  In [E. S. Gawlik, Zolotarev iterations for the matrix square root, arXiv
preprint 1804.11000, (2018)], a family of iterations for computing the matrix
square root was constructed by exploiting a recursion obeyed by Zolotarev's
rational minimax approximants of the function $z^{1/2}$. The present paper
generalizes this construction by deriving rational minimax iterations for the
matrix $p^{th}$ root, where $p \ge 2$ is an integer. The analysis of these
iterations is considerably different from the case $p=2$, owing to the fact
that when $p>2$, rational minimax approximants of the function $z^{1/p}$ do not
obey a recursion. Nevertheless, we show that several of the salient features of
the Zolotarev iterations for the matrix square root, including equioscillatory
error, order of convergence, and stability, carry over to case $p>2$. A key
role in the analysis is played by the asymptotic behavior of rational minimax
approximants on short intervals. Numerical examples are presented to illustrate
the predictions of the theory.
