Indexing¶
Array API specification for indexing arrays.
A conforming implementation of the array API standard must adhere to the following conventions.
Single-axis Indexing¶
To index a single array axis, an array must support standard Python indexing rules. Let n
be the axis (dimension) size.
An integer index must be an object satisfying operator.index (e.g.,
int
).Nonnegative indices must start at
0
(i.e., zero-based indexing).Valid nonnegative indices must reside on the half-open interval
[0, n)
.Note
This specification does not require bounds checking. The behavior for out-of-bounds integer indices is left unspecified.
Negative indices must count backward from the last array index, starting from
-1
(i.e., negative-one-based indexing, where-1
refers to the last array index).Note
A negative index
j
is equivalent ton-j
; the former is syntactic sugar for the latter, providing a shorthand for indexing elements that would otherwise need to be specified in terms of the axis (dimension) size.Valid negative indices must reside on the closed interval
[-n, -1]
.Note
This specification does not require bounds checking. The behavior for out-of-bounds integer indices is left unspecified.
A negative index
j
is related to a zero-based nonnegative indexi
viai = n+j
.Colons
:
must be used for slices:start:stop:step
, wherestart
is inclusive andstop
is exclusive.Note
The specification does not support returning scalar (i.e., non-array) values from operations, including indexing. In contrast to standard Python indexing rules, for any index, or combination of indices, which select a single value, the result must be a zero-dimensional array containing the selected value.
Slice Syntax¶
The basic slice syntax is i:j:k
where i
is the starting index, j
is the stopping index, and k
is the step (k != 0
). A slice may contain either one or two colons, with either an integer value or nothing on either side of each colon. The following are valid slices.
A[:]
A[i:]
A[:j]
A[i:k]
A[::]
A[i::]
A[:j:]
A[::k]
A[i:j:]
A[i::k]
A[:j:k]
A[i::k]
A[i:j:k]
Note
Slice syntax can be equivalently achieved using the Python built-in slice() API. From the perspective of A
, the behavior of A[i:j:k]
and A[slice(i, j, k)]
is indistinguishable (i.e., both retrieve the same set of items from __getitem__
).
Using a slice to index a single array axis must select m
elements with index values
i, i+k, i+2k, i+3k, ..., i+(m-1)k
where
m = q + r
and q
and r
(r != 0
) are the quotient and remainder obtained by dividing j-i
by k
j - i = qk + r
such that
j > i + (m-1)k
Note
For i
on the interval [0, n)
(where n
is the axis size), j
on the interval (0, n]
, i
less than j
, and positive step k
, a starting index i
is always included, while the stopping index j
is always excluded. This preserves x[:i]+x[i:]
always being equal to x
.
Note
Using a slice to index into a single array axis should select the same elements as using a slice to index a Python list of the same size.
Slice syntax must have the following defaults. Let n
be the axis (dimension) size.
If
k
is not provided (e.g.,0:10
),k
must equal1
.If
k
is greater than0
andi
is not provided (e.g.,:10:2
),i
must equal0
.If
k
is greater than0
andj
is not provided (e.g.,0::2
),j
must equaln
.If
k
is less than0
andi
is not provided (e.g.,:10:-2
),i
must equaln-1
.If
k
is less than0
andj
is not provided (e.g.,0::-2
),j
must equal-n-1
.
Using a slice to index a single array axis must adhere to the following rules. Let n
be the axis (dimension) size.
If
i
equalsj
, a slice must return an empty array, whose axis (dimension) size along the indexed axis is0
.Indexing via
:
and::
must be equivalent and have defaults derived from the rules above. Both:
and::
indicate to select all elements along a single axis (dimension).Note
This specification does not require “clipping” out-of-bounds slice indices. This is in contrast to Python slice semantics where
0:100
and0:10
are equivalent on a list of length10
.
The following ranges for the start and stop values of a slice must be supported. Let n
be the axis (dimension) size being sliced. For a slice i:j:k
, the behavior specified above should be implemented for the following:
i
orj
omitted (None
).-n <= i <= n
.For
k > 0
ork
omitted (None
),-n <= j <= n
.For
k < 0
,-n - 1 <= j <= max(0, n - 1)
.
The behavior outside of these bounds is unspecified.
Note
Rationale: this is consistent with bounds checking for integer indexing; the behavior of out-of-bounds indices is left unspecified. Implementations may choose to clip (consistent with Python list
slicing semantics), raise an exception, return junk values, or some other behavior depending on device requirements and performance considerations.
Multi-axis Indexing¶
Multi-dimensional arrays must extend the concept of single-axis indexing to multiple axes by applying single-axis indexing rules along each axis (dimension) and supporting the following additional rules. Let N
be the number of dimensions (“rank”) of a multi-dimensional array A
.
Each axis may be independently indexed via single-axis indexing by providing a comma-separated sequence (“selection tuple”) of single-axis indexing expressions (e.g.,
A[:, 2:10, :, 5]
).Note
In Python,
A[(exp1, exp2, ..., expN)]
is equivalent toA[exp1, exp2, ..., expN]
; the latter is syntactic sugar for the former.Accordingly, if
A
has rank1
, thenA[(2:10,)]
must be equivalent toA[2:10]
. IfA
has rank2
, thenA[(2:10, :)]
must be equivalent toA[2:10, :]
. And so on and so forth.Providing a single nonnegative integer
i
as a single-axis index must index the same elements as the slicei:i+1
.Providing a single negative integer
i
as a single-axis index must index the same elements as the slicen+i:n+i+1
, wheren
is the axis (dimension) size.Providing a single integer as a single-axis index must reduce the number of array dimensions by
1
(i.e., the array rank must decrease by one; ifA
has rank2
,rank(A)-1 == rank(A[0, :])
). In particular, a selection tuple with them
th element an integer (and all other entries:
) indexes a sub-array with rankN-1
.Note
When providing a single integer as a single-axis index to an array of rank
1
, the result should be an array of rank0
, not a NumPy scalar. Note that this behavior differs from NumPy.Providing a slice must retain array dimensions (i.e., the array rank must remain the same;
rank(A) == rank(A[:])
).Providing ellipsis must apply
:
to each dimension necessary to index all dimensions (e.g., ifA
has rank4
,A[1:, ..., 2:5] == A[1:, :, :, 2:5]
). Only a single ellipsis must be allowed. AnIndexError
exception must be raised if more than one ellipsis is provided.Providing an empty tuple or an ellipsis to an array of rank
0
must result in an array of the same rank (i.e., ifA
has rank0
,A == A[()]
andA == A[...]
).Note
This behavior differs from NumPy where providing an empty tuple to an array of rank
0
returns a NumPy scalar.Each
None
in the selection tuple must expand the dimensions of the resulting selection by one dimension of size1
. The position of the added dimension must be the same as the position ofNone
in the selection tuple.Note
Expanding dimensions can be equivalently achieved via repeated invocation of
expand_dims()
.Note
The constant
newaxis
is an alias ofNone
and can thus be used in a similar manner asNone
.Except in the case of providing a single ellipsis (e.g.,
A[2:10, ...]
orA[1:, ..., 2:5]
), the number of provided single-axis indexing expressions (excludingNone
) should equalN
. For example, ifA
has rank2
, a single-axis indexing expression should be explicitly provided for both axes (e.g.,A[2:10, :]
). AnIndexError
exception should be raised if the number of provided single-axis indexing expressions (excludingNone
) is less thanN
.Note
Some libraries, such as SymPy, support flat indexing (i.e., providing a single-axis indexing expression to a higher-dimensional array). That practice is not supported here.
To perform flat indexing, use
reshape(x, (-1,))[integer]
.An
IndexError
exception must be raised if the number of provided single-axis indexing expressions (excludingNone
) is greater thanN
.Note
This specification leaves unspecified the behavior of providing a slice which attempts to select elements along a particular axis, but whose starting index is out-of-bounds.
Rationale: this is consistent with bounds-checking for single-axis indexing. An implementation may choose to set the axis (dimension) size of the result array to
0
, raise an exception, return junk values, or some other behavior depending on device requirements and performance considerations.
Boolean Array Indexing¶
Data-dependent output shape
For common boolean array use cases (e.g., using a dynamically-sized boolean array mask to filter the values of another array), the shape of the output array is data-dependent; hence, array libraries which build computation graphs (e.g., JAX, Dask, etc.) may find boolean array indexing difficult to implement. Accordingly, such libraries may choose to omit boolean array indexing. See Data-dependent output shapes section for more details.
An array must support indexing where the sole index is an M
-dimensional boolean array B
with shape S1 = (s1, ..., sM)
according to the following rules. Let A
be an N
-dimensional array with shape S2 = (s1, ..., sM, ..., sN)
.
Note
The prohibition against combining boolean array indices with other single-axis indexing expressions includes the use of
None
. To expand dimensions of the returned array, use repeated invocation ofexpand_dims()
.
If
N >= M
, thenA[B]
must replace the firstM
dimensions ofA
with a single dimension having a size equal to the number ofTrue
elements inB
. The values in the resulting array must be in row-major (C-style order); this is equivalent toA[nonzero(B)]
.Note
For example, if
N == M == 2
, indexingA
via a boolean arrayB
will return a one-dimensional array whose size is equal to the number ofTrue
elements inB
.If
N < M
, then anIndexError
exception must be raised.The size of each dimension in
B
must equal the size of the corresponding dimension inA
or be0
, beginning with the first dimension inA
. If a dimension size does not equal the size of the corresponding dimension inA
and is not0
, then anIndexError
exception must be raised.The elements of a boolean index array must be iterated in row-major, C-style order, with the exception of zero-dimensional boolean arrays.
A zero-dimensional boolean index array (equivalent to
True
orFalse
) must follow the same axis replacement rules stated above. Namely, a zero-dimensional boolean index array removes zero dimensions and adds a single dimension of length1
if the index array’s value isTrue
and of length0
if the index array’s value isFalse
. Accordingly, for a zero-dimensional boolean index arrayB
, the result ofA[B]
has shapeS = (1, s1, ..., sN)
if the index array’s value isTrue
and has shapeS = (0, s1, ..., sN)
if the index array’s value isFalse
.
Return Values¶
The result of an indexing operation (e.g., multi-axis indexing, boolean array indexing, etc) must be an array of the same data type as the indexed array.
Note
The specified return value behavior includes indexing operations which return a single value (e.g., accessing a single element within a one-dimensional array).