On the Newton Method for the Matrix Pth Root

Authors:
Bruno Iannazzo
Affiliations:
-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2006

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Cited 3

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Abstract

Stable versions of Newton's iteration for computing the principal matrix pth root A1/p of an n x n matrix A are provided. In the case in which X0 is the identity matrix, it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real parts. Based on these results we provide a general algorithm for computing the principal pth root of any matrix A having no nonpositive real eigenvalues. The algorithm has quadratic convergence, is stable in a neighborhood of the solution, and has a cost of O(n3 log p) operations per step. Numerical experiments and comparisons are performed.