Function axis

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Function axis ([ax]) is a special syntax which modifies the behavior of a function, for example [2] to Rotate along the second axis. Axis specification was a feature of Iverson notation and was ubiquitous in early APLs; many newer APLs which adhere to leading axis theory reject the use of axis specification in favor of the Rank operator because it is a fully general operator while the behavior of functions with axis must be defined for each function separately.


Monadic functions

The following monads may allow an axis:

  • Mix accepts a list of axes to specify where the axes of argument elements will be placed in the result.
  • Ravel accepts a list of axes which are combined, or a single fractional number to add a length-1 axis.
  • Enclose accepts a list of axes. Each subarray along these axes is enclosed.
  • Split accepts a single axis, and encloses each vector along that axis.
  • Reverse reverses along the specified axis.

Dyadic functions

The following dyads may allow one:

  • Scalar dyadics accept a list of axes to override conformability rules: it specifies , for each axis in the lower-rank (or left, in case of a tie) argument, which axis in the other argument it is paired with.
  • Catenate combines along the selected axis, adding a new axis if a non-integer axis is given.
  • Rotate rotates the right argument along the selected axis.
  • Replicate and Expand work on the specified right argument axis.
  • Take and Drop modify the selected right argument axes.
  • Squad indexing takes axes to specify which axis of the right argument corresponds to each left argument element.
  • Partition and Partitioned Enclose have complicated and different behavior.


The following operators may admit axis specification:

  • Reduction removes the specified right argument axis.
  • Scan works on the specified right argument axis.

In SHARP APL, Replicate and Expand are also included in this category as they are operators and not functions.

Axis specification always modifies the derived function obtained from an operator, and not the operator itself. The exact syntax may vary: in most cases a set of brackets is parsed like a monadic operator and applies directly to the derived function; sometimes brackets can be applied directly to an operator, in which case the derived function produced by this operator is modified. In Dyalog APL, a slash with axis retains its function-operator overloading: it can be applied as an operator or as a dyadic function (Replicate or Expand).

External links



APL syntax [edit]
General Comparison with traditional mathematicsPrecedenceTacit programming
Array Numeric literalStringStrand notationObject literalArray notation
Function ArgumentFunction valenceDerived functionDerived operatorNiladic functionMonadic functionDyadic functionAmbivalent functionDefined function (traditional)DfnFunction train
Operator OperandOperator valenceTradopDopDerived operator
Assignment MultipleIndexedSelectiveModified
Other Function axisBranchQuad nameSystem commandUser commandKeywordDot notationFunction-operator overloading