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,
*J*_{ν}(x),
is performed.

**Accuracy and Relative Error**

Depending on the real parameters pair
(*ν,x*), different
algorithms are used to achieve an optimal
mix of accuracy and function evaluation time. This is achieved by utilizing
the computer's 80-bit FPU registers that can handle binary exponents of up to
2^{±14} and a mantissa of 64 bits. Values
exceeding these extremes are set to infinity and zero, respectively.

The relative error of the function
*f ≡ J*_{ν}(x)
is defined as
*ε = |Δf/f|*,
where *Δf*
is the absolute error to be *visually* determined. The analysis at the
convergence-point
*f = J*_{100}(95)
is captured below: