bessel functions

Hi,

I am trying to write a code to calculate the bessel functions of the first kind. The equation is given here: http://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_first_kind_:_J.CE.B1

I also have the following code:

beta = 4;
alpha = 1;
iteration = 3;

format long

for m = 1:iteration
    J(m) = (((-1)^m)/(factorial(m)*gamma(m+alpha+1)))*(beta/2)^(2*m+alpha);
end

out = sum(J)

However, when i compare my results with the built in function besselj, I cannot get the same result. Unfortunately, I cannot see where I am going wrong.

All assistance is appreciated.

tia

Answer 1

I have solved this issue. I should have started the iterations at 0 instead of 1.

Thread closed. Thanks

Answer 2

The iteration should go to infinity but yours stop at 3.

In fact, this kind of implementation has another numerical issue, see the notes below from factorial's reference page

Since double precision numbers only have about 15 digits, the answer is only accurate for n <= 21. For larger n, the answer will have the right magnitude, and is accurate for the first 15 digits.

hmm, i probably should have not left it at 3. However, I found that after any number greater than 8 for the number of iterations, produces the same result, so i'm not sure if an infinite sum is needed for an approximation.

Are you saying that for iteration < 21, the answer should still be accurate for the first 15 digits? for my needs, this is more than enough, but i found that even for iteration = 10, that the answer is significantly different than when i use the besselj( ) function call.

This is indeed the equation but I'm sure there are programming tricks to get around the numerical issue. I personally don't know what they are, but looks to me the brute force approach will hit the wall very soon due to numerical issues.

Answer 3

I want to also add that i tried to use the

doc besselj  

and then formula that was presented there. However, I still do not get the same result as besselj(), but I do get the same result as the formula i initially used.