  The numerical evaluation of an individual Bessel or Hankel function of large
order and large argument is a notoriously problematic issue in physics.
Recurrence relations are inefficient when an individual function of high order
and argument is to be evaluated. The coefficients in the well-known uniform
asymptotic expansions have a complex mathematical structure which involves Airy
functions. For Bessel and Hankel functions, we present an adapted algorithm
which relies on a combination of three methods: (i) numerical evaluation of
Debye polynomials, (ii) calculation of Airy functions with special emphasis on
their Stokes lines, and (iii) resummation of the entire uniform asymptotic
expansion of the Bessel and Hankel functions by nonlinear sequence
transformations.
  In general, for an evaluation of a special function, we advocate the use of
nonlinear sequence transformations in order to bridge the gap between the
asymptotic expansion for large argument and the Taylor expansion for small
argument ("principle of asymptotic overlap"). This general principle needs to
be strongly adapted to the current case, taking into account the complex phase
of the argument. Combining the indicated techniques, we observe that it
possible to extend the range of applicability of existing algorithms. Numerical
examples and reference values are given.
