[Jprogramming] Bell numbers

[Roger Hui]

See http://www.jsoftware.com/pipermail/programming/2007-December/009055.html
Basically, for extended precision arguments,

   x=: 1000 ?@$ 100x
   ts=: 6!:2 , 7!:2@]
   ts '+/\ x'
0.599717 154240
   ts '+/\.x'
0.000962413 217536

Fixed for the next J6.02 beta.



----- Original Message -----
From: "R.E. Boss" <r.e.boss at planet.nl>
Date: Wednesday, December 19, 2007 10:46
Subject: RE: [Jprogramming] Bell numbers
To: 'Programming forum' <programming at jsoftware.com>

> In my exploration of the Bell numbers I hit on something strange.
> 
>    ptr=: [:+/\ {:, ]
>    PTR=: [:+/\. ], {.
> 
> are two verbs which produce the Pierce numbers
> http://www.research.att.com/~njas/sequences/A011971
> 
>    ptr^:(i.`(x:@1:))5
>  1  0  0  0  0
>  1  2  0  0  0
>  2  3  5  0  0
>  5  7 10 15  0
> 15 20 27 37 52
> 
>    PTR^:(i.`(x:@1:))5
>  1  0  0  0  0
>  2  1  0  0  0
>  5  3  2  0  0
> 15 10  7  5  0
> 52 37 27 20 15
> 
> Notice the row wise relation between the two:
>    (ptr^:(i.`(x:@1:)) (-:|.)&{: PTR^:(i.`(x:@1:)))120
> 1
> 
> But look at the amazing difference in performance:
> 
>    5 ts'ptr^:(i.`(x:@1:))120'
> 0.40665763 1604992
>    5 ts'PTR^:(i.`(x:@1:))120'
> 0.014059384 1604992
> 
> Same amount of space, but almost 30 times as fast!!
> 
> Any explanation ???
> 
> 
> BTW, this leads to a (1.6 times) faster verb for Bell numbers:
> 
>    5 ts'Bn2 120'
> 0.014777791 1594880
> 
>    (Bn1-:1,Bn2) 120
> 1
> 
> although 10 times more space is used
> 
> 
> R.E. Boss