John D. Cook • 8/25/2026

Numerical (in)stability of recurrence relations

This article discusses the numerical stability of three-term recurrence relations, focusing on Bessel functions Jn and Yn. It explains that while these functions behave similarly as functions of x, as functions of n they diverge: Jn decays and Yn goes to -∞. This causes instability when computing one solution while inadvertently picking up the growing component due to rounding errors. The post clarifies why forward recurrence is stable for Yn but not Jn, and reverse is stable for Jn but not Yn, and mentions techniques like Miller's algorithm for computing the minimal solution.

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#Numerical Stability #Bessel Functions #Special Functions