A recursive algorithm for an efficient and accurate computation of incomplete Bessel functions

TL;DR

This paper presents a recursive algorithm that significantly improves the efficiency and accuracy of computing incomplete Bessel functions by reducing computational complexity from polynomial to linear time.

Contribution

The authors develop a recurrence relation for the sequences used in incomplete Bessel function computation, reducing complexity and enhancing accuracy over previous methods.

Findings

Achieves high accuracy in incomplete Bessel function computation.

Reduces computational complexity from O(n^4) to O(n).

Demonstrates effectiveness through error analysis and plotting.

Abstract

In a previous work, we developed an algorithm for the computation of incomplete Bessel functions, which pose as a numerical challenge, based on the $G_{n}^{(1)}$ transformation and Slevinsky-Safouhi formula for differentiation. In the present contribution, we improve this existing algorithm for incomplete Bessel functions by developing a recurrence relation for the numerator sequence and the denominator sequence whose ratio forms the sequence of approximations. By finding this recurrence relation, we reduce the complexity from ${\cal O}(n^4)$ to ${\cal O}(n)$. We plot relative error showing that the algorithm is capable of extremely high accuracy for incomplete Bessel functions.