(|**) gets rid of the rounding errors around 0. (|**) bernoulli i.9 1 _0.5 0.166667 0 _0.0333333 0 0.0238095 0 _0.0333333
>________________________________ > Fra: Raul Miller <rauldmil...@gmail.com> >Til: Programming forum <programming@jsoftware.com> >Sendt: 21:39 mandag den 21. maj 2012 >Emne: Re: [Jprogramming] Challenge 12(?) > >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 >---------------------------------------------------------------------- >For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
