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, $K_{\nu }(z)$

This paper, the second in a series, presents an efficient, self-contained algorithm for computing the modified Bessel function of the second kind, $K_{\nu }(z)$ for complex argument and real orders, building on Part~I (for $I_{\nu }$). The method adaptively selects among analytic representations such as power series, large-$\mathrm{\mid }z\mathrm{\mid }$ asymptotics, uniform asymptotics for large $\mathrm{\mid }\nu \mathrm{\mid }$, and numerically stable forward recurrence with region boundaries tuned for accuracy and efficiency. A robust $\mathtt{\text{Fortran}}$ 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 $\mathtt{\text{Maple}}$ 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 $\{I_{\nu },K_{\nu }\}$ across wide parameter ranges.