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Karim Belabas on Wed, 01 Feb 2023 10:14:42 +0100
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Re: Numerical instability of j_1(x)=sin(Pi*x)^2+sin(Pi*(gamma(x)+1)/x)^2
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- To: Georgi Guninski <gguninski@gmail.com>
- Subject: Re: Numerical instability of j_1(x)=sin(Pi*x)^2+sin(Pi*(gamma(x)+1)/x)^2
- From: Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>
- Date: Wed, 1 Feb 2023 10:13:27 +0100
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* Georgi Guninski [2023-02-01 08:51]:
> The function:
>
> j_1(x)=sin(Pi*x)^2+sin(Pi*(gamma(x)+1)/x)^2
>
> has real zeros at the primes for x>1.
[...]
> ? j_1(40.0+20.*I)
> fails in pari, but I don't care about this.
[...]
For the record, the failure is related to
(10:05) gp > sin(10^40 * I)
*** at top-level: sin(10^40*I)
*** ^------------
*** sin: overflow in expo().
The result is about -exp(10^40) / (2*I), but the 'exponent' of 10^40 is
so large that the resulting floating point number can't be represented in PARI.
(Exponents are coded by C longs, so the hardcoded limit for exponents is 2^63;
i.e. floats about 2^(2^63) ...)
Cheers,
K.B.
--
Pr Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/
`