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
