A generalized Steffensen's method for matrix sign function

A direct generalized Steffensen's method is proposed for solving the quadratic matrix equation F(X) ≔ X2 − I = 0. In this way, when the matrix A is nonsingular, we derive a new numerical scheme for finding matrix sign function. The local and global convergence of the proposed generalization is brought up via the concept of basins of attraction. We also report some numerical results of the proposed method, which show its applicability.