  We consider geometric numerical integration algorithms for differential
equations evolving on symmetric spaces. The integrators are constructed from
canonical operations on the symmetric space, its Lie triple system (LTS), and
the exponential from the LTS to the symmetric space. Examples of symmetric
spaces are n-spheres and Grassmann manifolds, the space of positive definite
symmetric matrices, Lie groups with a symmetric product, and elliptic and
hyperbolic spaces with constant sectional curvatures. We illustrate the
abstract algorithm with concrete examples. In particular for the n-sphere and
the n-dimensional hyperbolic space the resulting algorithms are very simple and
cost only O(n) operations per step.
