On the Computation of the Logarithm of the Modified Bessel Function of the Second Kind

TL;DR

This paper presents a stable recursive method for computing the logarithm of the modified Bessel function of the second kind, Kν, addressing overflow issues in large parameter regimes with practical applications demonstrated.

Contribution

It introduces a simple, stable recursion technique for directly computing the log of Kν, improving numerical stability in scientific computations.

Findings

Recursive method effectively avoids overflow for large ν

Method is validated with statistical examples using Gil-Pelaez inversion

Provides conditions for overflow and practical computational guidelines

Abstract

The modified Bessel function of the second kind K$\nu$ appears in a wide variety of applied scientific fields. While its use is greatly facilitated by an implementation in most numerical libraries, overflow issues can be encountered especially for large value of $\nu$. After giving some necessary and sufficient conditions for their occurrences, this technical note shows that they can mostly be avoided by directly computing the logarithm of K$\nu$ thanks to a simple and stable forward recursion. A statistical examples based on the Gil-Pelaez inversion formula is given to illustrate the recursive method.