Miller's recurrence algorithm is a procedure for the backward calculation of a rapidly decreasing solution of a three-term recurrence relation developed by J. C. P. Miller. It was originally developed to compute tables of the modified Bessel function but also applies to Bessel functions of the first kind and has other applications such as computation of the coefficients of Chebyshev expansions of other special functions. Many families of special functions satisfy a recurrence relation that relates the values of the functions of different orders with common argument x {\displaystyle x} . The modified Bessel functions of the first kind I n ( x ) {\displaystyle I_{n}(x)} satisfy the recurrence relation
I n − 1 ( x ) = 2 n x I n ( x ) + I n + 1 ( x ) {\displaystyle I_{n-1}(x)={\frac {2n}{x}}I_{n}(x)+I_{n+1}(x)} . However, the modified Bessel functions of the second kind K n ( x ) {\displaystyle K_{n}(x)} also satisfy the same recurrence relation
K n − 1 ( x ) = 2 n x K n ( x ) + K n + 1 ( x ) {\displaystyle K_{n-1}(x)={\frac {2n}{x}}K_{n}(x)+K_{n+1}(x)} . The first solution decreases rapidly with n {\displaystyle n} . The second solution increases rapidly with n {\displaystyle n} . Miller's algorithm provides a numerically stable procedure to obtain the decreasing solution. To compute the terms of a recurrence a 0 {\displaystyle a_{0}} through a N {\displaystyle a_{N}} according to Miller's algorithm, one first chooses a value M {\displaystyle M} much larger than N {\displaystyle N} and computes a trial solution taking initial condition a M {\displaystyle a_{M}} to an arbitrary non-zero value (such as 1) and taking a M + 1 {\displaystyle a_{M+1}} and later terms to be zero. Then the recurrence relation is used to successively compute trial values for a M − 1 {\displaystyle a_{M-1}} , a M − 2 {\displaystyle a_{M-2}} down to a 0 {\displaystyle a_{0}} . Noting that a second sequence obtained from the trial sequence by multiplication by a constant normalizing factor will still satisfy the same recurrence relation, one can then apply a separate normalizing relationship to determine the normalizing factor that yields the actual solution. In the example of the modified Bessel functions, a suitable normalizing relation is a summation involving the even terms of the recurrence:
I 0 ( x ) + 2 ∑ m = 1 ∞ ( − 1 ) m I 2 m ( x ) = 1 {\displaystyle I_{0}(x)+2\sum _{m=1}^{\infty }(-1)^{m}I_{2m}(x)=1}
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