Re: Comments on Special functions, Quaternions, Octonions

From: Hubert HOLIN (Hubert.Holin_at_[hidden])
Date: 2001-05-28 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:
> > > 1.0000000000000000818e-05       9.9999999998333335331e-06      
> > > 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 1.19209290e-07F  
> #define DBL_EPSILON 2.2204460492503131e-16 
> #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
> 1.1920928955078125e-07  2.2204460492503130808e-16 1.084202172485504434e-19  
> 
> >
> > 		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 "stop-gap" library, we will have
> > > > to fine-tune 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]