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

作者：Gomilko Oleksandr; Greco Federico; Zietak Krystyna^*

来源：Numerical Linear Algebra with Applications, 2012, 19(3): 585-605.

DOI：10.1002/nla.786

• matrix sign function
• matrix sector function
• Pade family of iterations
• local convergence
• Pade approximant
• hypergeometric function
• root of hypergeometric polynomial
• reciprocal of power series
• log-concavity of sequence

In this paper we consider the Pade family of iterations for computing the matrix sign function and the Pade family of iterations for computing the matrix p-sector function. We prove that all the iterations of the Pade family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Pade family. We show that the iterations of the Pade 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 Pade approximants to the 1-z)-s, 0%26lt;s%26lt;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.

• 出版日期2012-5