The author presents a convergence analysis of the Newton method for the inverse of the matrix $p$-th root, i.e., ${A}^{-1/p}$ for $p>2$. The $p$-th root arises in the computation of the matrix logarithm by the inverse scaling and squaring method. A quadratic convergence to ${A}^{-1/p}$ is proved provided the eigenvalues of $A$ lie in a wedge-shaped convex set within the disc $\{z:|z-{c}^{p}|<{c}^{p}\}$ with the initial matrix ${c}^{-1}I$, where $c$ is a positive scalar. It includes an optimal choice of $c$ for $A$ 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 ${A}^{1/p}$ or ${A}^{-1/p}$. It is more efficient than the Schur method of *M. I. Smith* [ibid. 24, No. 4, 971–989 (2003; Zbl 1040.65038)] for large $p$ 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.

##### MSC:

65F30 | Other matrix algorithms |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A51 | Stochastic matrices (MSC2000) |

15A24 | Matrix equations and identities |