The authors consider iteration methods for finding approximate inverse preconditioners to the inverse of a given -matrix . The iterative methods aim at the minimization of the functional
on the space of all -matrices, where denotes the Frobenius norm, is the Euclidean norm in , and are the th columns of the identity matrix and of the matrix , respectively. The authors propose and analyze several iterative methods (Newton, MR, GMRES) with several modifications (numerical dropping in the iterates or in the search directions, self-preconditioning etc.). The different techniques are compared numerically on several examples taken from the well-known Harwell-Boeing collection and from matrices generated by the fluid dynamics analysis package FIDAP.