  We introduce the use of the Fast Multipole Method (FMM) to speed up
gravitational lensing ray tracing calculations. The method allows very fast
calculation of ray deflections when a large number of deflectors, $N_*$, is
involved, while keeping rigorous control on the errors. In particular, we apply
this method, in combination with the Inverse Polygon Mapping technique (IPM),
to quasar microlensing to generate microlensing magnification maps with very
high workloads (high magnification, large size and/or high resolution) that
require a very large number of deflectors. Using, FMM-IPM, the computation time
can be reduced by a factor $\sim 10^5$ with respect to standard Inverse Ray
Shooting, making the use of this algorithm on a personal computer comparable to
the use of standard IRS on GPUs. We also provide a flexible web interface for
easy calculation of microlensing magnification maps using
FMM-IPM\footnote{http://gloton.ugr.es/microlensing/}. We exemplify the power of
this new method by applying it to some challenging interesting astrophysical
scenarios, including clustered primordial black holes, or extremely magnified
stars close to the giant arcs of galaxy clusters. We also show the
performance/use of FMM to calculate ray deflection for a halo resulting from
cosmological simulations composed by a large number ($N\gtrsim 10^7$) of
elements.
