Based on the mathworld page, this looks like an implementation of
Bernoulli numbers in J:
bernoulli=: ! * (% ^ - 1:) t.
Example use:
bernoulli i. 9
1 _0.5 0.166667 _4.16334e_17 _0.0333333 _1.30104e_17 0.0238095
_1.76074e_16 _0.0333333
The "zero" values are not quite zero, but that's a limitation of floating point.
--
Raul
On Sun, May 20, 2012 at 6:07 AM, Viktor Cerovski
<viktor.cerov...@gmail.com> wrote:
> R.E. Boss wrote:
>> What is the relation between the integrals (
>> http://www.jsoftware.com/jwiki/Pi/AConvergentSeries ) and the Bernoulli
>> numbers? Or
>> where can I find it?
>
> The first integral has the form:
>
> integral cos(2x) * product_n cos(x/n) dx
>
> When you write the product of cos(x/n) as exp of sum of ln cos(x/n),
> series expansion of ln cos(x/n) has Bernoulli numbers
> in its coefficients.
>
> Obtained double-sum series then should be transformed
> a bit and appropriately truncated so that it has sufficient precision,
> and then numerical integration performed.
>
> The second expression (expansion of pi/8) is written in terms
> of integrals with different cosine terms and the same infinite
> product term, so truncation and numerical integration has to be done
> with even higher precision for m=1, etc.
>
> In short, there is quite a bit of work to get this through in J.
>
>
>
>>
>> R.E. Boss
>>
>>
>>> -----Oorspronkelijk bericht-----
>>> Van: programming-boun...@jsoftware.com
>>> [mailto:programming-boun...@jsoftware.com] Namens Viktor Cerovski
>>> Verzonden: dinsdag 15 mei 2012 18:54
>>> Aan: programming@jsoftware.com
>>> Onderwerp: Re: [Jprogramming] Challenge 12(?)
>>>
>>>
>>>
>>> R.E. Boss wrote:
>>> >
>>> > I scanned a part of Exploratory Experimentation and Computation,
>>> > https://www.opendrive.com/files?57384074_sCfMP , where two
>>> > statements are made about a very rapidly convergent series.
>>> >
>>> > 1. The first term coincides with (pi % 8) in the first 42 digits
>>> > 2. The first 2 terms even give 500 digits
>>> >
>>> > How can these two statements be confirmed (or rejected) with J?
>>> >
>>> They can be confirmed by calculating the integrals.
>>>
>>> Bernoulli numbers are involved among other things, and Roger's essay
>>> http://www.jsoftware.com/jwiki/Essays/Bernoulli%20Numbers
>>> gives
>>>
>>> B0=: 3 : 0
>>> b=. ,1x
>>> for_m. }.i.x: y do. b=. b,(1+m)%~-+/b*(i.m)!1+m end.
>>> )
>>>
>>> which can be shortened to:
>>>
>>> B0t=: [: ([ , +/@:(* i.!>:) -@% 1+])~/ }:@i.@-,1:
>>>
>>> B0 20
>>> 1 _1r2 1r6 0 _1r30 0 1r42 0 _1r30 0 5r66 0 _691r2730 0 7r6 0 _3617r510 0
>>> 43867r798 0
>>>
>>> B0t 20x
>>> 1 _1r2 1r6 0 _1r30 0 1r42 0 _1r30 0 5r66 0 _691r2730 0 7r6 0 _3617r510 0
>>> 43867r798 0
>>>
>>> ts 'y0=.B0 200'
>>> 1.85257 362496
>>>
>>> ts 'y0t=.B0t 200x'
>>> 1.8853 293888
>>>
>>> y0-:y0t
>>> 1
>>>
>>>
>>>
>>>
>>>
>>> >
>>> > (No deadlines apply.)
>>> >
>>> >
>>> > R.E. Boss
>>> >
>>> > ----------------------------------------------------------------------
>>> > For information about J forums see http://www.jsoftware.com/forums.htm
>>> >
>>> >
>>>
>>> --
>>> View this message in context:
>>> http://old.nabble.com/Challenge-12%28-%29-tp33844136s24193p33849223.html
>>> Sent from the J Programming mailing list archive at Nabble.com.
>>>
>>> ----------------------------------------------------------------------
>>> For information about J forums see http://www.jsoftware.com/forums.htm
>>
>> ----------------------------------------------------------------------
>> For information about J forums see http://www.jsoftware.com/forums.htm
>>
>>
>
> --
> View this message in context:
> http://old.nabble.com/Challenge-12%28-%29-tp33844136s24193p33877568.html
> Sent from the J Programming mailing list archive at Nabble.com.
>
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
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