4,493
edits
Index: Difference between revisions
Jump to navigation
Jump to search
m
Text replacement - "</source>" to "</syntaxhighlight>"
Miraheze>Marshall |
m (Text replacement - "</source>" to "</syntaxhighlight>") |
||
| (8 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
:''This page is about the array theory concept. See [[Indexing]], [[Indices]], [[Index Generator]], [[Index of]], and [[Interval Index]] for operations named after indices.'' | :''This page is about the array theory concept. See [[Index (function)]], [[Indexing]], [[Indices]], [[Index Generator]], [[Index of]], and [[Interval Index]] for operations named after indices.'' | ||
In the APL [[array model]], an '''index''' is a [[vector]] of integers which indicates an [[element]] of an array. The term "index" may also refer to a [[scalar]] index ''along an [[axis]]'', or (particularly in [[leading axis theory]]) the index of a [[cell]]. Indices are subject to [[index origin]] in languages which have such a concept. | In the APL [[array model]], an '''index''' is a [[vector]] of integers which indicates an [[element]] of an array. The term "index" may also refer to a [[scalar]] index ''along an [[axis]]'', or (particularly in [[leading axis theory]]) the index of a [[cell]]. Indices are subject to [[index origin]] in languages which have such a concept. | ||
| Line 8: | Line 8: | ||
An index is a [[vector]] of integers each of which is greater than or equal to the [[index origin]]. For a particular array, the index of an element has length equal to that array's [[rank]], and it is less than the array's [[shape]] plus the index origin in every element. In APL we can write | An index is a [[vector]] of integers each of which is greater than or equal to the [[index origin]]. For a particular array, the index of an element has length equal to that array's [[rank]], and it is less than the array's [[shape]] plus the index origin in every element. In APL we can write | ||
< | <syntaxhighlight lang=apl> | ||
(⍴i) ≡ ⍴⍴A | (⍴i) ≡ ⍴⍴A | ||
∧/ (⎕IO≤i) ∧ (i<(⍴A)-⎕IO) | ∧/ (⎕IO≤i) ∧ (i<(⍴A)-⎕IO) | ||
</ | </syntaxhighlight> | ||
to describe the possible indices < | to describe the possible indices <syntaxhighlight lang=apl inline>i</syntaxhighlight> for an array <syntaxhighlight lang=apl inline>A</syntaxhighlight>. | ||
The elements of an array then correspond one-to-one with valid indices for that array (while two elements may have the same value as arrays, they are still considered to be different elements). Using [[Squad indexing]], the element at index < | The elements of an array then correspond one-to-one with valid indices for that array (while two elements may have the same value as arrays, they are still considered to be different elements). Using [[Squad indexing]], the element at index <syntaxhighlight lang=apl inline>i</syntaxhighlight> is the sole element of [[scalar]] <syntaxhighlight lang=apl inline>i⌷A</syntaxhighlight>. | ||
When indexing into a [[scalar]] array, there is only one valid index: the [[empty]] numeric vector < | When indexing into a [[scalar]] array, there is only one valid index: the [[empty]] numeric vector <syntaxhighlight lang=apl inline>⍴0</syntaxhighlight>, or [[Zilde]]. There are no valid indices for an [[empty]] array, because it has no elements. | ||
=== Index of an axis === | === Index of an axis === | ||
| Line 22: | Line 22: | ||
The indices and [[shape]] of an array both contain one element for each axis. Because these objects are [[vector]]s, each has an element ordering and indices of its own. These indices are called ''axis indices'', and in addition to their implicit use in ordering the shape and index vectors, they are also used directly to refer to axes, for example in [[dyadic Transpose]] and when specifying a [[function axis]]. | The indices and [[shape]] of an array both contain one element for each axis. Because these objects are [[vector]]s, each has an element ordering and indices of its own. These indices are called ''axis indices'', and in addition to their implicit use in ordering the shape and index vectors, they are also used directly to refer to axes, for example in [[dyadic Transpose]] and when specifying a [[function axis]]. | ||
In some cases, such as [[Laminate]], [[function axis]] may allow a non-integer value. Such a value indicates an axis that exists in the result but not in the argument. It must lie strictly between < | In some cases, such as [[Laminate]], [[function axis]] may allow a non-integer value. Such a value indicates an axis that exists in the result but not in the argument. It must lie strictly between <syntaxhighlight lang=apl inline>⎕IO-1</syntaxhighlight> and <syntaxhighlight lang=apl inline>⎕IO+r</syntaxhighlight> for an array of [[rank]] <syntaxhighlight lang=apl inline>r</syntaxhighlight>. For a specified axis <syntaxhighlight lang=apl inline>i</syntaxhighlight>, the axis will be inserted between axes <syntaxhighlight lang=apl inline>⌊i</syntaxhighlight> and <syntaxhighlight lang=apl inline>⌈i</syntaxhighlight>, or before every axis if <syntaxhighlight lang=apl inline>⌊i</syntaxhighlight> is not a valid axis and after every axis if <syntaxhighlight lang=apl inline>⌈i</syntaxhighlight> is not a valid axis (for an [[empty array]], both exceptions hold). | ||
=== Ravel index === | === Ravel index === | ||
The [[ravel]] of an array can be used to define another kind of index selecting one of its [[element]]s. The '''ravel index''' is a [[scalar]] (although it could equivalently be defined as a [[singleton]] [[vector]]) which corresponds to an ordinary index by the relation < | The [[ravel]] of an array can be used to define another kind of index selecting one of its [[element]]s. The '''ravel index''' is a [[scalar]] (although it could equivalently be defined as a [[singleton]] [[vector]]) which corresponds to an ordinary index by the relation <syntaxhighlight lang=apl inline>r⌷A</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>i⌷,A</syntaxhighlight> where <syntaxhighlight lang=apl inline>r</syntaxhighlight> is the ravel index and <syntaxhighlight lang=apl inline>i</syntaxhighlight> the ordinary index of an element in <syntaxhighlight lang=apl inline>A</syntaxhighlight>. Using [[Encode]] and [[Decode]] we may write <syntaxhighlight lang=apl inline>r</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>(⍴A)⊥i</syntaxhighlight> and <syntaxhighlight lang=apl inline>i</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>(⍴A)⊤r</syntaxhighlight> assuming index origin 0. | ||
The correspondence between indices and ravel indices is governed by [[ravel order]]. For more information, see [[Axis ordering]]. | The correspondence between indices and ravel indices is governed by [[ravel order]]. For more information, see [[Axis ordering]]. | ||
| Line 34: | Line 34: | ||
Each of the values in an index corresponds to one [[axis]] of the indexed array. When considered in isolation, one of these values (a [[scalar]] number) is called an index ''along'' the corresponding axis. Selecting from the array using this index produces an array whose [[rank]] is one smaller than the initial array (and it cannot be done to a [[scalar]], as there are no axes along which to index). This array is sometimes called a "slice" or [[hyperplane]] of the array. Selecting on the first axis gives a [[major cell]], one kind of hyperplane. | Each of the values in an index corresponds to one [[axis]] of the indexed array. When considered in isolation, one of these values (a [[scalar]] number) is called an index ''along'' the corresponding axis. Selecting from the array using this index produces an array whose [[rank]] is one smaller than the initial array (and it cannot be done to a [[scalar]], as there are no axes along which to index). This array is sometimes called a "slice" or [[hyperplane]] of the array. Selecting on the first axis gives a [[major cell]], one kind of hyperplane. | ||
The set of possible indices into an array as a whole is the [[wikipedia: | The set of possible indices into an array as a whole is the [[wikipedia:Cartesian product|Cartesian product]] of the possible indices into each axis. | ||
== Index of a cell == | == Index of a cell == | ||
| Line 40: | Line 40: | ||
In [[leading axis theory]] an array's [[shape]] may be split in two with leading [[Axis|axes]] forming the [[frame]] and trailing axes forming the shape of each [[cell]]. A [[vector]] of indices for the axes in the frame only selects a cell of the array. [[Squad indexing]] supports this kind of selection using a short [[simple]] left argument. Using a frame with the same length as that argument, the given indices correspond to leading axes of the right argument—the ones in the frame—and those axes are not present in the result. All the indices into a cell, corresponding to trailing axes, are implicit, and those axes are unchanged. | In [[leading axis theory]] an array's [[shape]] may be split in two with leading [[Axis|axes]] forming the [[frame]] and trailing axes forming the shape of each [[cell]]. A [[vector]] of indices for the axes in the frame only selects a cell of the array. [[Squad indexing]] supports this kind of selection using a short [[simple]] left argument. Using a frame with the same length as that argument, the given indices correspond to leading axes of the right argument—the ones in the frame—and those axes are not present in the result. All the indices into a cell, corresponding to trailing axes, are implicit, and those axes are unchanged. | ||
{{APL | == External links == | ||
* [https://www.sacrideo.us/apl-a-day-5-indexing-into-arrays/ Indexing into Arrays] (from [https://www.sacrideo.us/tag/apl-a-day/ APL a Day]) | |||
{{APL features}}[[Category:Arrays]] | |||