  Efficient numerical methods to approximate the parallel transport operators
of the induced connection on a sub-bundle of a vector bundle are presented.
These methods are simpler than naive applications of a Runge--Kutta algorithm,
and have accuracy up to order 4. They have the desirable property of being
insensitive to choices of trivialisation of the sub-bundle. The methods were
developed in order to solve a problem of computing skyrmions using the
Atiyah--Manton--Sutcliffe and Atiyah--Drinfeld--Hitchin--Manin constructions,
but are applicable to a broader range of problems in computational geometry.
