The Faster Methods for Computing Bessel Functions of the First Kind of an Integer Order with Application to Graphic Processors

Abstract

Algorithms for fast computations of the Bessel functions of an integer order with required accuracy are considered. The domain of functions is split into two intervals: 0 ≤ x ≤ 8 and x > 8. For the finite interval, expansion in the Chebyshev polynomials is applied. An optimal algorithm for computing functions J₀(x) and J₁(x) is presented. It is shown that the sufficient number of mathematical operations equals 15 for computing the function J₀(x) and 16 for computing the function J₁(x) in the interval x ≤ 8 with the approximation error O(10⁻⁶). Several algorithms for approximation of the functions J_n(x) at n > 1 are presented. The increase in speed of computations of the Bessel functions obtained through using our in-house methods in place of the Toolkit library is evaluated. Graphs showing the improvement of performance are presented.