@INPROCEEDINGS{ChevillardMezzarobba2013,
  author = {Chevillard, Sylvain and Mezzarobba, Marc},
  title = {{Multiple-Precision Evaluation of the Airy Ai Function with Reduced Cancellation}},
  booktitle = {{21st IEEE SYMPOSIUM on Computer Arithmetic}},
  year = {2013},
  editor = {Nannarelli, A. and Seidel, P.-M. and Tang, P.~T.~P.},
  pages = {175--182},
  address = {Los Alamitos, CA},
  month = {Apr},
  publisher = {IEEE Computer Society},
  hal = {hal-00767085},
  doi = {10.1109/ARITH.2013.33},
  abstract = {The series expansion at the origin of the Airy function Ai(x) is
                  alternating and hence problematic to evaluate for x > 0 due
                  to cancellation. Based on a method recently proposed by
                  Gawronski, Müller, and Reinhard, we exhibit two functions F
                  and G, both with nonnegative Taylor expansions at the origin,
                  such that Ai(x) = G(x)/F(x). The sums are now
                  well-conditioned, but the Taylor coefficients of G turn out
                  to obey an ill-conditioned three-term recurrence. We use the
                  classical Miller algorithm to overcome this issue. We bound
                  all errors and our implementation allows an arbitrary and
                  certified accuracy, that can be used, e.g., for providing
                  correct rounding in arbitrary precision.},
  keywords = {special functions, algorithm, numerical evaluation, arbitrary precision, Miller method, asymptotics, correct rounding, error bounds}
}