Rank (operator)


 * This article is about the operator. See Rank for the number associated with every array. For numbers associated with a function specifying its argument rank, see function rank.

Rank is a primitive dyadic operator which applies its left operand function to cells of its arguments specified by its right operand array.

Rank specification
The right operand specifies the rank of subarrays to which the left operand function is applied as follows: For left argument  and right argument  ,

A non-negative right operand specifies the number of final axes to which the function applies. A negative right operand specifies complementary rank, i.e. the number of leading axes to be excluded. Negative rank can also be thought of as rank specification relative to the overall rank of the argument array.

Since a rank specification greater than the rank of the argument array means to apply the function to the whole array,,   or  , depending on the implementation, is "rank infinity" and always specifies the whole argument array.

Examples
Rotate rows in matrices of a 3D array:

Laminate scalars from arrays of differing ranks:

Flat outer product:

History
The rank operator was invented by Ken Iverson 1982, based on Arthur Whitney's leading axis theory. Whitney was the first to implement the operator for SHARP APL in 1983. It has been described as "a microcosm of APL history", its evolution a progression from scalar extension, which has been in APL since its inception, through leading axis theory to a construct which is a generalisation of scalar extension, inner (matrix) product, outer product, maplist in LISP, map in modern functional programming languages and the broadcast facility in NumPy.

Rank vs Axis
Due to its ability to apply functions to specified subarrays, rank is frequently contrasted with bracket-axis. It provides nearly all of the functionality of the anomalous axis operator without its draw-backs.

One of these draw-backs is that bracket-axis is specified ad hoc for each of the specific primitives on which it applies. Rank benefits from consistent behaviour when applied to any function, including user-defined functions. The ad hoc nature of bracket-axis definitions means that a generalised axis operator which works on any function, but behaves just as bracket-axis on those particular primitives, is impossible to formulate.

Here we show some bracket-axis constructs and their equivalent expressions which use rank but do not use bracket-axis.

Sum along axis
Rank k-cells are defined for k trailing axes, whereas axes are numbered from most major (first axis i.e. axis number 1) to least major (last axis). This leads to a simple and symmetrical relationship.

Enclose axes
Enclose-with-axis is equivalent to transposing desired axes to the end of the array's shape and enclosing subarrays of a rank matching the number of axes.

Merge axes
Ravel-with-axis allows data to be merged along specified consecutive axes. The requirement that axes be consecutive is so that the data can remain in its original relative order.

Merging trailing axes is trivial.

Merging leading axes is more involved, but can be expressed in one line.

The general treatment benefits from being expanded.

Documentation

 * Dyalog
 * NARS2000

Publications

 * Dyalog '09 presentation by Roger Hui: The Rank Operator
 * Dyalog Webinars by Richard Park: Introduction, with Dyadic Transpose, advanced use
 * APL Cultivations: basic use, advanced use