.. _basics.indexing:

****************************************
Indexing on :class:`ndarrays <.ndarray>`
****************************************

.. seealso::

   :ref:`Indexing routines <routines.indexing>`

.. sectionauthor:: adapted from "Guide to NumPy" by Travis E. Oliphant

.. currentmodule:: numpy

.. index:: indexing, slicing

:class:`ndarrays <ndarray>` can be indexed using the standard Python
``x[obj]`` syntax, where *x* is the array and *obj* the selection.
There are different kinds of indexing available depending on *obj*:
basic indexing, advanced indexing and field access.

Most of the following examples show the use of indexing when
referencing data in an array. The examples work just as well
when assigning to an array. See :ref:`assigning-values-to-indexed-arrays` for
specific examples and explanations on how assignments work.

Note that in Python, ``x[(exp1, exp2, ..., expN)]`` is equivalent to
``x[exp1, exp2, ..., expN]``; the latter is just syntactic sugar
for the former.


Basic indexing
--------------

.. _single-element-indexing:

Single element indexing
^^^^^^^^^^^^^^^^^^^^^^^

Single element indexing works
exactly like that for other standard Python sequences. It is 0-based,
and accepts negative indices for indexing from the end of the array. ::

    >>> x = np.arange(10)
    >>> x[2]
    2
    >>> x[-2]
    8

It is not necessary to
separate each dimension's index into its own set of square brackets. ::

    >>> x.shape = (2, 5)  # now x is 2-dimensional
    >>> x[1, 3]
    8
    >>> x[1, -1]
    9

Note that if one indexes a multidimensional array with fewer indices
than dimensions, one gets a subdimensional array. For example: ::

    >>> x[0]
    array([0, 1, 2, 3, 4])

That is, each index specified selects the array corresponding to the
rest of the dimensions selected. In the above example, choosing 0
means that the remaining dimension of length 5 is being left unspecified,
and that what is returned is an array of that dimensionality and size.
It must be noted that the returned array is a :term:`view`, i.e., it is not a
copy of the original, but points to the same values in memory as does the
original array.
In  this case, the 1-D array at the first position (0) is returned.
So using a single index on the returned array, results in a single
element being returned. That is: ::

    >>> x[0][2]
    2

So note that ``x[0, 2] == x[0][2]`` though the second case is more
inefficient as a new temporary array is created after the first index
that is subsequently indexed by 2.

.. note::

    NumPy uses C-order indexing. That means that the last
    index usually represents the most rapidly changing memory location,
    unlike Fortran or IDL, where the first index represents the most
    rapidly changing location in memory. This difference represents a
    great potential for confusion.


Slicing and striding
^^^^^^^^^^^^^^^^^^^^

Basic slicing extends Python's basic concept of slicing to N
dimensions. Basic slicing occurs when *obj* is a :class:`slice` object
(constructed by ``start:stop:step`` notation inside of brackets), an
integer, or a tuple of slice objects and integers. :py:data:`Ellipsis`
and :const:`newaxis` objects can be interspersed with these as
well.

.. deprecated:: 1.15.0

  In order to remain backward compatible with a common usage in
  Numeric, basic slicing is also initiated if the selection object is
  any non-ndarray and non-tuple sequence (such as a :class:`list`) containing
  :class:`slice` objects, the :py:data:`Ellipsis` object, or the :const:`newaxis`
  object, but not for integer arrays or other embedded sequences.

.. index::
   triple: ndarray; special methods; getitem
   triple: ndarray; special methods; setitem
   single: ellipsis
   single: newaxis

The simplest case of indexing with *N* integers returns an :ref:`array
scalar <arrays.scalars>` representing the corresponding item.  As in
Python, all indices are zero-based: for the *i*-th index :math:`n_i`,
the valid range is :math:`0 \le n_i < d_i` where :math:`d_i` is the
*i*-th element of the shape of the array.  Negative indices are
interpreted as counting from the end of the array (*i.e.*, if
:math:`n_i < 0`, it means :math:`n_i + d_i`).


All arrays generated by basic slicing are always :term:`views <view>`
of the original array.

.. note::

    NumPy slicing creates a :term:`view` instead of a copy as in the case of
    built-in Python sequences such as string, tuple and list.
    Care must be taken when extracting
    a small portion from a large array which becomes useless after the
    extraction, because the small portion extracted contains a reference
    to the large original array whose memory will not be released until
    all arrays derived from it are garbage-collected. In such cases an
    explicit ``copy()`` is recommended.

The standard rules of sequence slicing apply to basic slicing on a
per-dimension basis (including using a step index). Some useful
concepts to remember include:

- The basic slice syntax is ``i:j:k`` where *i* is the starting index,
  *j* is the stopping index, and *k* is the step (:math:`k\neq0`).
  This selects the *m* elements (in the corresponding dimension) with
  index values *i*, *i + k*, ..., *i + (m - 1) k* where
  :math:`m = q + (r\neq0)` and *q* and *r* are the quotient and remainder
  obtained by dividing *j - i* by *k*: *j - i = q k + r*, so that
  *i + (m - 1) k < j*. 
  For example::

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

- Negative *i* and *j* are interpreted as *n + i* and *n + j* where
  *n* is the number of elements in the corresponding dimension.
  Negative *k* makes stepping go towards smaller indices.
  From the above example::

      >>> x[-2:10]
      array([8, 9])
      >>> x[-3:3:-1]
      array([7, 6, 5, 4])

- Assume *n* is the number of elements in the dimension being
  sliced. Then, if *i* is not given it defaults to 0 for *k > 0* and
  *n - 1* for *k < 0* . If *j* is not given it defaults to *n* for *k > 0*
  and *-n-1* for *k < 0* . If *k* is not given it defaults to 1. Note that
  ``::`` is the same as ``:`` and means select all indices along this
  axis.
  From the above example::

      >>> x[5:]
      array([5, 6, 7, 8, 9])

- If the number of objects in the selection tuple is less than
  *N*, then ``:`` is assumed for any subsequent dimensions.
  For example::

      >>> x = np.array([[[1],[2],[3]], [[4],[5],[6]]])
      >>> x.shape
      (2, 3, 1)
      >>> x[1:2]
      array([[[4],
              [5],
              [6]]])

- An integer, *i*, returns the same values as ``i:i+1``
  **except** the dimensionality of the returned object is reduced by
  1. In particular, a selection tuple with the *p*-th
  element an integer (and all other entries ``:``) returns the
  corresponding sub-array with dimension *N - 1*. If *N = 1*
  then the returned object is an array scalar. These objects are
  explained in :ref:`arrays.scalars`.

- If the selection tuple has all entries ``:`` except the
  *p*-th entry which is a slice object ``i:j:k``,
  then the returned array has dimension *N* formed by
  concatenating the sub-arrays returned by integer indexing of
  elements *i*, *i+k*, ..., *i + (m - 1) k < j*,

- Basic slicing with more than one non-``:`` entry in the slicing
  tuple, acts like repeated application of slicing using a single
  non-``:`` entry, where the non-``:`` entries are successively taken
  (with all other non-``:`` entries replaced by ``:``). Thus,
  ``x[ind1, ..., ind2,:]`` acts like ``x[ind1][..., ind2, :]`` under basic
  slicing.

  .. warning:: The above is **not** true for advanced indexing.

- You may use slicing to set values in the array, but (unlike lists) you
  can never grow the array. The size of the value to be set in
  ``x[obj] = value`` must be (broadcastable) to the same shape as
  ``x[obj]``.

- A slicing tuple can always be constructed as *obj*
  and used in the ``x[obj]`` notation. Slice objects can be used in
  the construction in place of the ``[start:stop:step]``
  notation. For example, ``x[1:10:5, ::-1]`` can also be implemented
  as ``obj = (slice(1, 10, 5), slice(None, None, -1)); x[obj]`` . This
  can be useful for constructing generic code that works on arrays
  of arbitrary dimensions. See :ref:`dealing-with-variable-indices`
  for more information.

.. index::
   pair: ndarray; view


Dimensional indexing tools
^^^^^^^^^^^^^^^^^^^^^^^^^^

There are some tools to facilitate the easy matching of array shapes with
expressions and in assignments.

:py:data:`Ellipsis` expands to the number of ``:`` objects needed for the
selection tuple to index all dimensions. In most cases, this means that the
length of the expanded selection tuple is ``x.ndim``. There may only be a
single ellipsis present.
From the above example::

    >>> x[..., 0]
    array([[1, 2, 3],
          [4, 5, 6]])

This is equivalent to::

    >>> x[:, :, 0]
    array([[1, 2, 3],
          [4, 5, 6]])

Each :const:`newaxis` object in the selection tuple serves to expand
the dimensions of the resulting selection by one unit-length
dimension.  The added dimension is the position of the :const:`newaxis`
object in the selection tuple. :const:`newaxis` is an alias for
``None``, and ``None`` can be used in place of this with the same result.
From the above example::

    >>> x[:, np.newaxis, :, :].shape
    (2, 1, 3, 1)
    >>> x[:, None, :, :].shape
    (2, 1, 3, 1)

This can be handy to combine two
arrays in a way that otherwise would require explicit reshaping
operations. For example::

    >>> x = np.arange(5)
    >>> x[:, np.newaxis] + x[np.newaxis, :]
    array([[0, 1, 2, 3, 4],
          [1, 2, 3, 4, 5],
          [2, 3, 4, 5, 6],
          [3, 4, 5, 6, 7],
          [4, 5, 6, 7, 8]])


.. _advanced-indexing:

Advanced indexing
-----------------

Advanced indexing is triggered when the selection object, *obj*, is a
non-tuple sequence object, an :class:`ndarray` (of data type integer or bool),
or a tuple with at least one sequence object or ndarray (of data type
integer or bool). There are two types of advanced indexing: integer
and Boolean.

Advanced indexing always returns a *copy* of the data (contrast with
basic slicing that returns a :term:`view`).

.. warning::

   The definition of advanced indexing means that ``x[(1, 2, 3),]`` is
   fundamentally different than ``x[(1, 2, 3)]``. The latter is
   equivalent to ``x[1, 2, 3]`` which will trigger basic selection while
   the former will trigger advanced indexing. Be sure to understand
   why this occurs.

   Also recognize that ``x[[1, 2, 3]]`` will trigger advanced indexing,
   whereas due to the deprecated Numeric compatibility mentioned above,
   ``x[[1, 2, slice(None)]]`` will trigger basic slicing.

Integer array indexing
^^^^^^^^^^^^^^^^^^^^^^

Integer array indexing allows selection of arbitrary items in the array
based on their *N*-dimensional index. Each integer array represents a number
of indices into that dimension.

Negative values are permitted in the index arrays and work as they do with
single indices or slices::

    >>> x = np.arange(10, 1, -1)
    >>> x
    array([10,  9,  8,  7,  6,  5,  4,  3,  2])
    >>> x[np.array([3, 3, 1, 8])]
    array([7, 7, 9, 2])
    >>> x[np.array([3, 3, -3, 8])]
    array([7, 7, 4, 2])

If the index values are out of bounds then an ``IndexError`` is thrown::

    >>> x = np.array([[1, 2], [3, 4], [5, 6]])
    >>> x[np.array([1, -1])]
    array([[3, 4],
          [5, 6]])
    >>> x[np.array([3, 4])]
    IndexError: index 3 is out of bounds for axis 0 with size 3

When the index consists of as many integer arrays as dimensions of the array
being indexed, the indexing is straightforward, but different from slicing.

Advanced indices always are :ref:`broadcast<basics.broadcasting>` and
iterated as *one*::

     result[i_1, ..., i_M] == x[ind_1[i_1, ..., i_M], ind_2[i_1, ..., i_M],
                                ..., ind_N[i_1, ..., i_M]]

Note that the resulting shape is identical to the (broadcast) indexing array
shapes ``ind_1, ..., ind_N``. If the indices cannot be broadcast to the
same shape, an exception ``IndexError: shape mismatch: indexing arrays could
not be broadcast together with shapes...`` is raised. 

Indexing with multidimensional index arrays tend
to be more unusual uses, but they are permitted, and they are useful for some
problems. We’ll start with the simplest multidimensional case::

    >>> y = np.arange(35).reshape(5, 7)
    >>> y
    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, 24, 25, 26, 27],
           [28, 29, 30, 31, 32, 33, 34]])
    >>> y[np.array([0, 2, 4]), np.array([0, 1, 2])]
    array([ 0, 15, 30])

In this case, if the index arrays have a matching shape, and there is an
index array for each dimension of the array being indexed, the resultant
array has the same shape as the index arrays, and the values correspond
to the index set for each position in the index arrays. In this example,
the first index value is 0 for both index arrays, and thus the first value
of the resultant array is ``y[0, 0]``. The next value is ``y[2, 1]``, and
the last is ``y[4, 2]``.

If the index arrays do not have the same shape, there is an attempt to
broadcast them to the same shape. If they cannot be broadcast to the same
shape, an exception is raised::

    >>> y[np.array([0, 2, 4]), np.array([0, 1])]
    IndexError: shape mismatch: indexing arrays could not be broadcast
    together with shapes (3,) (2,)

The broadcasting mechanism permits index arrays to be combined with
scalars for other indices. The effect is that the scalar value is used
for all the corresponding values of the index arrays::

    >>> y[np.array([0, 2, 4]), 1]
    array([ 1, 15, 29])

Jumping to the next level of complexity, it is possible to only partially
index an array with index arrays. It takes a bit of thought to understand
what happens in such cases. For example if we just use one index array
with y::

    >>> y[np.array([0, 2, 4])]
    array([[ 0,  1,  2,  3,  4,  5,  6],
          [14, 15, 16, 17, 18, 19, 20],
          [28, 29, 30, 31, 32, 33, 34]])

It results in the construction of a new array where each value of the
index array selects one row from the array being indexed and the resultant
array has the resulting shape (number of index elements, size of row).

In general, the shape of the resultant array will be the concatenation of
the shape of the index array (or the shape that all the index arrays were
broadcast to) with the shape of any unused dimensions (those not indexed)
in the array being indexed.

.. rubric:: Example

From each row, a specific element should be selected. The row index is just
``[0, 1, 2]`` and the column index specifies the element to choose for the
corresponding row, here ``[0, 1, 0]``. Using both together the task
can be solved using advanced indexing::

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

To achieve a behaviour similar to the basic slicing above, broadcasting can be
used. The function :func:`ix_` can help with this broadcasting. This is best
understood with an example.

.. rubric:: Example

From a 4x3 array the corner elements should be selected using advanced
indexing. Thus all elements for which the column is one of ``[0, 2]`` and
the row is one of ``[0, 3]`` need to be selected. To use advanced indexing
one needs to select all elements *explicitly*. Using the method explained
previously one could write::

    >>> x = np.array([[ 0,  1,  2],
    ...               [ 3,  4,  5],
    ...               [ 6,  7,  8],
    ...               [ 9, 10, 11]])
    >>> rows = np.array([[0, 0],
    ...                  [3, 3]], dtype=np.intp)
    >>> columns = np.array([[0, 2],
    ...                     [0, 2]], dtype=np.intp)
    >>> x[rows, columns]
    array([[ 0,  2],
           [ 9, 11]])

However, since the indexing arrays above just repeat themselves,
broadcasting can be used (compare operations such as
``rows[:, np.newaxis] + columns``) to simplify this::

    >>> rows = np.array([0, 3], dtype=np.intp)
    >>> columns = np.array([0, 2], dtype=np.intp)
    >>> rows[:, np.newaxis]
    array([[0],
           [3]])
    >>> x[rows[:, np.newaxis], columns]
    array([[ 0,  2],
           [ 9, 11]])

This broadcasting can also be achieved using the function :func:`ix_`:

    >>> x[np.ix_(rows, columns)]
    array([[ 0,  2],
           [ 9, 11]])

Note that without the ``np.ix_`` call, only the diagonal elements would
be selected::

    >>> x[rows, columns]
    array([ 0, 11])

This difference is the most important thing to remember about
indexing with multiple advanced indices.

.. rubric:: Example

A real-life example of where advanced indexing may be useful is for a color
lookup table where we want to map the values of an image into RGB triples for
display. The lookup table could have a shape (nlookup, 3). Indexing
such an array with an image with shape (ny, nx) with dtype=np.uint8
(or any integer type so long as values are with the bounds of the
lookup table) will result in an array of shape (ny, nx, 3) where a
triple of RGB values is associated with each pixel location.


Boolean array indexing
^^^^^^^^^^^^^^^^^^^^^^

This advanced indexing occurs when *obj* is an array object of Boolean
type, such as may be returned from comparison operators. A single
boolean index array is practically identical to ``x[obj.nonzero()]`` where,
as described above, :meth:`obj.nonzero() <ndarray.nonzero>` returns a
tuple (of length :attr:`obj.ndim <ndarray.ndim>`) of integer index
arrays showing the :py:data:`True` elements of *obj*. However, it is
faster when ``obj.shape == x.shape``.

If ``obj.ndim == x.ndim``, ``x[obj]`` returns a 1-dimensional array
filled with the elements of *x* corresponding to the :py:data:`True`
values of *obj*.  The search order will be :term:`row-major`,
C-style. If *obj* has :py:data:`True` values at entries that are outside
of the bounds of *x*, then an index error will be raised. If *obj* is
smaller than *x* it is identical to filling it with :py:data:`False`.

A common use case for this is filtering for desired element values.
For example, one may wish to select all entries from an array which
are not :const:`NaN`::

    >>> x = np.array([[1., 2.], [np.nan, 3.], [np.nan, np.nan]])
    >>> x[~np.isnan(x)]
    array([1., 2., 3.])

Or wish to add a constant to all negative elements::

    >>> x = np.array([1., -1., -2., 3])
    >>> x[x < 0] += 20
    >>> x
    array([1., 19., 18., 3.])

In general if an index includes a Boolean array, the result will be
identical to inserting ``obj.nonzero()`` into the same position
and using the integer array indexing mechanism described above.
``x[ind_1, boolean_array, ind_2]`` is equivalent to
``x[(ind_1,) + boolean_array.nonzero() + (ind_2,)]``.

If there is only one Boolean array and no integer indexing array present,
this is straightforward. Care must only be taken to make sure that the
boolean index has *exactly* as many dimensions as it is supposed to work
with.

In general, when the boolean array has fewer dimensions than the array being
indexed, this is equivalent to ``x[b, ...]``, which means x is indexed by b
followed by as many ``:`` as are needed to fill out the rank of x. Thus the
shape of the result is one dimension containing the number of True elements of
the boolean array, followed by the remaining dimensions of the array being
indexed::

    >>> x = np.arange(35).reshape(5, 7)
    >>> b = x > 20
    >>> b[:, 5]
    array([False, False, False,  True,  True])
    >>> x[b[:, 5]]
    array([[21, 22, 23, 24, 25, 26, 27],
          [28, 29, 30, 31, 32, 33, 34]])

Here the 4th and 5th rows are selected from the indexed array and
combined to make a 2-D array.

.. rubric:: Example

From an array, select all rows which sum up to less or equal two::

    >>> x = np.array([[0, 1], [1, 1], [2, 2]])
    >>> rowsum = x.sum(-1)
    >>> x[rowsum <= 2, :]
    array([[0, 1],
           [1, 1]])


Combining multiple Boolean indexing arrays or a Boolean with an integer
indexing array can best be understood with the
:meth:`obj.nonzero() <ndarray.nonzero>` analogy. The function :func:`ix_`
also supports boolean arrays and will work without any surprises.

.. rubric:: Example

Use boolean indexing to select all rows adding up to an even
number. At the same time columns 0 and 2 should be selected with an
advanced integer index. Using the :func:`ix_` function this can be done
with::

    >>> x = np.array([[ 0,  1,  2],
    ...               [ 3,  4,  5],
    ...               [ 6,  7,  8],
    ...               [ 9, 10, 11]])
    >>> rows = (x.sum(-1) % 2) == 0
    >>> rows
    array([False,  True, False,  True])
    >>> columns = [0, 2]
    >>> x[np.ix_(rows, columns)]
    array([[ 3,  5],
           [ 9, 11]])

Without the ``np.ix_`` call, only the diagonal elements would be
selected.

Or without ``np.ix_`` (compare the integer array examples)::

    >>> rows = rows.nonzero()[0]
    >>> x[rows[:, np.newaxis], columns]
    array([[ 3,  5],
           [ 9, 11]])

.. rubric:: Example

Use a 2-D boolean array of shape (2, 3)
with four True elements to select rows from a 3-D array of shape
(2, 3, 5) results in a 2-D result of shape (4, 5)::

    >>> x = np.arange(30).reshape(2, 3, 5)
    >>> x
    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, 24],
            [25, 26, 27, 28, 29]]])
    >>> b = np.array([[True, True, False], [False, True, True]])
    >>> x[b]
    array([[ 0,  1,  2,  3,  4],
          [ 5,  6,  7,  8,  9],
          [20, 21, 22, 23, 24],
          [25, 26, 27, 28, 29]])


.. _combining-advanced-and-basic-indexing:

Combining advanced and basic indexing
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

When there is at least one slice (``:``), ellipsis (``...``) or :const:`newaxis`
in the index (or the array has more dimensions than there are advanced indices),
then the behaviour can be more complicated. It is like concatenating the
indexing result for each advanced index element.

In the simplest case, there is only a *single* advanced index combined with
a slice. For example::

    >>> y = np.arange(35).reshape(5,7)
    >>> y[np.array([0, 2, 4]), 1:3]
    array([[ 1,  2],
           [15, 16],
           [29, 30]])

In effect, the slice and index array operation are independent. The slice
operation extracts columns with index 1 and 2, (i.e. the 2nd and 3rd columns),
followed by the index array operation which extracts rows with index 0, 2 and 4
(i.e the first, third and fifth rows). This is equivalent to::

    >>> y[:, 1:3][np.array([0, 2, 4]), :]
    array([[ 1,  2],
           [15, 16],
           [29, 30]])

A single advanced index can, for example, replace a slice and the result array
will be the same. However, it is a copy and may have a different memory layout.
A slice is preferable when it is possible.
For example::

    >>> x = np.array([[ 0,  1,  2],
    ...               [ 3,  4,  5],
    ...               [ 6,  7,  8],
    ...               [ 9, 10, 11]])
    >>> x[1:2, 1:3]
    array([[4, 5]])
    >>> x[1:2, [1, 2]]
    array([[4, 5]])

The easiest way to understand a combination of *multiple* advanced indices may
be to think in terms of the resulting shape. There are two parts to the indexing
operation, the subspace defined by the basic indexing (excluding integers) and
the subspace from the advanced indexing part. Two cases of index combination
need to be distinguished:

* The advanced indices are separated by a slice, :py:data:`Ellipsis` or
  :const:`newaxis`. For example ``x[arr1, :, arr2]``.
* The advanced indices are all next to each other.
  For example ``x[..., arr1, arr2, :]`` but *not* ``x[arr1, :, 1]``
  since ``1`` is an advanced index in this regard.

In the first case, the dimensions resulting from the advanced indexing
operation come first in the result array, and the subspace dimensions after
that.
In the second case, the dimensions from the advanced indexing operations
are inserted into the result array at the same spot as they were in the
initial array (the latter logic is what makes simple advanced indexing
behave just like slicing). 

.. rubric:: Example

Suppose ``x.shape`` is (10, 20, 30) and ``ind`` is a (2, 3, 4)-shaped
indexing :class:`intp` array, then ``result = x[..., ind, :]`` has
shape (10, 2, 3, 4, 30) because the (20,)-shaped subspace has been
replaced with a (2, 3, 4)-shaped broadcasted indexing subspace. If
we let *i, j, k* loop over the (2, 3, 4)-shaped subspace then
``result[..., i, j, k, :] = x[..., ind[i, j, k], :]``. This example
produces the same result as :meth:`x.take(ind, axis=-2) <ndarray.take>`.

.. rubric:: Example

Let ``x.shape`` be (10, 20, 30, 40, 50) and suppose ``ind_1``
and ``ind_2`` can be broadcast to the shape (2, 3, 4). Then
``x[:, ind_1, ind_2]`` has shape (10, 2, 3, 4, 40, 50) because the
(20, 30)-shaped subspace from X has been replaced with the
(2, 3, 4) subspace from the indices. However,
``x[:, ind_1, :, ind_2]`` has shape (2, 3, 4, 10, 30, 50) because there
is no unambiguous place to drop in the indexing subspace, thus
it is tacked-on to the beginning. It is always possible to use
:meth:`.transpose() <ndarray.transpose>` to move the subspace
anywhere desired. Note that this example cannot be replicated
using :func:`take`.

.. rubric:: Example

Slicing can be combined with broadcasted boolean indices::

    >>> x = np.arange(35).reshape(5, 7)
    >>> b = x > 20
    >>> b
    array([[False, False, False, False, False, False, False],
          [False, False, False, False, False, False, False],
          [False, False, False, False, False, False, False],
          [ True,  True,  True,  True,  True,  True,  True],
          [ True,  True,  True,  True,  True,  True,  True]])
    >>> x[b[:, 5], 1:3]
    array([[22, 23],
          [29, 30]])


.. _arrays.indexing.fields:

Field access
-------------

.. seealso:: :ref:`structured_arrays`

If the :class:`ndarray` object is a structured array the :term:`fields <field>`
of the array can be accessed by indexing the array with strings,
dictionary-like.

Indexing ``x['field-name']`` returns a new :term:`view` to the array,
which is of the same shape as *x* (except when the field is a
sub-array) but of data type ``x.dtype['field-name']`` and contains
only the part of the data in the specified field. Also,
:ref:`record array <arrays.classes.rec>` scalars can be "indexed" this way.

Indexing into a structured array can also be done with a list of field names,
e.g. ``x[['field-name1', 'field-name2']]``. As of NumPy 1.16, this returns a
view containing only those fields. In older versions of NumPy, it returned a
copy. See the user guide section on :ref:`structured_arrays` for more
information on multifield indexing.

If the accessed field is a sub-array, the dimensions of the sub-array
are appended to the shape of the result.
For example::

   >>> x = np.zeros((2, 2), dtype=[('a', np.int32), ('b', np.float64, (3, 3))])
   >>> x['a'].shape
   (2, 2)
   >>> x['a'].dtype
   dtype('int32')
   >>> x['b'].shape
   (2, 2, 3, 3)
   >>> x['b'].dtype
   dtype('float64')

.. _flat-iterator-indexing:

Flat Iterator indexing
----------------------

:attr:`x.flat <ndarray.flat>` returns an iterator that will iterate
over the entire array (in C-contiguous style with the last index
varying the fastest). This iterator object can also be indexed using
basic slicing or advanced indexing as long as the selection object is
not a tuple. This should be clear from the fact that :attr:`x.flat
<ndarray.flat>` is a 1-dimensional view. It can be used for integer
indexing with 1-dimensional C-style-flat indices. The shape of any
returned array is therefore the shape of the integer indexing object.

.. index::
   single: indexing
   single: ndarray


.. _assigning-values-to-indexed-arrays:

Assigning values to indexed arrays
----------------------------------

As mentioned, one can select a subset of an array to assign to using
a single index, slices, and index and mask arrays. The value being
assigned to the indexed array must be shape consistent (the same shape
or broadcastable to the shape the index produces). For example, it is
permitted to assign a constant to a slice: ::

 >>> x = np.arange(10)
 >>> x[2:7] = 1

or an array of the right size: ::

 >>> x[2:7] = np.arange(5)

Note that assignments may result in changes if assigning
higher types to lower types (like floats to ints) or even
exceptions (assigning complex to floats or ints): ::

 >>> x[1] = 1.2
 >>> x[1]
 1
 >>> x[1] = 1.2j
 TypeError: can't convert complex to int


Unlike some of the references (such as array and mask indices)
assignments are always made to the original data in the array
(indeed, nothing else would make sense!). Note though, that some
actions may not work as one may naively expect. This particular
example is often surprising to people: ::

 >>> x = np.arange(0, 50, 10)
 >>> x
 array([ 0, 10, 20, 30, 40])
 >>> x[np.array([1, 1, 3, 1])] += 1
 >>> x
 array([ 0, 11, 20, 31, 40])

Where people expect that the 1st location will be incremented by 3.
In fact, it will only be incremented by 1. The reason is that
a new array is extracted from the original (as a temporary) containing
the values at 1, 1, 3, 1, then the value 1 is added to the temporary,
and then the temporary is assigned back to the original array. Thus
the value of the array at ``x[1] + 1`` is assigned to ``x[1]`` three times,
rather than being incremented 3 times.

.. _dealing-with-variable-indices:

Dealing with variable numbers of indices within programs
--------------------------------------------------------

The indexing syntax is very powerful but limiting when dealing with
a variable number of indices. For example, if you want to write
a function that can handle arguments with various numbers of
dimensions without having to write special case code for each
number of possible dimensions, how can that be done? If one
supplies to the index a tuple, the tuple will be interpreted
as a list of indices. For example::

 >>> z = np.arange(81).reshape(3, 3, 3, 3)
 >>> indices = (1, 1, 1, 1)
 >>> z[indices]
 40

So one can use code to construct tuples of any number of indices
and then use these within an index.

Slices can be specified within programs by using the slice() function
in Python. For example: ::

 >>> indices = (1, 1, 1, slice(0, 2))  # same as [1, 1, 1, 0:2]
 >>> z[indices]
 array([39, 40])

Likewise, ellipsis can be specified by code by using the Ellipsis
object: ::

 >>> indices = (1, Ellipsis, 1)  # same as [1, ..., 1]
 >>> z[indices]
 array([[28, 31, 34],
        [37, 40, 43],
        [46, 49, 52]])

For this reason, it is possible to use the output from the 
:meth:`np.nonzero() <ndarray.nonzero>` function directly as an index since
it always returns a tuple of index arrays.

Because the special treatment of tuples, they are not automatically
converted to an array as a list would be. As an example: ::

 >>> z[[1, 1, 1, 1]]  # produces a large array
 array([[[[27, 28, 29],
          [30, 31, 32], ...
 >>> z[(1, 1, 1, 1)]  # returns a single value
 40


Detailed notes
--------------

These are some detailed notes, which are not of importance for day to day
indexing (in no particular order):

* The native NumPy indexing type is ``intp`` and may differ from the
  default integer array type. ``intp`` is the smallest data type
  sufficient to safely index any array; for advanced indexing it may be
  faster than other types.
* For advanced assignments, there is in general no guarantee for the
  iteration order. This means that if an element is set more than once,
  it is not possible to predict the final result.
* An empty (tuple) index is a full scalar index into a zero-dimensional array.
  ``x[()]`` returns a *scalar* if ``x`` is zero-dimensional and a view
  otherwise. On the other hand, ``x[...]`` always returns a view.
* If a zero-dimensional array is present in the index *and* it is a full
  integer index the result will be a *scalar* and not a zero-dimensional array.
  (Advanced indexing is not triggered.)
* When an ellipsis (``...``) is present but has no size (i.e. replaces zero
  ``:``) the result will still always be an array. A view if no advanced index
  is present, otherwise a copy.
* The ``nonzero`` equivalence for Boolean arrays does not hold for zero
  dimensional boolean arrays.
* When the result of an advanced indexing operation has no elements but an
  individual index is out of bounds, whether or not an ``IndexError`` is
  raised is undefined (e.g. ``x[[], [123]]`` with ``123`` being out of bounds).
* When a *casting* error occurs during assignment (for example updating a
  numerical array using a sequence of strings), the array being assigned
  to may end up in an unpredictable partially updated state.
  However, if any other error (such as an out of bounds index) occurs, the
  array will remain unchanged.
* The memory layout of an advanced indexing result is optimized for each
  indexing operation and no particular memory order can be assumed.
* When using a subclass (especially one which manipulates its shape), the
  default ``ndarray.__setitem__`` behaviour will call ``__getitem__`` for
  *basic* indexing but not for *advanced* indexing. For such a subclass it may
  be preferable to call ``ndarray.__setitem__`` with a *base class* ndarray
  view on the data. This *must* be done if the subclasses ``__getitem__`` does
  not return views.
