  We consider the problem of determination of a magnetic field from three
dimensional polarimetric neutron tomography data. We see that this is an
example of a non-Abelian ray transform and that the problem has a globally
unique solution for smooth magnetic fields with compact support, and a locally
unique solution for less smooth fields. We derive the linearization of the
problem and note that the derivative is injective. We go on to show that the
linearised problem about a zero magnetic field reduces to plane Radon
transforms and suggest a modified Newton Kantarovich method (MNKM) type
algorithm for the numerical solution of the non-linear problem, in which the
forward problem is re-solved but the same derivative used each time. Numerical
experiments demonstrate that MNKM works for small enough fields (or large
enough velocities) and show an example where it fails to reconstruct a slice of
the simulated data set. Lastly we show viewed as an optimization problem the
inverse problem is non-convex so we expect gradient based methods may fail.
