Re: [boost] Re: Comments on Special functions, Quaternions, Octonions

From: boost (boost_at_[hidden])
Date: 2001-05-25 20:19:57

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
8.0000000000000013321e-06       7.9999999999146678448e-06       0.99999999998933331025  0.99999999998933331025  0.99999999998933331025
6.4000000000000014045e-06       6.399999999956310598e-06        0.99999999999317334964  0.99999999999317334964  0.99999999999317334964
5.1200000000000018012e-06       5.1199999999776325081e-06       0.99999999999563093933  0.99999999999563093933  0.99999999999563093933
4.0960000000000019492e-06       4.0959999999885483697e-06       0.99999999999720379229  0.99999999999720379229  0.99999999999720379229
3.2768000000000018134e-06       3.2767999999941378044e-06       0.99999999999821043151  0.99999999999821043151  0.99999999999821043151
2.6214400000000015355e-06       2.6214399999969992272e-06       0.99999999999885469393  0.99999999999885469393  0.99999999999885469393
2.0971520000000012284e-06       2.0971519999984638636e-06       0.99999999999926703076  0.99999999999926703076  0.99999999999926703076
1.6777216000000011097e-06       1.6777215999992140044e-06       0.99999999999953081975  0.99999999999953081975  0.99999999999953081975
1.3421772800000010572e-06       1.3421772799995980813e-06       0.99999999999969979569  0.99999999999969979569  0.99999999999969979569
1.0737418240000009305e-06       1.0737418239997946779e-06       0.99999999999980782039  0.99999999999980782039  0.99999999999980782039
8.5899345920000082908e-07       8.5899345919989516172e-07       0.99999999999987698729  0.99999999999987698729  0.99999999999987698729
6.8719476736000068444e-07       6.8719476735994658021e-07       0.99999999999992128519  0.99999999999992128519  0.99999999999992128519
5.4975581388800056873e-07       5.497558138879728284e-07        0.99999999999994959587  0.99999999999994959587  0.99999999999994959587
4.3980465111040046557e-07       4.3980465111038627777e-07       0.99999999999996780353  0.99999999999996780353  0.99999999999996780353
3.5184372088832040422e-07       3.518437208883131515e-07        0.99999999999997934985  0.99999999999997934985  0.99999999999997934985
2.8147497671065634455e-07       2.8147497671065263878e-07       0.99999999999998678835  0.99999999999998678835  0.99999999999998678835
2.2517998136852507564e-07       2.2517998136852316982e-07       0.99999999999999156231  0.99999999999999156231  0.99999999999999156231
1.801439850948200711e-07        1.8014398509481909172e-07       0.99999999999999455991  0.99999999999999455991  0.99999999999999455991
1.4411518807585607276e-07       1.4411518807585556984e-07       0.99999999999999655831  0.99999999999999655831  0.99999999999999655831
1.1529215046068486086e-07       1.1529215046068460939e-07       0.99999999999999777955  0.99999999999999777955  0.99999999999999777955
9.2233720368547893979e-08       9.2233720368547761631e-08       0.99999999999999855671  0.99999999999999855671  0.99999999999999855671
7.3786976294838315184e-08       7.3786976294838249009e-08       0.99999999999999911182  0.99999999999999911182  0.99999999999999911182
5.9029581035870656117e-08       5.902958103587062303e-08        0.99999999999999944489  0.99999999999999944489  0.99999999999999944489
4.7223664828696528864e-08       4.7223664828696509012e-08       0.99999999999999966693  0.99999999999999966693  0.99999999999999966693
3.7778931862957224415e-08       3.7778931862957217797e-08       0.99999999999999977796  0.99999999999999977796  0.99999999999999977796
3.0223145490365778208e-08       3.0223145490365771591e-08       0.99999999999999988898  0.99999999999999988898  0.99999999999999988898
2.4178516392292624552e-08       2.4178516392292621243e-08       0.99999999999999988898  0.99999999999999988898  0.99999999999999988898
1.9342813113834100965e-08       1.9342813113834100965e-08       0.99999999999999988898  0.99999999999999988898  0.99999999999999988898
1.5474250491067282096e-08       1.5474250491067282096e-08       1       1       1
1.2379400392853826007e-08       1.2379400392853826007e-08       1       1       1
 
long double:
1.0000000000000000818e-05       9.9999999998333335331e-06       0.99999999998333333331  0.99999999998333333331  0.99999999998333333331
8.0000000000000010988e-06       7.9999999999146678448e-06       0.9999999999893333625   0.99999999998933333334  0.99999999998933333334
6.4000000000000012345e-06       6.399999999956310598e-06        0.99999999999317335989  0.99999999999317333333  0.99999999999317333333
5.1200000000000012718e-06       5.1199999999776316611e-06       0.99999999999563087124  0.99999999999563093331  0.99999999999563093331
4.0960000000000012448e-06       4.0959999999885475226e-06       0.99999999999720376247  0.99999999999720379733  0.99999999999720379733
3.2768000000000011778e-06       3.2767999999941369573e-06       0.9999999999982103658   0.99999999999821043031  0.99999999999821043031
2.6214400000000010877e-06       2.6214399999969988037e-06       0.99999999999885468466  0.99999999999885467539  0.99999999999885467539
2.0971520000000009866e-06       2.0971519999984634401e-06       0.99999999999926690553  0.99999999999926699227  0.99999999999926699227
1.6777216000000008825e-06       1.6777215999992137926e-06       0.99999999999953088426  0.99999999999953087504  0.99999999999953087504
1.3421772800000007805e-06       1.3421772799995978695e-06       0.99999999999969980838  0.99999999999969976002  0.99999999999969976002
1.073741824000000684e-06        1.0737418239997944662e-06       0.99999999999980787878  0.99999999999980784642  0.99999999999980784642
8.5899345920000059488e-07       8.5899345919989494996e-07       0.99999999999987704784  0.99999999999987702171  0.99999999999987702171
6.8719476736000051404e-07       6.8719476735994636845e-07       0.99999999999992123369  0.99999999999992129392  0.99999999999992129392
5.4975581388800044175e-07       5.4975581388797272252e-07       0.99999999999994966646  0.99999999999994962808  0.99999999999994962808
4.3980465111040037781e-07       4.3980465111038617189e-07       0.99999999999996772075  0.99999999999996776195  0.99999999999996776195
3.5184372088832032179e-07       3.5184372088831304562e-07       0.99999999999997930101  0.99999999999997936769  0.99999999999997936769
2.8147497671065627305e-07       2.8147497671065258584e-07       0.99999999999998686126  0.99999999999998679528  0.99999999999998679528
2.2517998136852503095e-07       2.2517998136852311688e-07       0.99999999999999151238  0.99999999999999154897  0.99999999999999154897
1.8014398509482003476e-07       1.8014398509481906525e-07       0.99999999999999464621  0.99999999999999459135  0.99999999999999459135
1.4411518807585603581e-07       1.4411518807585554337e-07       0.99999999999999661122  0.99999999999999653847  0.99999999999999653847
1.1529215046068483505e-07       1.1529215046068458292e-07       0.9999999999999977789   0.9999999999999977846   0.9999999999999977846
9.2233720368547873158e-08       9.2233720368547735161e-08       0.99999999999999852093  0.99999999999999858213  0.99999999999999858213
7.3786976294838302621e-08       7.3786976294838235774e-08       0.99999999999999908347  0.99999999999999909258  0.99999999999999909258
5.9029581035870645374e-08       5.9029581035870609795e-08       0.99999999999999937702  0.99999999999999941925  0.99999999999999941925
4.7223664828696518919e-08       4.7223664828696495777e-08       0.99999999999999955868  0.99999999999999962834  0.99999999999999962834
3.7778931862957217232e-08       3.777893186295721118e-08        0.99999999999999977709  0.99999999999999976213  0.99999999999999976213
3.0223145490365775462e-08       3.0223145490365771591e-08       0.99999999999999993863  0.99999999999999984778  0.99999999999999984778
2.4178516392292621712e-08       2.4178516392292617935e-08       0.99999999999999988318  0.99999999999999990258  0.99999999999999990258
1.9342813113834098443e-08       1.9342813113834097656e-08       0.99999999999999989695  0.99999999999999993766  0.99999999999999993766
1.5474250491067279614e-08       1.5474250491067278787e-08       0.99999999999999990665  0.9999999999999999601   0.9999999999999999601
1.2379400392853824379e-08       1.2379400392853824353e-08       0.99999999999999997241  0.99999999999999997447  0.99999999999999997447

> 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?

>
> > 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 :)

Best wishes,
Peter

P.S: interval< octernion<double> > will be funny