Aitken's and Steffensen's accelerations in several variables
type: Item
instance of: scholarly article
title: Aitken's and Steffensen's accelerations in several variables
zbMATH Open document ID: 0712.65037
DOI: 10.1007/BF01385782
author: Yves Nievergelt
published in: Numerische Mathematik
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133551
review text: Aitken's acceleration of scalar sequences extends to sequences of vectors that behave asymptotically as iterations of a linear transformation. However, the minimal and characteristic polynomials of that transformation must coincide (but the initial sequence of vectors need not converge) for a numerically stable convergence of Aitken's acceleration to occur. Similar results hold for Steffensen's acceleration of the iterations of a function of several variables. First, the iterated function need not be a contracting map in any neighbourhood of its fixed point. Instead, the second partial derivatives need only remain bounded in such a neighbourhood for Steffensen's acceleration to converge quadratically, even if ordinary iterations diverge. Second, at the fixed point the minimal and characteristic polynomials of the Jacobian matrix must coincide to ensure a numerically stable convergence. By generalizing the work that Noda did on the subject between 1981 and 1986, the results presented here explain the numerical observations reported by Henrici in 1964 and 1982.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65B05
zbMATH DE Number: 4172264
zbMATH Keywords: nonlinear systems
zbMATH Keywords: Aitken's acceleration
zbMATH Keywords: iterations of a linear transformation
zbMATH Keywords: convergence
zbMATH Keywords: Steffensen's acceleration
MaRDI profile type: Publication
cites work: Q3359644
cites work: Q5528098
cites work: Q3707245
cites work: Numerical Linear Algebra
cites work: The Steffensen iteration method for systems of nonlinear equations
cites work: The Aitken-Steffensen formula for systems of nonlinear equations. III
OpenAlex ID: W1979173135
description: scientific article; zbMATH DE number 4172264