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How can I solve a recurrence relation with variable coefficient like the one below:

$$J_{n+1}(x) = \frac{2n}{x}J_{n}(x) - J_{n-1}(x)$$

for simplicity assume $x=1$.

I need to solve this in order to analyse the error trend of $\in_{n+1}(x) = |J_{n+1} - \overline{J}_{n+1}|$ where $J_{n+1}$ is the actual value and $\overline{J}_{n+1}$ is the value computed by a computer with precision error.

Upperwal
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1 Answers1

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This solve is the difficult task of computing Bessel functions. The recurrence relation is stable only in a specific $(n,x)$ region. See e.g. Gautschi, Computational aspects of three-term recurrence relations. Your example for $J_n(1)$ is listed in Table 1.

If $n$ not too large you can compute $J_n(1)$ by Miller’s backward recurrence algorithm.

gammatester
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