Fix Bug 83566 - cyl_bessel_j returns wrong result for x>1000 for high orders

[Michele Pezzutti]

On 01/04/2018 06:16 PM, Ed Smith-Rowland wrote:
> On 01/03/2018 02:49 PM, Michele Pezzutti wrote:
>>
>> Hi.
>>
>> On 01/02/2018 05:43 PM, Michele Pezzutti wrote:
>>>
>>> On 01/02/2018 02:28 AM, Ed Smith-Rowland wrote:
>>>> I like the patch.
>>>>
>>>> I have a similar one in the tr29124 branch.
>>>>
>>>> Anyway, I got held up and I think it's good to have new folks 
>>>> looking into this.
>>>>
>>>> It looks good except that you need to uglify k.
>>>
>>> I looked at the GSL implementation, based on same reference, and 
>>> their loop is cleaner. What about porting that implementation here? 
>>> Possible?
>>>
>>> My implementation is also using one more term for P than for Q, 
>>> which is discouraged in GSL, according to their comments.
>>
>> Ed, do you have any comment about this point?
>> Regarding the 2nd line, after my observations, usually term stops 
>> contributing to P, Q before k > nu/2, so actually, an offset by one 
>> is most likely without consequences.
>> GSL implementation is nevertheless more elegant.
>>
> I've seen their implementation.  It's tight.  Feel free to port it.
> I assume this is it:
> int
> gsl_sf_bessel_Jnu_asympx_e(const double nu, const double x, 
> gsl_sf_result * result);
>
> You do want to give Q or b one more term so as to make that last 
> factor as small as possible.  They're right.
>

Here it is.

diff --git a/libstdc++-v3/include/tr1/bessel_function.tcc 
b/libstdc++-v3/include/tr1/bessel_function.tcc
index 7ac733d..7842350 100644
--- a/libstdc++-v3/include/tr1/bessel_function.tcc
+++ b/libstdc++-v3/include/tr1/bessel_function.tcc
@@ -353,21 +353,47 @@ namespace tr1
       *   @param  __x   The argument of the Bessel functions.
       *   @param  __Jnu  The output Bessel function of the first kind.
       *   @param  __Nnu  The output Neumann function (Bessel function 
of the second kind).
+Â Â Â Â  *
+Â Â Â Â  *Â  Adapted for libstdc++ from GNU GSL version 2.4 specfunc/bessel_j.c
+Â Â Â Â  *Â  Copyright (C) 1996,1997,1998,1999,2000,2001,2002,2003 Gerard 
Jungman
 Â Â Â Â Â  */
 Â Â Â Â  template <typename _Tp>
 Â Â Â Â  void
 Â Â Â Â  __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu)
 Â Â Â Â  {
        const _Tp __mu   = _Tp(4) * __nu * __nu;
-Â Â Â Â Â  const _Tp __mum1 = __mu - _Tp(1);
-Â Â Â Â Â  const _Tp __mum9 = __mu - _Tp(9);
-Â Â Â Â Â  const _Tp __mum25 = __mu - _Tp(25);
-Â Â Â Â Â  const _Tp __mum49 = __mu - _Tp(49);
-Â Â Â Â Â  const _Tp __xx = _Tp(64) * __x * __x;
-Â Â Â Â Â  const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx)
-Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx));
-Â Â Â Â Â  const _Tp __Q = __mum1 / (_Tp(8) * __x)
-Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx));
+Â Â Â Â Â  const _Tp __8x = _Tp(8) * __x;
+
+Â Â Â Â Â  _Tp __P = _Tp(0);
+Â Â Â Â Â  _Tp __Q = _Tp(0);
+
+Â Â Â Â Â  _Tp k = _Tp(0);
+Â Â Â Â Â  _Tp __term = _Tp(1);
+
+Â Â Â Â Â  int __epsP = 0;
+Â Â Â Â Â  int __epsQ = 0;
+
+Â Â Â Â Â  _Tp __eps = std::numeric_limits<_Tp>::epsilon();
+
+Â Â Â Â Â  do
+Â Â Â Â Â Â Â  {
+Â Â Â Â Â Â Â Â Â  __term *= (k == 0) ? _Tp(1) : -(__mu - (2 * k - 1) * (2 * k - 
1)) / (k * __8x);
+Â Â Â Â Â Â Â Â Â  __epsP = std::abs(__term) < std::abs(__eps * __P);
+Â Â Â Â Â Â Â Â Â  __P += __term;
+
+Â Â Â Â Â Â Â Â Â  k++;
+
+Â Â Â Â Â Â Â Â Â  __term *= (__mu - (2 * k - 1) * (2 * k - 1)) / (k * __8x);
+Â Â Â Â Â Â Â Â Â  __epsQ = std::abs(__term) < std::abs(__eps * __Q);
+Â Â Â Â Â Â Â Â Â  __Q += __term;
+
+Â Â Â Â Â Â Â Â Â  if (__epsP && __epsQ && k > __nu / 2.)
+Â Â Â Â Â Â Â Â Â Â Â  break;
+
+Â Â Â Â Â Â Â Â Â  k++;
+Â Â Â Â Â Â Â  }
+Â Â Â Â Â  while (k < 1000);
+

 Â Â Â Â Â Â  const _Tp __chi = __x - (__nu + _Tp(0.5L))
 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  * __numeric_constants<_Tp>::__pi_2();



> In principal, these ideas should be ported to I,K but I think (and 
> IIRC GSL agrees) these under,over-flow before they have much effect.
> I think I put the full series in there.  They could use a similar 
> clean-up.
>
> But let's get this in there first.
>
> Ed
>
>