for the
    lower triangular part starting at the main diagonal, and one starting two
    diagonals further right:

    >>> il1 = np.tril_indices(4)
    >>> il2 = np.tril_indices(4, 2)

    Here is how they can be used with a sample array:

    >>> 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]])

    Both for indexing:

    >>> a[il1]
    array([ 0,  4,  5, ..., 13, 14, 15])

    And for assigning values:

    >>> a[il1] = -1
    >>> a
    array([[-1,  1,  2,  3],
           [-1, -1,  6,  7],
           [-1, -1, -1, 11],
           [-1, -1, -1, -1]])

    These cover almost the whole array (two diagonals right of the main one):

    >>> a[il2] = -10
    >>> a
    array([[-10, -10, -10,   3],
           [-10, -10, -10, -10],
           [-10, -10, -10, -10],
           [-10, -10, -10, -10]])

    """
    tri_ = tri(n, m, k=k, dtype=bool)

    return tuple(broadcast_to(inds, tri_.shape)[tri_]
                 for inds in indices(tri_.shape, sparse=True))


def _trilu_indices_form_dispatcher(arr, k=None):
    return (arr,)


@array_function_dispatch(_trilu_indices_form_dispatcher)
def tril_indices_from(arr, k=0):
    """
    Return the indices for the lower-triangle of arr.

    See `tril_indices` for full details.

    Parameters
    ----------
    arr : array_like
        The indices will be valid for square arrays whose dimensions are
        the same as arr.
    k : int, optional
        Diagonal offset (see `tril` for details).

    Examples
    --------

    Create a 4 by 4 array.

    >>> 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]])

    Pass the array to get the indices of the lower triangular elements.

    >>> trili = np.tril_indices_from(a)
    >>> trili
    (array([0, 1, 1, 2, 2, 2, 3, 3, 3, 3]), array([0, 0, 1, 0, 1, 2, 0, 1, 2, 3]))

    >>> a[trili]
    array([ 0,  4,  5,  8,  9, 10, 12, 13, 14, 15])

    This is syntactic sugar for tril_indices().

    >>> np.tril_indices(a.shape[0])
    (array([0, 1, 1, 2, 2, 2, 3, 3, 3, 3]), array([0, 0, 1, 0, 1, 2, 0, 1, 2, 3]))

    Use the `k` parameter to return the indices for the lower triangular array
    up to the k-th diagonal.

    >>> trili1 = np.tril_indices_from(a, k=1)
    >>> a[trili1]
    array([ 0,  1,  4,  5,  6,  8,  9, 10, 11, 12, 13, 14, 15])

    See Also
    --------
    tril_indices, tril, triu_indices_from

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

    """
    if arr.ndim != 2:
        raise ValueError("input array must be 2-d")
    return tril_indices(arr.shape[-2], k=k, m=arr.shape[-1])


@set_module('numpy')
def triu_indices(n, k=0, m=None):
    """
    Return the indices for the upper-triangle of an (n, m) array.

    Parameters
    ----------
    n : int
        The size of the arrays for which the returned indices will
        be valid.
    k : int, optional
        Diagonal offset (see `triu` for details).
    m : int, optional
        .. versionadded:: 1.9.0

        The column dimension of the arrays for which the returned
        arrays will be valid.
        By default `m` is taken equal to `n`.


    Returns
    -------
    inds : tuple, shape(2) of ndarrays, shape(`n`)
        The indices for the triangle. The returned tuple contains two arrays,
        each with the indices along one dimension of the array.  Can be used
        to slice a ndarray of shape(`n`, `n`).

    See also
    --------
    tril_indices : similar function, for lower-triangular.
    mask_indices : generic function accepting an arbitrary mask function.
    triu, tril

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

    Examples
    --------
    Compute two different sets of indices to access 4x4 arrays, one for the
    upper triangular part starting at the main diagonal, and one starting two
    diagonals further right:

    >>> iu1 = np.triu_indices(4)
    >>> iu2 = np.triu_indices(4, 2)

    Here is how they can be used with a sample array:

    >>> 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]])

    Both for indexing:

    >>> a[iu1]
    array([ 0,  1,  2, ..., 10, 11, 15])

    And for assigning values:

    >>> a[iu1] = -1
    >>> a
    array([[-1, -1, -1, -1],
           [ 4, -1, -1, -1],
           [ 8,  9, -1, -1],
           [12, 13, 14, -1]])

    These cover only a small part of the whole array (two diagonals right
    of the main one):

    >>> a[iu2] = -10
    >>> a
    array([[ -1,  -1, -10, -10],
           [  4,  -1,  -1, -10],
           [  8,   9,  -1,  -1],
           [ 12,  13,  14,  -1]])

    """
    tri_ = ~tri(n, m, k=k - 1, dtype=bool)

    return tuple(broadcast_to(inds, tri_.shape)[tri_]
                 for inds in indices(tri_.shape, sparse=True))


@array_function_dispatch(_trilu_indices_form_dispatcher)
def triu_indices_from(arr, k=0):
    """
    Return the indices for the upper-triangle of arr.

    See `triu_indices` for full details.

    Parameters
    ----------
    arr : ndarray, shape(N, N)
        The indices will be valid for square arrays.
    k : int, optional
        Diagonal offset (see `triu` for details).

    Returns
    -------
    triu_indices_from : tuple, shape(2) of ndarray, shape(N)
        Indices for the upper-triangle of `arr`.

    Examples
    --------

    Create a 4 by 4 array.

    >>> 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]])

    Pass the array to get the indices of the upper triangular elements.

    >>> triui = np.triu_indices_from(a)
    >>> triui
    (array([0, 0, 0, 0, 1, 1, 1, 2, 2, 3]), array([0, 1, 2, 3, 1, 2, 3, 2, 3, 3]))

    >>> a[triui]
    array([ 0,  1,  2,  3,  5,  6,  7, 10, 11, 15])

    This is syntactic sugar for triu_indices().

    >>> np.triu_indices(a.shape[0])
    (array([0, 0, 0, 0, 1, 1, 1, 2, 2, 3]), array([0, 1, 2, 3, 1, 2, 3, 2, 3, 3]))

    Use the `k` parameter to return the indices for the upper triangular array
    from the k-th diagonal.

    >>> triuim1 = np.triu_indices_from(a, k=1)
    >>> a[triuim1]
    array([ 1,  2,  3,  6,  7, 11])


    See Also
    --------
    triu_indices, triu, tril_indices_from

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

    """
    if arr.ndim != 2:
        raise ValueError("input array must be 2-d")
    return triu_indices(arr.shape[-2], k=k, m=arr.shape[-1])