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

Fri Jan 5 14:54:00 GMT 2018

On 01/04/2018 03:54 PM, Michele Pezzutti wrote:
>
>
> 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.Â  It's tight.Â  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.Â  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
> Â Â Â Â Â  *Â Â  @paramÂ  __xÂ Â  The argument of the Bessel functions.
> Â Â Â Â Â  *Â Â  @paramÂ  __JnuÂ  The output Bessel function of the first kind.
> Â Â Â Â Â  *Â Â  @paramÂ  __NnuÂ  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  {
> Â Â Â Â Â Â  const _Tp __muÂ Â  = _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.Â  They could use a similar 
>> clean-up.
>>
>> But let's get this in there first.
>>
>> Ed
>>
>>
>
>
This looks good to me.Â  The only nit is that you have to uglify the k 
loop variable.

(I hope we can un-uglify our variables when modules come!)

Do you have copyright assignment in place and all that?

Ed