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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

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.