  A hybrid MPI+OpenMP strategy for parallelizing multiple precision Taylor
series method is proposed, realized and tested. To parallelize the algorithm we
combine MPI and OpenMP parallel technologies together with GMP library (GNU
miltiple precision libary) and the tiny MPIGMP library. The details of the
parallelization are explained on the paradigmatic model of the Lorenz system.
We succeed to obtain a correct reference solution in the rather long time
interval - [0,7000]. The solution is verified by comparing the results for
2700-th order Taylor series method and precision of ~ 3374 decimal digits, and
those with 2800-th order and precision of ~ 3510 decimal digits. With 192 CPU
cores in Nestum cluster, Sofia, Bulgaria, the 2800-th order computation was ~
145 hours with speedup ~ 105.

Mathematical Software;Distributed, Parallel, and Cluster Computing;Numerical Analysis;Dynamical Systems;Numerical Analysis