[R] Error in besselJ(pi/2, 1E-15)?

[Richard O'Keefe]

 besselJ(pi/2, sqrt(0.1)^(24:40))
 [1] 4.720012e-01 4.720012e-01 4.720012e-01 4.720012e-01 4.720012e-01
     1e-12                                      1e-13
                    1e-14
 [6] 4.720012e-01 4.720012e-01 1.492599e+15 1.000000e+00 1.000000e+00
                             1e-15                                       1e-16
[11] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
      1 e-17                                       1e-18
                        1e-19
[16] 1.000000e+00 1.000000e+00
                               1e-20
> besselJ(pi/1, 1E-15)
[1] -3.042422e+14

For what it's worth, the besselj function in Octave 8.4 doesn't do this.
octave:4> besselj(0.1.^(12:20), pi/2)
ans =

   0.4720   0.4720   0.4720   0.4720   0.4720   0.4720   0.4720
0.4720   0.4720

Bessel functions are most commonly wanted at integer and half-integer orders,
so it's not surprising that a problem like this could go unnoticed for a while.

On Mon, 29 Dec 2025 at 05:29, Leo Mada via R-help <r-help using r-project.org> wrote:
>
> Dear R-Users,
>
> besselJ(pi/2, 0)
> # 0.4720012
> besselJ(pi/2, 1E-14)
> # 0.4720012
> besselJ(pi/2, 1E-15)
> # 4.720012e+14
>
> There seems to be an error in besselJ(pi/2, eps), where eps is close to 0; although besselJ(pi/2, 0) is well behaved.
>
> I am not an expert in the field - but it doesn't seem right.
>
> Sincerely,
>
> Leonard
>
>
>         [[alternative HTML version deleted]]
>
> ______________________________________________
> R-help using r-project.org mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide https://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.