  We focus on inverse preconditioners based on minimizing $F(X) =
1-\cos(XA,I)$, where $XA$ is the preconditioned matrix
  and $A$ is symmetric and positive definite. We present and analyze
gradient-type methods to minimize $F(X)$
  on a suitable compact set. For that we use the geometrical properties of the
non-polyhedral
  cone of symmetric and positive definite matrices, and also the special
properties of $F(X)$ on the feasible set.
  Preliminary and encouraging numerical results are also presented
  in which dense and sparse approximations are included.
