  A new method for the numerical computation of the lifetimes of magnetic
states within harmonic transition state theory (HTST) has been developed. In
the simplest case, the system is described by a Heisenberg-like Hamiltonian
with short-range interaction. Calculations are performed in Cartesian
coordinates. Constraints on the values of magnetic moments are taken into
account using Lagrange multipliers. The pre-exponential factor in the Arrhenius
law in HTST is written in terms of the determinants of the Hessian of energy at
the minima and saddle points on the multidimensional energy surface. An
algorithm for calculating these determinants without searching for eigenvalues
of the Hessian but using recursive relations is proposed. The method allows
calculating determinants for systems containing millions of magnetic moments.
This makes it possible to calculate the pre-exponential factor and estimate the
lifetimes of micron-scale topological structures with atomic resolution, which
until now has been impossible using standard approaches. The accuracy of the
method is demonstrated by calculating 2D and 3D skyrmionic structures.
