A Padé family of iterations for the matrix sign function and related problems |
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Authors: | Oleksandr Gomilko Federico Greco Krystyna Ziȩtak |
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Institution: | 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;3. Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy;4. Institute of Mathematics and Computer Science, Wroc?aw University of Technology, 50‐370 Wroc?aw, Poland |
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Abstract: | 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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Keywords: | matrix sign function matrix sector function Padé family of iterations local convergence Padé approximant hypergeometric function root of hypergeometric polynomial reciprocal of power series log‐concavity of sequence |
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