Fast.ai APL study session 12

Array programming
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Discussion of the 12th study group can go here. Overview

[ <<< session 11 | session 13 >>>]

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While you’ve not really covered trains yet (though you’ve seen the mean←+⌿÷≢ example), sign←÷∘|⍨ can be written as the train (fork) sign←⊢÷| as follows.

Without going into all the details, think of the traditional mathematical notation (TMN) where (f+g)(x)=f(x)+g(x). Forks (3-trains) are just a generalisation of this pattern to all functions (though the middle one has to be dyadic). For n÷|n we have the + become a ÷, while g is |, but what is f‽ We just want n unmodified. This calls for an identity function à la f(x)=x, which is APL’s ⊢. So…
TMN: f(x)=x, g(x)=|x|, sgn(x)=(f÷g)(x)
APL: f←⊢ ⋄ g←| ⋄ sgn←f÷g or simply sgn←⊢÷|

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So
1 2 3 +⊂ 4 5 6
┌─────┬─────┬─────┐
│5 6 7│6 7 8│7 8 9│
└─────┴─────┴─────┘
Is an APL loop equivalent?

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Dyadic + is a scalar function meaning that it will pervasively apply to corresponding elements. Since there are 3 elements on the left, but only 1 on the right (⊂ always gives a scalar result) it will “broadcast” that single element to all of them, so 1 2 3 + ⊂4 5 6 is equivalent to 1 2 3 + (4 5 6)(4 5 6)(4 5 6). Now it pairs up 1 with its 4 5 6 and 2 with its 4 5 6 and 3 with its 4 5 6, giving (1+4 5 6)(2+4 5 6)(3+4 5 6). Call it a loop if you want :slight_smile:.

P.S. Put lines with triple-backticks before and after your code block to make it format in monospace font and respect leading spaces.

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Here is an explaination of the DOMAIN ERROR encountered in the session:

    (÷|)⍨ 3j4      ⍝ Duplicate input f⍨Y ⇔ Y f⍨ Y
3j4 (÷|)  3j4      ⍝ "Two Train" ⍺ (g h) ⍵ ⇔ g (⍺ h ⍵)
     ÷   (3j4|3j4) ⍝ Self modulus is always zero
     ÷   (0)       ⍝ DOMAIN ERROR: Divide by zero

I have been practicing my “Trainspotting” :slight_smile:

The APL Wiki on Tacit Programming is a great reference.

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