A modified Newton method with cubic convergence: the multivariate case
type: Item
instance of: scholarly article
title: A modified Newton method with cubic convergence: the multivariate case
zbMATH Open document ID: 1059.65044
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-16T00:00:00Z
review text: Let \(m,n\in\mathbb{N}\), \(F: \mathbb{R}^m\to\mathbb{R}^n\) be a sufficiently differentiable mapping, \(x_0, x_n\in\mathbb{R}^m\), \(F_n:= F(x_n)\), and \(F_n':= F'(x_n)\) the Jacobi matrix at \(x_n\). The author considers the following multipoint method for the approximate computation of a zero of \(F\): \[ \text{For }n=1,2,\dots: F_n+ F_n'c_n= 0\Rightarrow c_n,\;F_n+ F'(x_n+\tfrac12 c_n) d_n= 0\Rightarrow d_n,\;x_{n+1}:= x_n+ d_n.\tag{+} \] He shows that (+) under appropiate conditions converges locally with order three to a simple zero of \(F\). Moreover, he uses two nontrivial examples to compare the computational results of (+) with those of Newton's method but not, e.g. with the respective results of a similar multipoint method of -- under appropiate assumptions -- order three which differs from (+) in the second equation which is replaced by \(F(x_n+ c_n)+ F_n'c_n= 0\) (and then \(x_{n+1}:= x_n+ c_n+ d_n)\). In this case, the two systems of linear equations in question to be solved in each iteration step have the same coefficient matrix, respectively [Comp., e.g., \textit{W. E. Bosarge, jun.} and \textit{P. L. Falb}, J. Optimization Theory Appl. 4, 155--166 (1969; Zbl 0172.18703)]. Finally, the author shortly discusses how certain numerical difficulties -- e.g., if the Jadobi matrix at the zero of \(F\) is singular -- could be dealt with, and under what conditions (+) may converge globally.
reviewed by: Boro Döring
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 2091432
zbMATH Keywords: systems of nonlinear equations
zbMATH Keywords: multipoint method of order three
zbMATH Keywords: generalized Bloch equation
zbMATH Keywords: convergence
zbMATH Keywords: comparison of methods
zbMATH Keywords: rootfinding
zbMATH Keywords: Newton method
zbMATH Keywords: coupled algebraic equations
zbMATH Keywords: affine-invariant Newton technique
zbMATH Keywords: hybrid method
author: Herbert H. H. Homeier
describes a project that uses: Maple
describes a project that uses: GIANT
MaRDI profile type: Publication
full work available at URL: j.cam.2003.12.041
OpenAlex ID: W2073765564
cites work: Q4002478
cites work: Q4002421
cites work: Q4954490
cites work: Recent advances in solvers for nonlinear algebraic equations
cites work: Q4433920
cites work: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting
cites work: Q4070229
cites work: A modified Newton method for rootfinding with cubic convergence.
cites work: Q4403648
cites work: Q4386703
DOI: 10.1016/J.CAM.2003.12.041
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description: scientific article; zbMATH DE number 2091432