Yes, that's exactly what happened. Thanks for the correction.
On Mon, May 21, 2012 at 11:08 PM, Bo Jacoby <bojac...@yahoo.dk> wrote: > Roger, you first example below should have been (bernoulli i.9) rather > than (bernoulli i.9x). You modified the line before typing Enter, I guess. > > > > >________________________________ > > Fra: Roger Hui <rogerhui.can...@gmail.com> > >Til: Programming forum <programming@jsoftware.com> > >Sendt: 5:16 tirsdag den 22. maj 2012 > >Emne: Re: [Jprogramming] Challenge 12(?) > > > >Or use extended precision arguments. e.g. > > > > bernoulli=: ! * (% ^ - 1:) t. > > > > bernoulli i.9x > >1 _0.5 0.166667 _4.16334e_17 _0.0333333 _1.30104e_17 0.0238095 > _1.76074e_16 > >_0.0333333 > > > > bernoulli i.9x > >1 _1r2 1r6 0 _1r30 0 1r42 0 _1r30 > > > > > > > >On Mon, May 21, 2012 at 4:12 PM, Bo Jacoby <bojac...@yahoo.dk> wrote: > > > >> (|**) 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 > >> > >---------------------------------------------------------------------- > >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
