Looking for fortran subroutine for I Bessel function with negative order

[Fr4ct4l]

Hello all, I am developing a new analytical solution for a problem in flow in porous media, and I need to write it in Fortran.

This solution contains the modified Bessel function of the first kind, I_n(x).

The order n is a real number, and it can be both negative and positive.
The argument x is real, and is always positive. Correct me if I'm wrong, but I believe this means I will never encounter a complex result.

I have looked in netlib, Numerical Recipes books, and an assortment of personal webpages (Alan Miller, JP Moreau), and I still can't find a Bessel_I subroutine that will compute for negative non-integer n. Obviously they all do a splendid job for positive n.

Alternatively, (and I have looked in functions.wolfram.com website) are you aware how I can compute this from other functions ? Gamma func ?

Thanks,
Manuel

 

[jvicens]

Dear Manuel,

Thank you for reaching out about your research project. I understand your frustration in finding a Fortran subroutine that can compute the modified Bessel function for negative non-integer n. After reviewing the resources you have mentioned, it seems that there are limited options for this specific case.

One approach you could take is to use the recurrence relation for computing I_n(x) for negative n, which is:

I_n(x) = I_{n+1}(x) + \frac{2n}{x}I_n(x)

This can be implemented in Fortran and used to compute the function for negative non-integer n.

Another possibility is to use the gamma function, as you have mentioned. The modified Bessel function can be expressed in terms of the gamma function as:

I_n(x) = \frac{1}{\pi} \left(\frac{x}{2}\right)^n \int_0^\pi e^{x\cos\theta}\cos(n\theta)d\theta

This integral can be evaluated numerically using standard methods, such as the trapezoidal rule or Simpson's rule.

I hope these suggestions are helpful in your research. If you have any further questions, please do not hesitate to reach out.