A Schur-Newton Method for the Matrix \lowercase{\boldmath$p$}th Root and its Inverse

Newton’s method for the inverse matrix $p$th root, $A^{-1/p}$, has the attraction that it involves only matrix multiplication. We show that if the starting matrix is $c^{-1}I$ for $c\in\mathbb{R}^+$ then the iteration converges quadratically to $A^{-1/p}$ if the eigenvalues of $A$ lie in a wedge-shaped convex set containing the disc $\{z: |z-c^p| < c^p\}$. We derive an optimal choice of $c$ for the case where $A$ has real, positive eigenvalues. An application is described to roots of transition matrices from Markov models, in which for certain problems the convergence condition is satisfied with $c=1$. Although the basic Newton iteration is numerically unstable, a coupled version is stable and a simple modification of it provides a new coupled iteration for the matrix $p$th root. For general matrices we develop a hybrid algorithm that computes a Schur decomposition, takes square roots of the upper (quasi-) triangular factor, and applies the coupled Newton iteration to a matrix for which fast convergence is guaranteed. The new algorithm can be used to compute either $A^{1/p}$ or $A^{-1/p}$, and for large $p$ that are not highly composite it is more efficient than the method of Smith based entirely on the Schur decomposition.