Abstract:This paper, the second in a series, presents an efficient, self-contained algorithm for computing the modified Bessel function of the second kind, \(K_{\nu}(z)\) 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-\(|z|\) asymptotics, uniform asymptotics for large \(|\nu|\), 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 \(\{I_{\nu},K_{\nu}\}\) across wide parameter ranges.
From: Mofreh Zaghloul [view email]
[v1]
Fri, 12 Jun 2026 17:20:14 UTC (3,719 KB)
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