[Speex-dev] A technical question about the speex preprocessor.

[John Ridges]

By my reckoning the confluent hypergoemetric functions should have the 
following values:

M(-.25;1;-.5) = 1.11433
M(-.25;1;-1) = 1.21088
M(-.25;1;-1.5) = 1.29385
M(-.25;1;-2) = 1.36627
M(-.25;1;-2.5) = 1.43038
M(-.25;1;-3) = 1.48784
M(-.25;1;-3.5) = 1.53988
M(-.25;1;-4) = 1.58747
M(-.25;1;-4.5) = 1.63134
M(-.25;1;-5) = 1.67206
M(-.25;1;-5.5) = 1.71009
M(-.25;1;-6) = 1.74579
M(-.25;1;-6.5) = 1.77947
M(-.25;1;-7) = 1.81136
M(-.25;1;-7.5) = 1.84167
M(-.25;1;-8) = 1.87056
M(-.25;1;-8.5) = 1.89818
M(-.25;1;-9) = 1.92466
M(-.25;1;-9.5) = 1.95009
M(-.25;1;-10) = 1.97456

Which would give table values of:

   static const float table[21] = {
      0.82157f, 0.91549f, 0.99482f, 1.06298f, 1.12248f, 1.17515f, 1.22235f,
      1.26511f, 1.30421f, 1.34025f, 1.37371f, 1.40495f, 1.43428f, 1.46195f,
      1.48815f, 1.51305f, 1.53679f, 1.55948f, 1.58123f, 1.60212f, 1.62223f};
 
There is also a formula which asymptotically approaches M(a;b;-x) for 
high values of x that is:

(gamma(b)/gamma(b-a))*(x^-a)

Which for M(-.25;1;-x) is sqrt(sqrt(x))*1.10326

at x=10 this gives a value of 1.96191 (vs. what I think is the true 
value of 1.97456).

The reason I've gotten into all this is I'm trying to vectorize with SSE 
intrinsics some of the slower loops in the preprocessor, and the 
hypergeom_gain function is the only thing stopping me from removing all 
the branches in the loops. I don't know how critical the accuracy of the 
function is to the performance of the preprocessor, but if that 
aforementioned approximation was good enough for all the values of x, it 
would really speed the loops up.

Let me know what you think. I hope I'm helping out here (and not just 
confusing things).

John Ridges


Jean-Marc Valin wrote:
> OK, so the problem is that the table you see does not match the definition?
> y = gamma(1.25)^2 * M(-.25;1;-x) / sqrt(x)
> Note that the table data has an interval of .5 for the x axis.
>
> How far are your results from the data in the table?
>
> Cheers,
>
>    Jean-Marc
>
> Quoting John Ridges <jridges at masque.com>:
>
>   
>> Thanks for the confirmation  Jean-Marc. I kind of suspected from the
>> comments that it was the confluent hypergoemetric function, which I was
>> trying to evaluate using Kummer's equation, namely:
>>
>> M(a;b;x) is the sum from n=0 to infinity of (a)n*x^n / (b)n*n!
>> where (a)n = a(a+1)(a+2) ... (a+n-1)
>>
>> But when I use Kummer's equation, I don't get the values in the
>> "hypergeom_gain" table. Did you use a different solution to the
>> confluent hypergoemetric function when you created the table?
>>
>> John Ridges
>>
>>
>> Jean-Marc Valin wrote:
>>     
>>> Hi John,
>>>
>>> M(;;) is the confluent hypergeometric function.
>>>
>>> 	Jean-Marc
>>>
>>> John Ridges a écrit :
>>>
>>>       
>>>> Hi,
>>>>
>>>> I've been trying to re-create the table in the function "hypergeom_gain"
>>>> in preprocess.c, and I just simply can't get the same values. I get the
>>>> same value for the first element, so I know I'm computing gamma(1.25)^2
>>>> correctly, but I can't get the same numbers for M(-.25;1;-x), which I
>>>> assume is Kummer's function. Is it possible that the comment is out of
>>>> date and the values of Kummer's function used to make the table were
>>>> different? Any help would be greatly appreciated.
>>>>
>>>> John Ridges
>>>>
>>>>
>>>> _______________________________________________
>>>> Speex-dev mailing list
>>>> Speex-dev at xiph.org
>>>> http://lists.xiph.org/mailman/listinfo/speex-dev
>>>>
>>>>
>>>>
>>>>         
>>>
>>>       
>>
>>     
>
>
>
>
>