Accelerated Sign-Function-Based Iterations for Matrix Square Roots with Fourth-Order Convergence

Motivated by the close relationship between the matrix square root and the matrix sign function, this paper develops a new high-order iterative framework for computing the principal square root of a matrix and its inverse. The proposed approach is derived from a rational fixed-point iteration associated with a scalar nonlinear equation and is extended consistently to the matrix setting. The method is shown to be globally convergent and to achieve fourth-order convergence. Numerical experiments demonstrate that the new scheme outperforms several classical and Padé-based methods.