  The time integration of semilinear parabolic problems by exponential methods
of different kinds is considered. A new algorithm for the implementation of
these methods is proposed. The algorithm evaluates the operators required by
the exponential methods by means of a quadrature formula that converges like
$O(e^{-cK/\ln K})$, with $K$ the number of quadrature nodes. The algorithm
allows also the evaluation of the associated scalar mappings and in this case
the quadrature converges like $O(e^{-cK})$. The technique is based on the
numerical inversion of sectorial Laplace transforms. Several numerical
illustrations are provided to test the algorithm.
