Efficient Multi-Precision Computation of Bessel Functions for Real Orders and Complex Arguments with Fortran Implementation -- Part II: The Modified Bessel Function of the Second Kind, $K_ν(z)$

Abstract

This paper, the second in a series, presents an efficient, self-contained algorithm for computing the modified Bessel function of the second kind, \(K_ν(z)\) for complex argument and real orders, building on Part~I (for \(I_ν\)). The method adaptively selects among analytic representations such as power series, large-\(|z|\) asymptotics, uniform asymptotics for large \(|ν|\), and numerically stable forward recurrence with region boundaries tuned for accuracy and efficiency. A robust \texttt{Fortran} implementation supports double precision and quadruple precision. The use of quadruple precision extends the reliable computational domain and improves stability in challenging regimes. Accuracy is validated against high-precision \texttt{Maple} results, and benchmarks show runtimes significantly superior to those of established methods, in the literature, while avoiding their numerical failure modes across several decades of the parameter domain. Together with Part~I, this work provides a comprehensive, multiple-precision toolkit for \(\{I_ν,K_ν\}\) across wide parameter ranges.