A Padé family of iterations for the matrix sign function and related problems

Authors

  • Oleksandr Gomilko,

    1. Faculty of Mathematics and Computer Science, Nicolas Copernicus University, 87-100 Toruń, Poland
    2. Institute of Mathematics, Polish Academy of Sciences, 00-956 Warszawa, Poland
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  • Federico Greco,

    1. Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy
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  • Krystyna Ziȩtak

    Corresponding author
    1. Institute of Mathematics and Computer Science, Wrocław University of Technology, 50-370 Wrocław, Poland
    • Institute of Mathematics and Computer Science, Wrocław University of Technology, 50-370 Wrocław, Poland
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SUMMARY

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p-sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p-sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1−z)−σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log-concavity property. Copyright © 2011 John Wiley & Sons, Ltd.

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