Functions preserving matrix groups and iterations for the matrix square root

For which functions f does A ε script G sign ⇒ f(A) ε script G sign when script G sign is the matrix automorphism group associated with a bilinear or sesquilinear form? For example, if A is symplectic when is f(A) symplectic? We show that group structure is preserved precisely when f(A -1) = f(A) -1 for bilinear forms and when f(A -*) = f(A) -* for sesquilinear forms. Meromorphic functions that satisfy each of these conditions are characterized. Related to structure preservation is the condition f(Ā) = f(A)̄, and analytic functions and rational functions satisfying this condition are also characterized. These results enable us to characterize all meromorphic functions that map every script G sign into itself as the ratio of a polynomial and its "reversal," up to a monomial factor and conjugation. The principal square root is an important example of a function that preserves every automorphism group script G sign. By exploiting the matrix sign function, a new family of coupled iterations for the matrix square root is derived. Some of these iterations preserve every script G sign all of them are shown, via a novel Fréchet derivative-based analysis, to be numerically stable. A rewritten form of Newton's method for the square root of A ε scripy G sign is also derived. Unlike the original method, this new form has good numerical stability properties, and we argue that it is the iterative method of choice for computing A 1/2 when A ε script G sign. Our tools include a formula for the sign of a certain block 2 ×2 matrix, the generalized polar decomposition along with a wide class of iterations for computing it, and a connection between the generalized polar decomposition of I + A and the square root of A ε script G sign. © 2005 Society for Industrial and Applied Mathematics.