GoKawiilWhy Bessel function recurrence relations become numerically unstable
A technical blog post explains why three-term recurrence relations used to compute Bessel functions can fail depending on direction and function type. Forward recurrence works for second-kind Bessel functions but not first-kind, while reverse recurrence works for first-kind but not second-kind functions.
GoKawiil's interpretation of the reporting above, not reported fact.
The explanation traces the instability to how the two independent solutions of the underlying difference equation grow or decay differently as the index n increases, causing rounding errors to amplify the wrong component. This matters for anyone writing numerical software that relies on such recurrences, since naive implementations can silently produce garbage results.
- Recurrence direction determines whether Bessel function computations are stable or unstable.
- Instability arises because one solution decays and another grows as n increases, letting rounding errors dominate.
- Similar risks apply broadly whenever two solutions of a recurrence diverge in scale as a function of index.
Accuracy and Stability of Numerical Algorithms by Nicholas Higham — If this post's dive into why Bessel recurrences blow up in one direction but not the other hooked you, Higham's classic is the deep-dive companion. It covers rounding error, error propagation, and the stability analysis behind exactly these kinds of numerical algorithms, making it a staple reference for anyone writing serious floating-point code.
See Accuracy and Stability of Numerical Algorithms by Nicholas Higham on Amazon → Affiliate link — we may earn a commission on purchases, at no extra cost to you. Product picked by AI based on this article; it is not a tested recommendation.Source: johndcook.com, 2026-08-27
Published there as: “Numerical (In)Stability of Recurrence Relations”
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