Researchers Develop Efficient Multi-Precision Algorithm for Computing Bessel Functions with Complex Arguments

Researchers have completed a three-part series on developing numerically stable and efficient algorithms for computing Bessel functions with real orders and complex arguments, with implementations in Fortran. The final installment, submitted to arXiv in June 2026, focuses on regular Bessel functions of the first and second kinds, denoted J_ν(z) and Y_ν(z), complementing earlier work on modified Bessel functions I_ν(z) and K_ν(z).

The proposed algorithm represents a significant advancement in computational mathematics by supporting both positive and negative real orders with complex arguments, implemented in native double and quadruple precision arithmetic. Quadruple precision substantially increases dynamic range and accuracy by approximately an order of magnitude in reliably computable parameters, extending applicability to problems requiring 20-30 digits of precision—a substantial improvement over traditional methods.

Performance evaluations demonstrate marked improvements over established computational approaches. Compared to the widely-used Algorithm 644 (restricted to double-precision), the new implementation achieves execution times of 35-67% for J_ν(z) and 44-72% for Y_ν(z) while producing reliable results in cases where Algorithm 644 fails. When benchmarked against the more recent Algorithm 912, which supports both double and quad-precision as well as complex orders, the new algorithm delivers comparable accuracy in double-precision and significantly higher accuracy in quad-precision calculations while requiring only a small fraction of the computational cost—typically from a few thousandths to a few hundredths depending on precision and parameter regime.

A key advantage of the new algorithm is its broader applicability. Unlike Algorithm 912, which has restricted domains of validity across the (Re(ν), z) plane, the present algorithm maintains stability and accuracy over the full tested domain where reliable reference values are available. This expanded coverage makes it particularly valuable for scientific and engineering applications requiring precise Bessel function computations across diverse parameter spaces.