A continuation procedure based on projected Newton steps
type: Item
instance of: scholarly article
title: A continuation procedure based on projected Newton steps
zbMATH Open document ID: 0695.65039
DOI: 10.1007/BF02243802
published in: Computing
publication date: 1989-01-01T00:00:00Z
review text: Consider the fixed point equation \(u=GKu\) with a linear operator K on a real Hilbert space into a real Banach space and a nonlinear operator G. Hammerstein integral equations can be transformed into this (a bit unusual) form. The author considers an imbedding T(v,t) for \(t\in [0,1]\) such that \(T(Kz,1)=GKz.\) A sequence of pairs \((z_ n,t_ n)\) is constructed such that the inequality \(\| z_{n+1}-T(Kz_{n+1},t_{n+1})\| \leq \| z_ n-T(Kz_ n,t_ n)\| +(t_{n+1}-t_ n)\alpha\) with a prescribed real constant \(\alpha >0\) holds. The construction is on the basis of projected Newton steps and it is proved that there exists a natural number N such that \(\| z_ N-GKz_ N\| \leq \alpha\) holds. It is shown that the differences \(t_{n+1}-t_ n\) are uniformly bounded below so that the interval [0,1] is eventually covered and the endpoint \(t=1\) is reached. A central assumption lies on the error term of the forward finite difference quotient for T(Kz,t) with respect to (z,t). It allows the author to predict the next \(t_{n+1}\) from \(t_ n\). A worked numerical example concludes the paper.
reviewed by: E. Bohl
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 4139211
zbMATH Keywords: continuation method
zbMATH Keywords: Newton's method
zbMATH Keywords: fixed point equation
zbMATH Keywords: Hilbert space
zbMATH Keywords: Banach space
zbMATH Keywords: Hammerstein integral equations
zbMATH Keywords: numerical example
author: Igor Moret
MaRDI profile type: Publication
cites work: Q4191525
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description: scientific article; zbMATH DE number 4139211