Rational Minimax Iterations for Computing the Matrix $p$th Root
Provides a novel method for computing matrix p-th roots, which is a known bottleneck in numerical linear algebra, but the extension is incremental as it builds directly on prior work for the square root.
This paper generalizes rational minimax iterations for computing the matrix square root to the matrix p-th root for p≥2, showing that key properties like equioscillatory error, order of convergence, and stability carry over despite the lack of a recursion for approximants. Numerical examples confirm the theoretical predictions.
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.