Efficient Calculation of Modified Bessel Functions of the First Kind, $I_{\nu} (z)$, for Real Orders and Complex Arguments: Fortran Implementation with Double and Quadruple Precision

TL;DR

This paper introduces a robust, efficient Fortran algorithm for computing the modified Bessel function of the first kind, supporting double and quadruple precision, with improved accuracy, stability, and performance over previous methods.

Contribution

The paper presents a new algorithm and Fortran implementation that surpasses Algorithm 644 in efficiency and stability, especially for large parameters and high-precision calculations.

Findings

Execution time reduced to 54%-80% of Algorithm 644

Supports both double and quadruple precision

Validated accuracy against high-precision Maple calculations

Abstract

We present an efficient self-contained algorithm for computing the modified Bessel function of the first kind $I_{\nu}(z)$, implemented in a robust Fortran code supporting double and quadruple (quad) precision. The algorithm overcomes the limitations of Algorithm 644, which is restricted to double precision and applies overly conservative underflow and overflow thresholds, leading to failures in large parameter regions. Accuracy is validated against high-precision Maple calculations, and benchmarking shows execution time reductions to 54%-80% of Algorithm 644 (in double precision). Quad precision enhances numerical stability and broadens the domain of computations, making the implementation well suited for high-precision applications in physics and engineering. This work also provides a foundation for the development of efficient algorithms for other Bessel functions.