A study of Schröder’s method for the matrix pth root using power series expansions

Abstract

When A is a matrix with all eigenvalues in the disk |z −1| < 1, the principal pth root of A can be computed by Schröder's method, among many other methods. In this paper we present a further study of Schröder's method for the matrix pth root, through an examination of power series expansions of some sequences of scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schröder's method, a monotonic convergence result when A is a nonsingular M -matrix, and a structure preserving result when A is a nonsingular M -matrix or a real nonsingular H-matrix with positive diagonal entries.