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

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.

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


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