Generating the rational fraction series is pretty easy:
c=. 1, x: 1 % ". _1}.,(100 7$'(*/1+i.'),"1(": 100 1$1+i.100),"1(100
3$'x),')
c
1 1 1r2 1r6 1r24 1r120 1r720 1r5040 1r40320 1r362880 1r3628800 1r39916800
1r479001600 1r6227020800 1r87178291200 1r1307674368000 1r20922789888000
1r355687428096000 1r6402373705728000 1r121645100408832000
1r2432902008176640000 1r51090942171709440000 1r11240...
I just want to know how to combine those rational fractions into Raul's
equation?
Skip
Skip Cave
Cave Consulting LLC
On Tue, Aug 5, 2014 at 11:08 AM, Dan Bron <[email protected]> wrote:
> Are they good enough to eliminate the ! ? That's cheating :)
>
> -Dan
>
> ----- Original Message ---------------
>
> Subject: Re: [Jprogramming] Power for the powerless
> From: Skip Cave <[email protected]>
> Date: Tue, 5 Aug 2014 10:59:59 -0500
> To: "[email protected]" <[email protected]>
>
> 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
> > >
> > >
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