On Newton's method and Halley's method for the principal $p$th root of a matrix

Date
2010
Authors
Guo, Chun-Hua
Journal Title
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Volume Title
Publisher
Elsevier
Abstract

If A is a matrix with no negative real eigenvalues
and all zero eigenvalues of A are semisimple, the principal pth root of A can be computed by Newton's method or Halley's method, with a preprocessing procedure if necessary. We prove a new convergence result for Newton's method, and discover an interesting property of Newton's method and Halley's method in terms of series expansions.
We explain how the convergence of Newton's method and Halley's method can be improved when the eigenvalues of A are known or when A is a singular matrix.
We also prove new results on pth roots of M-matrices and H-matrices, and consider the application of Newton's method and Halley's method to find the principal pth roots of these special matrices.

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Citation
Linear Algebra Appl.
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