And, of course,
diff =: 2&(-~/\) NB. differences of a vector
(For the notes at the end)
Kip Murray wrote:
> In (long sentence here, watch for the period)
>
> ]data =: 3 ^~ i. 11
> 0 1 8 27 64 125 216 343 512 729 1000
>
> p =: 3 interp data
>
> p NB. conjunction interp embedded
> (3) (2 : '(y !~ i. >: m) +/ . * {. (>: m) difftb n') 0 1 8 27 64 125 216
> 343 512 729 1000
>
> the conjunction interp is supposed to look at the difference table
>
> 4 difftb data NB. successive differences of first column, see note [1]
> 0 1 6 6
> 1 7 12 0
> 8 19 0 0
> 27 0 0 0
>
> and, reading the first line, produce the equivalent of
>
> q =: 0 1 6 6 +/ . * (i.4) ! ] NB. A Newton series, see note [2]
> .
>
> Conjunction interp _does_ produce an equivalent:
>
> (p"0 ,: q"0) i. 11
> 0 1 8 27 64 125 216 343 512 729 1000
> 0 1 8 27 64 125 216 343 512 729 1000
>
> But I don't want p to recreate the difference table every time it is
> called. Is there a way to rewrite conjunction interp so that
>
> q =: 3 interp1 data
>
> q
> 0 1 6 6 +/ .* 0 1 2 3 ! ]
>
> ?
>
> Kip
>
> ----------
>
>
> [1] x difftb y is an x by x table whose kth column has
> kth differences for vector x {. y (y is a vector and x <: #y)
>
> difftb =: 4 : '|: diff^:(i.x) x {. y'
>
>
> [2] In algebra notation q is the third degree Newton series
>
> q(x) = 0 + 1 x/1 + 6 (x*(x-1))/(1*2) + 6 (x*(x-1)*(x-2))/(1*2*3)
>
> This one simplifies to q(x) = x^3 .
>
> For Newton series, see pages 190-191 of Graham, Knuth, and Patashnik,
> Concrete Mathematics Second Edition.
>
>
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