I think this is the right automated function?
eval_z_ =: 1 : ' a: 1 : m'
('"_' ,~ (":@[ , ' + ] * ' , ":@])/ x: 1 % !i.10x) eval
1 + ] * 1 + ] * 1r2 + ] * 1r6 + ] * 1r24 + ] * 1r120 + ] * 1r720 + ] * 1r5040 +
] * 1r40320 + ] * 1r362880"_
x: inv ('"_' ,~ (":@[ , ' + [ * ' , ":@])/ x: 1 % !i.10x) eval 2 3 4
7.38871 20.0634 54.1541
appologies if it is supposed to be dyadic.
----- Original Message -----
From: Skip Cave <[email protected]>
To: "[email protected]" <[email protected]>
Cc:
Sent: Tuesday, August 5, 2014 11:59:59 AM
Subject: Re: [Jprogramming] Power for the powerless
I think Raul missed a term:
exp=: 1 + (* 1+ (* 0.5 + (*1r6+(*1r24+(*1r120+(*1r720+(*1r5040+
(*1r40320+(*1r362880"_)))))))))
In any case, the list of constants is easy to generate. This generates a
hundred numbers in the series, in rational fractions::
c =. x: 1 % !i.100x
c
1 1 1r2 1r6 1r24 1r120 1r720 1r5040 1r40320 1r362880 1r3628800 1r39916800
1r479001600 1r6227020800 1r87178291200 1r1307674368000 1r20922789888000
1r355687428096000 1r6402373705728000 1r121645100408832000
1r2432902008176640000 1r51090942171709440000 1r11240...
Unfortunately my J skills aren't good enough to insert all 100 of these
values in to Raul's equation.
Skip
Skip Cave
Cave Consulting LLC
On Tue, Aug 5, 2014 at 10:16 AM, Raul Miller <[email protected]> wrote:
> This is getting silly, but:
>
> ^T.10
>
> 1 1 0.5 0.16666666666666666 0.041666666666666664 0.0083333333333333332
> 0.0013888888888888889 0.00019841269841269841 2.4801587301587302e_5
> 2.7557319223985893e_6&p.
>
>
> exp=: 1 + (* 1+ (* 0.5 +
> (*1r6+(*1r24+(*1r720+(*1r5040+(*1r40320+(*1r362880"_))))))))
>
>
> That's not accurate enough, except for exponents near zero, but it's enough
> to illustrate the principle.
>
>
> Thanks,
>
>
> --
>
> Raul
>
>
>
>
> On Tue, Aug 5, 2014 at 9:49 AM, Dan Bron <[email protected]> wrote:
>
> > 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
> >
> >
> > ----------------------------------------------------------------------
> > For information about J forums see http://www.jsoftware.com/forums.htm
> >
> ----------------------------------------------------------------------
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>
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