Approximate Newton methods and homotopy for stationary operator equations

Joseph W. Jerome*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


A quadratically convergent algorithm based on a Newton-type iteration is defined to approximate roots of operator equations in Banach spaces. Fréchet derivative operator invertibility is not required; approximate right inverses are used in a neighborhood of the root. This result, which requires an initially small residual, is sufficiently robust to yield existence; it may be viewed as a generalized version of the Kantorovich theorem. A second algorithm, based on continuation via single, Euler-predictor-Newton-corrector iterates, is also presented. It has the merit of controlling the residual until the homotopy terminates, at which point the first algorithm applies. This method is capable of yielding existence of a solution curve as well. An application is given for operators described by compact perturbations of the identity.

Original languageEnglish (US)
Pages (from-to)271-285
Number of pages15
JournalConstructive Approximation
Issue number1
StatePublished - Dec 1985


  • AMS classification: 41A25, 47H15, 47H17, 40A05, 65J15, 65M15
  • Approximate Newton method
  • Bootstrapping lemma
  • Continuation, Eigenvalues
  • Euler-predictor
  • Newton-corrector
  • R-quadratic convergence
  • Solution set

ASJC Scopus subject areas

  • Analysis
  • General Mathematics
  • Computational Mathematics


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