  We analyse three related preconditioned steepest descent algorithms, which
are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant
subspace computations, from the viewpoint of minimization of the corresponding
functionals, constrained by orthogonality conditions. We exploit the geometry
of the of the admissible manifold, i.e. the invariance with respect to unitary
transformations, to reformulate the problem on the Grassmann manifold as the
admissible set. We then prove asymptotical linear convergence of the algorithms
under the condition that the Hessian of the corresponding Lagrangian is
elliptic on the tangent space of the Grassmann manifold at the minimizer.
