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A study of Schröder’s method for the matrix pth root using power series expansions

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Abstract

When A is a matrix with all eigenvalues in the disk |z - 1| < 1, the principal pth root of A can be computed by Schröder’s method, among many other methods. In this paper, we present a further study of Schröder’s method for the matrix pth root, through an examination of power series expansions of some scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schröder’s method, a monotonic convergence result when A is a nonsingular M-matrix, and a structure preserving result when A is a nonsingular M-matrix or a real nonsingular H-matrix with positive diagonal entries. We also explain how a convergence region larger than the disk |z - 1| < 1 can be obtained for Schröder’s method.

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Funding

The research of Chun-Hua Guo was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

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  1. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

    Chun-Hua Guo & Di Lu

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  1. Chun-Hua Guo
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  2. Di Lu
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Correspondence to Chun-Hua Guo.

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Guo, CH., Lu, D. A study of Schröder’s method for the matrix pth root using power series expansions. Numer Algor 83, 265–279 (2020). https://doi.org/10.1007/s11075-019-00681-2

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  • DOI: https://doi.org/10.1007/s11075-019-00681-2

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