Bessel functions' analytical and numerical characteristics are well documented.[1,2] When getting into the particulars of the numerical evaluation of different algorithms, however, error analyses are sometimes cumbersome or only partially available. Using the capabilities of computer graphics, it is possible to visualize errors and estimate their magnitude "empirically". To demonstrate the power of this method, graphical error analysis of the procedure _JaX [3] that evaluates the Bessel Function of the First Kind, Jν(x), is performed.
Accuracy and Relative Error
The relative error of the function f ≡ Jν(x) is defined as ε = |Δf/f|, where Δf is the absolute error to be visually determined. The analysis at the convergence-point f = J100(95) is captured below: |
Convergence-point at J100(95)
From this graph one finds that ε ≤ 4x10-16. The time required to evaluate the function,[4] left and right of x = 95, is approximately 25 µs and 3 µs respectively (1 µs = 0.000001 seconds). Function Evaluation Map The procedure _JaX evaluates Jν(x) for a wide range of arguments and indices. The Function Evaluation Map below highlights several results:
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