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

When A is a matrix with all eigenvalues in the disk |z - 1| < 1, the principal pth root of A can be computed by Schroder’s method, among many other methods. In this paper, we present a further study of Schroder’s method for the matrix pth root, through an examination of power series expansions of some scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schroder’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. We also explain how a convergence region larger than the disk |z - 1| < 1 can be obtained for Schroder’s method.