# Rational Minimax Iterations for Computing the Matrix $p$th Root

## TL;DR

This paper extends rational minimax iteration methods from the matrix square root to the matrix pth root for integers p ≥ 2, analyzing their convergence, stability, and error characteristics.

## Contribution

It generalizes Zolotarev's rational minimax iterations to compute matrix pth roots, addressing the lack of recursion for p > 2 and analyzing key properties.

## Key findings

- Iterations exhibit equioscillatory error behavior.
- Convergence order and stability are preserved for p > 2.
- Numerical examples confirm theoretical predictions.

## Abstract

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.