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arXiv · 1807.04251

A study of Schr\"oder's method for the matrix $p$th root using power series expansions

Chun-Hua Guo·Di Lu
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Abstract

When $A$ is a matrix with all eigenvalues in the disk $|z-1|<1$, the principal $p$th root of $A$ can be computed by Schr\"oder's method, among many other methods. In this paper we present a further study of Schr\"oder's method for the matrix $p$th root, through an examination of power series expansions of some sequences of scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schr\"oder'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.

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Chun-Hua Guo, Di Lu. 2018-07-11. A study of Schr\"oder's method for the matrix $p$th root using power series expansions. https://arxiv.org/abs/1807.04251

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