A hyperbolic tangent identity and the geometry of Padé sign function iterations
type: Item
instance of: scholarly article
title: A hyperbolic tangent identity and the geometry of Padé sign function iterations
zbMATH Open document ID: 0812.65035
DOI: 10.1007/BF02140677
published in: Numerical Algorithms
publication date: 1995-05-14T00:00:00Z
review text: This paper consists of two parts. First, the rational iterations obtained from certain Padé approximations associated with computing the matrix sign function are shown to be equivalent to iterations involving the hyperbolic tangent and its inverse. Using this equivalent formulation many results about these Padé iterations, such as global convergence, semigroup property, and partial fraction decompositions can be obtained easily. In the second part of the paper it is shown that the behaviour of points under the Padé iterations can be expressed, via Cayley transform, as the combined result of a completely regular iteration and a chaotic iteration. Thereby, the chaotic iteration takes the form of a multiplicative linear congruential random number generator.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 65C10
Mathematics Subject Classification ID: 65E05
zbMATH DE Number: 640736
zbMATH Keywords: rational iterations
zbMATH Keywords: Padé approximations
zbMATH Keywords: matrix sign function
zbMATH Keywords: hyperbolic tangent
zbMATH Keywords: Padé iterations
zbMATH Keywords: global convergence
zbMATH Keywords: partial fraction decompositions
zbMATH Keywords: Cayley transform
zbMATH Keywords: regular iteration
zbMATH Keywords: chaotic iteration
zbMATH Keywords: multiplicative linear congruential random number generator
author: Charles S. Kenney
author: Alan J. Laub
reviewed by: W. Gander
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02140677
OpenAlex ID: W2013346099
description: scientific article; zbMATH DE number 640736