Raul wrote:
> A really simple approach would be to use T.
> pow=: ^ T. 99
First: probably a better name for your function is "exp" instead of "pow"
("exp" is a monad related to the dyad "pow" by fixing its left argument to
1x1).
But certainly exp is a welcome improvement over my formulation, with a
couple caveats. Specifically, even if we write off the presence of the
primitive ^ in ^T.99 as permissible due to the use-mention distinction,
the result of T., as you point out:
> gives you a polynomial expression
i.e., is an expression involving p., which is not one of the specifically
enumerated operations. More problematic, it has an implicit use of ^ (the
DoJ explicitly defines x p. y as +/x*y^i.#x). This is why I explicitly
excluded T.'s cousin t. in the message you replied to:
> can you calculate the power series without using ^
> explicitly or implicitly (e.g. via t. or #: etc)?
With that said, you made me realize that the Taylor coefficients for ^y are
fairly straightforward, and I can calculate them without t. or T. or even
! :
exp =: 250 & ( # +/ . % &(_1 |.!.1 */\) >:@:i.@:[ )
[I have no idea how many terms I should include, but 250 seemed sufficient
for the spot checks I did.]
Which is a lot more precise than my previous *:
> exp =: 1e5 spow~ 1 + %&1e5
> spow =: */@:#~
Unfortunately, since most of the complexity of my original function is in
the estimatation of ^.y (that is, log), it mutes the impact of this
improvement to ^y (unless someone can show me how to generalize it to x^y
i.e., truly pow, not exp?).
However, Jon Hough's suggestion looks very promising:
> x^y = (1 +(x-1))^y
> e^pi = (1+(e-1))^pi = 1+ pi*e + pi*(pi - 1)*e*e/2! +...
> http://en.wikipedia.org/wiki/Binomial_series
Let me play with that a bit.
-Dan
* Speaking of spow, Pascal wrote:
> "A really simple approach would be"
> for integer powers,
> pow =: [: */ #~
Yes, spow is a core function of the more general pow; but note it's not for
all "integer powers", it's for non-negative integer powers. See also the
RosettaCode "exponentiation" task [1], which shows how to generalize this
to negative powers simply:
> exp =: *@:] %: */@:(#~|)
But unfortunately, this uses %: which is not in our toolbox either.
[1] RC: Exponentiation in J
http://rosettacode.org/wiki/Exponentiation_operator#J
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