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From: Hubert HOLIN (Hubert.Holin_at_[hidden])
Date: 20010528 16:12:44
Paris (U.E.), le 28/05/2001
 In boost_at_y..., Peter Schmitteckert (boost) <boost_at_s...> wrote:
> Salut,
>
> On Saturday 26 May 2001 23:44, you wrote:
> [...]
>
> > > double:
> > > 1.0000000000000000818e05 9.9999999998333335331e06
> > > 0.99999999998333333195 0.99999999998333333195 0.99999999998333333195
> >
> > [SNIP]
> >
> > Thanks, but could you in addition give me the output of
> >
> > ::std::cout << ::std::numeric_limits<float>::epsilon() << ::std::endl;
> > ::std::cout << ::std::numeric_limits<double>::epsilon() << ::std::endl;
> > ::std::cout << ::std::numeric_limits<long double>::epsilon() <<
> > ::std::endl;
> >
> > for your platform?
> Well,can't find numerics<..> for ma poor man gcc, but from float,h I have the
> following:
> #define FLT_EPSILON 1.19209290e07F
> #define DBL_EPSILON 2.2204460492503131e16
> #define LDBL_EPSILON (__extension__ ((union __convert_long_double)
> {__convert_long_double_i: {0x0, 0x80000000, 0x3fc0,
> 0x0}}).__convert_long_double_d))
>
>
> and
> cout << FLT_EPSILON << " " << DBL_EPSILON << " " <<LDBL_EPSILON << endl;
>
> yields
> 1.1920928955078125e07 2.2204460492503130808e16 1.084202172485504434e19
>
> >
> > If these are comparable to the discrepancy you indicated,
> > then i guess I'll let things stand as they are. Otherwise, I'll aim for
> > a more ambitious (annd somewhet slower) aproach: instead of just two
> > zones (smaller than epsilon, or not), I'll add an intermediate (say
> > between epsilon and sqrt(epsilon)), with perhaps small continuity
> > matching zones.
> In a region between zero and epsilon my solution will give you the same
> answer as your solution: 1.
>
> for x=sqrt( sqrt(epsilon) ), the x*x will be of the order of epsilon,
> due to the small prefactor of the next term, you could extend my idea
> even to pow( eps, 1/6 ), when x^6 = epsilon.
OK. I'll add one or more zones where I compute the result via
Taylor expansion That really seems necessary. I'll see if the
continuity matching zones are really necessary too, however.
> > > > 2) this is, as noted in the docs, a "stopgap" library, we will have
> > > > to finetune it for various hardware/software combinations... so this
> > > > situation will probably require some extensive list of ifdefs.
> six(x) is well behaved at x=0, and so is sinc(x). The only problem is that
> sometime trigonometric functio are poorly designed..
> Note that since you elready need the if .. else clause my solution
> will have better performance or small argument, i.e.x < pow(x, 1/6),
> since most processor need much more time to evaluate sin(x) than
> 1 (1/6) * x *x.
> I guess the problem with long douible is an internal rounding issue.
> But with arctan functions one has to be more careful.
>
> > http://www.7stones.com/Homepage/history.html
> Has a link to a nice article by John Baez (I like his book on knots)
> http://math.ucr.edu/home/baez/oct.ps
>
>
> > FWIW, when I use that function for my math, I spell it Argth
> > (argument tangente hyperbolique)... And let's not enter the highly
> > controvertial log/Log/ln/Ln... In the end, it boils down to a matter of
> Someone once said that theoretical physicist (yes, at least I was one)
> will rather use the underwear of their follow worker that using their naming
> for functions/constant/variables.
>
>
> > taste. I would rather keep the name sinc_pi (I changed it to that from
> > just sinc for the reasons I gave), and later introduce the one
> > parameter family under the name sinc, but frankly I'll go along with
> > whatever proves most popular.
> I was only confused, that's all. SInce you introduced the sinc_a family,
> I was always searching for one a != pi.
Slated for later, provided this library doe make the cut.
> > For that matter, what would interval< complex< float> > mean?
> A rectangle enclosing your soulution, like interval<double> is
> an 1D section that enclose your solution.
Looks nice. I'll have to take a closer look at these
interval<T>s.
> Best wishes,
> Peter
Amicalement
Hubert Holin
Hubert.Holin_at_[hidden]
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