# 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 to  n-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 index  i  via  i = n+j  .

• Colons  :  must be used for slices :  start:stop:step  , where  start  is inclusive and  stop  is exclusive.

### 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 equal  1  .

• If  k  is greater than  0  and  i  is not provided (e.g.,  :10:2  ),  i  must equal  0  .

• If  k  is greater than  0  and  j  is not provided (e.g.,  0::2  ),  j  must equal  n  .

• If  k  is less than  0  and  i  is not provided (e.g.,  :10:-2  ),  i  must equal  n-1  .

• If  k  is less than  0  and  j  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  equals  j  , a slice must return an empty array, whose axis (dimension) size along the indexed axis is  0  .

• 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  and  0:10  are equivalent on a list of length  10  .

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  or  j  omitted (  None  ).

•  -n <= i <= n  .

• For  k > 0  or  k  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,  x[(exp1, exp2, ..., expN)]  is equivalent to  x[exp1, exp2, ..., expN]  ; the latter is syntactic sugar for the former.

• Providing a single nonnegative integer  i  as a single-axis index must index the same elements as the slice  i:i+1  .

• Providing a single negative integer  i  as a single-axis index must index the same elements as the slice  n+i:n+i+1  , where  n  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 should decrease by one; if  A  has rank  2  ,  rank(A)-1 == rank(A[0, :])  ). In particular, a selection tuple with the  m  th element an integer (and all other entries  :  ) indexes a sub-array with rank  N-1  .

• 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., if  A  has rank  4  ,  A[1:, ..., 2:5] == A[1:, :, :, 2:5]  ). Only a single ellipsis must be allowed. An  IndexError  exception must be raised if more than one ellipsis is provided.

• Except in the case of providing a single ellipsis (e.g.,  A[2:10, ...]  or  A[1:, ..., 2:5]  ), the number of provided single-axis indexing expressions should equal  N  . For example, if  A  has rank  2  , a single-axis indexing expression should be explicitly provided for both axes (e.g.,  A[2:10, :]  ). An  IndexError  exception should be raised if the number of provided single-axis indexing expressions is less than  N  .

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 is greater than  N  .

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)  .

• If  N >= M  , then  A[B]  must replace the first  M  dimensions of  A  with a single dimension having a size equal to the number of  True  elements in  B  . The values in the resulting array must be in row-major (C-style order); this is equivalent to  A[nonzero(B)]  .

Note

For example, if  N == M == 2  , indexing  A  via a boolean array  B  will return a one-dimensional array whose size is equal to the number of  True  elements in  B  .

• If  N < M  , then an  IndexError  exception must be raised.

• The size of each dimension in  B  must equal the size of the corresponding dimension in  A  or be  0  , beginning with the first dimension in  A  . If a dimension size does not equal the size of the corresponding dimension in  A  and is not  0  , then an  IndexError  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  or  False  ) 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 length  1  if the index array’s value is  True  and of length  0  if the index array’s value is  False  . Accordingly, for a zero-dimensional boolean index array  B  , the result of  A[B]  has shape  S = (1, s1, ..., sN)  if the index array’s value is  True  and has shape  S = (0, s1, ..., sN)  if the index array’s value is  False  .

## 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).