Re: Beta[x,a,b]: how to get around bug
- To: mathgroup at smc.vnet.net
- Subject: [mg94423] Re: Beta[x,a,b]: how to get around bug
- From: dimitris <dimmechan at yahoo.com>
- Date: Fri, 12 Dec 2008 06:57:19 -0500 (EST)
- References: <ghqjrv$11e$1@smc.vnet.net>
On 11 =C4=E5=EA, 10:43, "g... at iap.fr" <g... at iap.fr> wrote:
> I've encountered a bug with the incomplete Beta function (both
> versions 6.0 and 7.0), which I just submitted to support.wolfram.com.
>
> Beta[-1.7, 1.39646, -0.37188]
> returns 0. + 0. i
>
> however
>
> Beta[N[-17/10, 10], N[139646/100000, 10], N[-37188/100000, 10]=
]
> provides the correct answer: -0.20657 - 0.6125 i
>
> There is a relation between Beta[z,a,b] and Hypergeometric functions
> Beta[z,a,b] = z^a/a Hypergeometric2F1[a,1-b,a+1,z] = z^a/a
> Hypergeometric1F1[1-b,a+1,z].
>
> If I estimate Hypergeometric2F1[a,1-b,a+1,z] for my values of z=-1.7,
> a=1.39646, b=-0.37188, I get
> Hypergeometric2F1[1.39646, 1.37188, 2.39646, -1.7]
> returns 0
>
> However, if I use Hypergeometric1F1[1-b,a+1,z]:
> Hypergeometric1F1[1.37188, 2.39646, -1.7]
> this returns a plausible value of 0.420134
>
> So I believe that Beta and Hypergeometric2F1 are bugged, but not
> Hypergeometric1F1.
I don't think this is a bug. You have entered Mathematica's arithmetic world!
In[77]:=
Rationalize /@ {-1.7, 1.39646, -0.37188}
(N[#1, 100] & )[Beta[Sequence @@ %]]
Out[77]=
{-(17/10), 69823/50000, -(9297/25000)}
Out[78]=
-0.206565532195211828488245686509461713110661823003533791150338364094234323=
8445737049165731802724786248249403389045196`99.655087\
24163801 -
0.6124813174705197196509036033954859795594341071520172702600419005284902885=
805220759649991747874535605198632071032405`100.1271\
2223132085*I
When you enter real numbers by hand Mathematica assumes that they are known
within MachinePrecision
In[79]:=
Precision /@ {-1.7, 1.39646, -0.37188}
Out[79]=
{MachinePrecision, MachinePrecision, MachinePrecision}
In[80]:=
$MachinePrecision
Out[80]=
15.9546
It is obvious that this number of precision cannot be used in order to
evaluate your Betta function.
Another way to work is, as follows, e.g.
In[94]:=
(SetPrecision[#1, 10] & ) /@ {-1.7, 1.39646, -0.37188}
Beta[Sequence @@ %]
Out[94]=
{-1.699999999999999955591079015`9.999999999999998,
1.3964600000000000346034312315`9.999999999999998,
-0.3718799999999999883470991335`10.}
Out[95]=
-0.2065655321737375277`4.417380954358203 -
0.6124813174068471289`4.417393963471557*I
Dimitris