Newton’s method for the matrix square root

One approach to computing a square root of a matrix A is to apply Newton’s method to the quadratic matrix equation F ( X ) ≡ X 2 − A = 0 F(X) \equiv {X^2} - A = 0 . Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simplified iterations are numerically unstable. A further variant of Newton’s method for the matrix square root, recently proposed in the literature, is shown to be, for practical purposes, numerically stable.