Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

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 this, 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.