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Rational Minimax Iterations for Computing the Matrix pth Root

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Abstract

In a previous paper by the author, 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 pth 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 the 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.

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Acknowledgements

The author was supported in part by NSF Grant DMS-1703719.

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  1. Department of Mathematics, University of Hawaii at Manoa, Honolulu, USA

    Evan S. Gawlik

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  1. Evan S. Gawlik
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Correspondence to Evan S. Gawlik.

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Communicated by Wolfgang Dahmen.

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Gawlik, E.S. Rational Minimax Iterations for Computing the Matrix pth Root. Constr Approx 54, 1–34 (2021). https://doi.org/10.1007/s00365-020-09504-3

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