  An efficient approximate version of implicit Taylor methods for initial-value
problems of systems of ordinary differential equations (ODEs) is introduced.
The approach, based on an approximate formulation of Taylor methods, produces a
method that requires less evaluations of the function that defines the ODE and
its derivatives than the usual version. On the other hand, an efficient
numerical solution of the equation that arises from the discretization by means
of Newton's method is introduced for an implicit scheme of any order. Numerical
experiments illustrate that the resulting algorithm is simpler to implement and
has better performance than its exact counterpart.
