The author presents a convergence analysis of the Newton method for the inverse of the matrix -th root, i.e., for . The -th root arises in the computation of the matrix logarithm by the inverse scaling and squaring method. A quadratic convergence to is proved provided the eigenvalues of lie in a wedge-shaped convex set within the disc with the initial matrix , where is a positive scalar. It includes an optimal choice of for having real positive eigenvalues.
The analysis leads to a hybrid algorithm for general matrices employing a Schur decomposition, matrix square roots of the upper (quasi-) triangular roots, and two coupled versions of the Newton iteration. The algorithm is stable and computes either or . It is more efficient than the Schur method of M. I. Smith [ibid. 24, No. 4, 971–989 (2003; Zbl 1040.65038)] for large that are not highly composite. An application is supplied to roots of transition matrices for a time-homogeneous continuous-time Markov process. Numerically illustrative experiments are supplied.