Monotonicity and positivity of coefficients of power series expansions associated with Newton and Halley methods for the matrix pth root

If A is a matrix with all eigenvalues in the disk |z−1|<1, the principal pth root of A can be computed by Newton's method or Halley's method. The study of Newton's method and Halley's method for the matrix pth root can be done through a study of power series expansions of some sequences of scalar functions. In this paper, we prove monotonicity results for the coefficients in the power series expansions for both Newton's method and Halley's method, and for all integer p≥2. The sign patterns of these coefficients can be seen directly from the monotonicity results. We then use these monotonicity results and their consequences to obtain some new error estimates in the matrix case, to obtain a monotonic convergence result when A is a nonsingular M-matrix, and to obtain a structure preserving result when A is a nonsingular M-matrix or a real nonsingular H-matrix with positive diagonal entries.