A family of iterations for the sector function based on the Padé 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 Padé table preserve the structure of a group of automorphisms associated with a scalar product.The regions of convergence of the Padé 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.
Two matrix approximation problems are considered: approximation of a rectangular complex matrix by subunitary matrices with respect to unitarily invariant norms and a minimal rank approximation with respect to the spectral norm. A characterization of a subunitary approximant of a square matrix with respect to the Schatten norms, given by Maher, is extended to the case of rectangular matrices and arbitrary unitarily invariant norms. Iterative methods, based on the family of Gander methods and on Higham's scaled method for polar decomposition of a matrix, are proposed for computing subunitary and minimal rank approximants. Properties of Gander methods are investigated in details. (2000): 65F30, 15A18.
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