BibTeX
@ARTICLE{
         Barton1980OTs,
       AUTHOR = "Barton, David",
       TITLE = "On {T}aylor series and stiff equations",
       JOURNAL = "ACM Trans. Math. Software",
       VOLUME = "6",
       NUMBER = "3",
       YEAR = "1980",
       PAGES = "280--294",
       REFERRED = "MR 82e 65078; [Corliss1982SOD]; [Gupt85a]; [Halin1983Tao].",
       KEYWORDS = "point algorithm; Taylor series; stiff; automatic differentiation.",
       ABSTRACT = "Surveys the work of Barton, Willers, and Zahar on Taylor series methods. Gives a
         predictor-corrector algorithm based on Taylor series. Parts: \begin{enumerate} \item
         Estimate the local error by matching the series at $ t_r $ and the series at $ t_{r+1} $ at the
         midpoint. \item Predict stepsize (iteratively) to achieve a fraction of the desired tolerance.
         This is very expensive. \item The predictor formula is an ordinary Taylor polynomial. The
         corrector formula uses eigenvalues of the Jacobian to distort the series at $ t_{r+1} $ to compute
         the actual local contributions of the transients. \item Determine the order of the series.
         \item Detect when a transient has died away, or when it has reappeared.\end{enumerate} A
         preprocessor written in PL/1 uses automatic differentiation to write a Fortran program which uses
         recurrence relations to compute the series."
}