Steffensen’s approach for the matrix square root

In this study, we propose two novel iterative methods for computing the matrix square root. The first method employs Steffensen’s iteration and we provide a detailed analysis of its quadratic convergence and stability. The second method explores the mathematical relationship between the matrix sign and square root functions, applying a specialized Steffensen-based iteration tailored for the matrix sign function. Both approaches are compared with the well-known Denman-Beavers iteration from the literature. The comparison evaluates their accuracy, computational efficiency and ability to preserve structural properties.