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Rank Operator: An Idea Worth Stealing Borrowing
Roger Hui
September, 2009


Abstract
1. Examples
2. Model
3. Applications
4. I.R.S.
5. History
References


Abstract

2. Examples

      x←2 3 4⍴⍳24
      (⍉rank 2) x
 0  4  8
 1  5  9
 2  6 10
 3  7 11
        
12 16 20
13 17 21
14 18 22
15 19 23

      ⍉[1]x
SYNTAX ERROR
      ⍉[1]x
     ∧

      (+/∘, rank 1) x
 6 22 38
54 70 86
      (+/∘, rank 2) x
66 210

      ⎕←a←↑ 'the' 'time' 'has' 'come' 'for' 'rank'
the 
time
has 
come
for 
rank
      ⍋ a
3 4 2 5 0 1
      (⍋ rank 1) a
3 2 1 0
3 1 2 0
3 1 0 2
0 3 2 1
3 0 1 2
1 3 2 0
      ({⍵[⍋⍵]} rank 1) a
 eht
eimt
 ahs
cemo
 for
aknr

2. Model

effrank ←{(0≤⍺)⊃(0⌈⍺+⍴⍴⍵)(⍺⌊⍴⍴⍵)}
cells   ←{⊂[(-⍺ effrank ⍵)↑⍳⍴⍴⍵] ⍵}
assemble←{↑ ((1⍴¨⍨r-⍨⌈/,r←⍴∘⍴¨⍵),¨⍴¨⍵)⍴¨⍵}

rank←{                               
     (m l r)←⌽3⍴⌽⍵⍵                   
     0=⎕NC'⍺' : assemble ⍺⍺¨ (m cells ⍵) 
     assemble (l cells ⍺) ⍺⍺¨ (r cells ⍵)
}

3. Applications





outer product

inner product

⍪⍵ ←→ , rank ¯1 ⍵  for non-scalar ⍵
commabar←{,rank ¯1⍣(2-×⍬⍴⍴⍴⍵){⍵}⍵}

 





4. I.R.S.



Integrated Rank Support (IRS) was 




5. History



Iverson introduced function rank  
in section 6 of his seminal 
Operators and Functions research report 
[2] 
on 1978-04-26.
The rank operator was invented by Arthur Whitney 
[3, 
4]
in July 1982 while on a train to the APL82 conference in Heidelberg.
It was first published in 
Rationalized APL 
[5] 
on 1983-01-06, and the first implementation 
was released in SHARP APL on 1983-05-02
[6].
It is described in Sections 9.3.3, .4, and .5 of
the ISO Standard on Extended APL
[7].

 






References





[0] 
 Hui, Roger K.W., 
 Rank and Uniformity, APL95,
 Volume 25, Number 4, 1995-06-04.


[1] 
 Bernecky, Robert, and Greg Bezoff, I.P. Sharp Associates, 1986.


[2] 
 Iverson, Kenneth E., 
 Operators and Functions, 
 Research Report #RC7091, IBM Corporation, 1978-04-26.



[3] 
 Whitney, Arthur, private e-mail, 2004-11-11.


[4] 
 Pesch, Roland, private e-mail, 2004-11-11.


[5] 
 Iverson, Kenneth E., 
 Rationalized APL, 
 I.P. Sharp Associates, 1983-01-06; revision 1, 1983-04-04.


[6] 
 Bernecky, Robert, Kenneth E. Iverson, Eugene McDonnell, Robert Metzger, 
 and Henri Schueler, 
 Language Extensions of May 1983, 
 SATN 45, 
 1983-05-02.


[7] 
 ISO/IEC 13571:2001 (E), Programming Language APL, Extended,
 2001-02-01.













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original writing: 2009-07-03 11:20; last updated: 2009-09-10 14:20