Roger Hui-2 wrote:
>
> Generating functions are valuable for the insights they provide.
>
Yes, I fully agree.
> On Tcd ../ue, 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.
>>>>>>
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>>>>>> http://www.jsoftware.com/forums.htm
>>>>>
>>>>> ----------------------------------------------------------------------
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>>>>>
>>>>>
>>>>
>>>> --
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>>>>
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>>> ----------------------------------------------------------------------
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>>>
>>
>> --
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>>
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