A Padé family of iterations for the matrix sector function and the matrix pth root

A family of iterations for the sector function based on the Pade table of a certain hypergeometric function is derived and investigated. This generalizes a result of Kenney and Laub for the sign function and yields a whole family of iterative methods for computing the matrix pth root. It is proved that the iterations for the matrix sector function corresponding to the main diagonal of the Pade table preserve the structure of a group of automorphisms associated with a scalar product. The regions of convergence of the Pade iterations for the matrix sector function are investigated theoretically and experimentally. Certain conjectures formulated on the regions of convergence have been verified in particular cases. Copyright © 2009 John Wiley & Sons, Ltd.

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