Numerical (In)Stability of Recurrence Relations

Recurrence relations for special functions like Bessel functions can be numerically unstable depending on the direction of computation and the function type. The instability arises because second-order linear difference equations have two independent solutions—one that grows and one that decays with n—and rounding errors can cause unwanted amplification of the growing solution when trying to compute the decaying one. Specialized techniques like Miller's algorithm can help compute the desired minimal solution reliably.