  This work presents an algorithm for calculating high temperature series
expansions (HTSE) of Heisenberg spin models with spin $S=1/2$ in the
thermodynamic limit. This algorithm accounts for the presence of a magnetic
field. The paper begins with a comprehensive introduction to HTSE and then
focuses on identifying the bottlenecks that limit the computation of higher
order coefficients. HTSE calculations involve two key steps: graph enumeration
on the lattice and trace calculations for each graph. The introduction of a
non-zero magnetic field adds complexity to the expansion because previously
irrelevant graphs must now be considered: bridged graphs. We present an
efficient method to deduce the contribution of these graphs from the
contribution of sub-graphs, that drastically reduces the time of calculation
for the last order coefficient (in practice increasing by one the order of the
series at almost no cost). Previous articles of the authors have utilized HTSE
calculations based on this algorithm, but without providing detailed
explanations. The complete algorithm is publicly available, as well as the
series on many lattice and for various interactions.
