Steffensen’s approach for the matrix square root

Abstract

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.