Re: Comments on Special functions, Quaternions, Octonions

From: Hubert HOLIN (Hubert.Holin_at_[hidden])
Date: 2001-05-26 16:44:53

Paris (U.E.), le 26/05/2001

        Bonjour!

--- In boost_at_y..., Peter Schmitteckert (boost) <boost_at_s...> wrote:
> Salut,
> 
> On Saturday 26 May 2001 00:49, you wrote:
> > Paris (U.E.), le 26/05/2001
> >
> > --- In boost_at_y..., Peter Schmitteckert (boost) <boost_at_s...> wrote:
> > > Salut,
> > >
> > > first of all I would see the library in boost, but I suggest that
> > > problems
> >
> > 	Thanks!
> Well, thanks for your quaternion/octernion pdf files. I more interested in these than the code. 
> 
> > 	Two remarks here:
> >
> > 1)	could you compare the discrepancy to numeric_limits<T>::epsilon()
> > for small values of the argument.
> The discrepancy is is the last few digits only.  I problem is that sin(1e-8) is inaccurate. But I guess this will dependent on compiler/processor.
> Note that for double everythin agrees.
> 
>  peter_at_baxter:/samba/Boost/spefquatocto_sc2 > t
> 8       12
> 
> 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?

                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.

> > 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.
> BTW, do you have an example for octernions in physics?

                Well, I'me no physicist, but for this part, I get my cue 
from:

        http://www.7stones.com/Homepage/history.html

> >
> > > By the way, why is no-one using the correct name:
> > > arctan == arcus tangens; artanh = area tangens hyperbolicus
> >
> > 	Histerical raisins ;-)
> >
> > > Inaddition, what is the deeper meening of sinc_pi, it behaves
> > > like sinc. It confused me quite some time.
> >
> > 	"sinc" is sometimes the name of a one-parameter family of
> > functions, of which "sinc_pi" is the most famous (this is the approach
> > we have taken here,  see docs). At other times (f.i. MATLAB), "sinc"
> > just designates what I called here "sinc_pi". Just one more unfortunate
> > ambiguity of mathematics...
> I never saw a sinc_pi before, but sinc is an very ordinary function in  physics.
> So I give the "Histerical raisins" back., since you use "sinc_pi for  unfortunate
> ambiguity of mathematics", while denying artanh the correct name :)

                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 
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.

> Best wishes,
> Peter
> 
> P.S: interval< octernion<double> > will be funny

                For that matter, what would interval< complex< float> > mean?

        Merci

                        Hubert Holin
                        Hubert.Holin_at_[hidden]