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.

In this work, we revisit the two-step Jarratt method from the perspective of numerical stability. While high-order iterative schemes are often examined in terms of convergence rate and computational efficiency, their backward stability properties have received comparatively less attention. We begin by establishing the method’s strong consistency. Next, we provide a quantitative backward stability assessment within the standard floating-point arithmetic framework, deriving explicit perturbation bounds that show that the iteration errors remain proportional to machine precision. To support the theoretical findings, we present numerical experiments—including tests under finite-precision perturbations—as well as Python implementations and visualizations of the numerical examples. The results illustrate that the two-step Jarratt method not only achieves a high convergence order but also remains numerically robust for well-conditioned nonlinear systems.