The problem of finding a geodesic joining given points

*x*_{0},

*x*_{1}in a connected complete Riemannian manifold requires much more effort than
determining a geodesic from initial data. Boundary value problems of this type
are sometimes solved using shooting methods, which work best when good initial
guesses are available, especially when

*x*_{0},

*x*_{1} are nearby. Galerkin
methods have their drawbacks too. The situation is much more difficult with
general variational problems, which is why
we focus on the Riemannian case.

Our global algorithm is very simple to implement, and works well in
practice, with no need for an initial guess. The proof of convergence is
elementary and
very carefully stated, with a view to possible generalizations later on.
We have in mind the much larger class of interesting problems
arising in optimal control especially from mechanical engineering.