Generating functions are valuable for the insights they provide.
On Tue, May 22, 2012 at 7:45 AM, Viktor Cerovski
<viktor.cerov...@gmail.com> wrote:
>
>
> Raul Miller-4 wrote:
>>
>> Based on the mathworld page, this looks like an implementation of
>> Bernoulli numbers in J:
>>
>> bernoulli=: ! * (% ^ - 1:) t.
>>
>>
> Sure, you can calculate them via generating function, that's very elegant
> and very slow:
>
> B0t=: [: ([ , +/@:(* i.!>:) -@% 1+])~/ }:@i.@-,1:
>
> ts'b0 =. bernoulli i.100x'
> 23.3222 4.87437e6
>
> ts'B0 =. B0t 100x'
> 0.181124 119168
>
> b0-:B0
> 1
>
> _2{b0
> 67908260672905495624051117546403605607342195728504487509073961249992947058239r6
>
>
>
>> 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
>>>>
>>>>
>>>
>>> --
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>>> Sent from the J Programming mailing list archive at Nabble.com.
>>>
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>> ----------------------------------------------------------------------
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>>
>
> --
> View this message in context:
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> Sent from the J Programming mailing list archive at Nabble.com.
>
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