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The [[bound]] of an array is the number of [[ | The [[bound]] of an array is the number of [[element]]s it contains, or equivalently the length ([[Tally]]) of its [[ravel]] vector or the product (<source lang=apl inline>×/</source>) of its [[shape]]. | ||
The bound can be derived axiomatically as a function of the [[shape]] from two rules: | The bound can be derived axiomatically as a function of the [[shape]] from two rules: | ||
* The bound of a [[vector]] is its length, and | * The bound of a [[vector]] is its length, and | ||
* If two shapes are [[ | * If two shapes are [[catenate]]d (for instance by an [[outer product]]), the resulting bound is the product of their individual bounds. | ||
A scalar has [[rank]] 0, or empty shape. From the above axioms we can deduce that it has bound 1: catenating <source lang=apl inline>⍬</source> with some other shape leaves that shape unchanged, so multiplying any bound by a scalar's bound cannot change it either. Therefore a scalar's bound must be the multiplicative identity, 1. | A scalar has [[rank]] 0, or empty shape. From the above axioms we can deduce that it has bound 1: catenating <source lang=apl inline>⍬</source> with some other shape leaves that shape unchanged, so multiplying any bound by a scalar's bound cannot change it either. Therefore a scalar's bound must be the multiplicative identity, 1. | ||
Combining [[Axis|axes]] of an array, for instance by using [[Table]] or [[Ravel]] with [[Function axis|axis]], leaves the array's bound unchanged, despite changing its shape, rank, and possibly [[Tally]]. | Combining [[Axis|axes]] of an array, for instance by using [[Table]] or [[Ravel]] with [[Function axis|axis]], leaves the array's bound unchanged, despite changing its shape, rank, and possibly [[Tally]]. | ||
{{APL programming language}} | |||
Revision as of 14:08, 14 November 2019
The bound of an array is the number of elements it contains, or equivalently the length (Tally) of its ravel vector or the product (×/) of its shape.
The bound can be derived axiomatically as a function of the shape from two rules:
- The bound of a vector is its length, and
- If two shapes are catenated (for instance by an outer product), the resulting bound is the product of their individual bounds.
A scalar has rank 0, or empty shape. From the above axioms we can deduce that it has bound 1: catenating ⍬ with some other shape leaves that shape unchanged, so multiplying any bound by a scalar's bound cannot change it either. Therefore a scalar's bound must be the multiplicative identity, 1.
Combining axes of an array, for instance by using Table or Ravel with axis, leaves the array's bound unchanged, despite changing its shape, rank, and possibly Tally. Template:APL programming language