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.

中文翻译：

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用于计算矩阵 pth 根的有理极小极大迭代

在 [ES Gawlik, Zolotarev 矩阵平方根迭代，arXiv 预印本 1804.11000, (2018)] 中，通过利用 Zolotarev 函数 $z^ 的有理极小极大逼近所遵循的递归构造了一系列用于计算矩阵平方根{1/2}$。本文通过推导矩阵 $p^{th}$ 根的有理极小极大迭代来概括这种构造，其中 $p \ge 2$ 是一个整数。这些迭代的分析与 $p=2$ 的情况有很大不同，因为当 $p>2$ 时，函数 $z^{1/p}$ 的有理极小极大逼近不服从递归。尽管如此，我们证明了矩阵平方根的 Zolotarev 迭代的几个显着特征，包括等振误差、收敛阶数和稳定性，会延续到情况 $p>2$。分析中的一个关键作用是由有理极小极大逼近在短时间间隔内的渐近行为发挥的作用。给出了数值例子来说明理论的预测。