Indices

This function describes a primitive whose result is a list of indices. See Index for the page on indices themselves.

Indices (⍸), or Where, is a monadic primitive function which returns the indices of all ones in a boolean array. More generally, Indices accepts an array of non-negative integers and copies each index the corresponding number of times.

Indices was introduced in J as I. and is present in K as &.

Examples

In all implementations, Indices gives the indices of ones in a boolean vector.

      ⍸ 0 0 1 0 0 0 1 0
3 7

Works in: Dyalog APL,dzaima/APL,NARS2000

In nested APLs it returns nested indices when passed a matrix or higher-dimensional array.

      ⍸ 3 3⍴0 0 1 0 0 0 1 0
┌───┬───┐
│1 3│3 1│
└───┴───┘
      ⍸1   ⍝ An index into a scalar is empty!
┌┐
││
└┘

Works in: Dyalog APL,dzaima/APL,NARS2000

If numbers higher than 1 are allowed, they indicate that the index of the number is repeated. Negative numbers are never allowed.

      ⍸ 3 0 2
1 1 1 3 3

Works in: dzaima/APL,NARS2000

Description and APL model

Indices replicates each index in the argument by the number of times it appears. It is identical to the APL function:

Where ← {(,⍵)⌿,⍳⍴⍵}

Works in: Dyalog APL,dzaima/APL,NARS2000,ngn/apl

The argument is restricted to be an array of non-negative integers, or, in Dyalog APL, booleans.

The only flat array language which implements Indices is J. Because J's iota does not return multidimensional indices, J defines Indices to have function rank 1 so that only vector indices are used.

Mathematical interpretation

Indices may be viewed as a way to convert between two ways of writing multisets of array indices. The argument uses a dense representation indicating for each index the number of times it appears, and the result uses a sparse representation which lists all the indices contained in the set.

Documentation

Dyalog

NARS2000

J Dictionary, NuVoc

Other resources

APL Cultivation