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.

We develop a novel Steffensen-type iterative solver to solve nonlinear scalar equations without requiring derivatives. A two-parameter one-step scheme without memory is first introduced and analyzed. Its optimal quadratic convergence is then established. To enhance the convergence rate without additional functional evaluations, we extend the scheme by incorporating memory through adaptively updated accelerator parameters. These parameters are approximated by Newton interpolation polynomials constructed from previously computed values, yielding a derivative-free method with R-rate of convergence of approximately 3.56155. A dynamical system analysis based on attraction basins demonstrates enlarged convergence regions compared to Steffensen-type methods without memory. Numerical experiments further confirm the accuracy of the proposed scheme for solving nonlinear equations.