Convergence of Newton-like methods for singular operator equations using outer inverses

M. Z. Nashed, Xiaojun Chen

Research output: Journal article publicationJournal articleAcademic researchpeer-review

148 Citations (Scopus)


We present a (semilocal) Kantorovich-type analysis for Newton-like methods for singular operator equations using outer inverses. We establish sharp generalizations of the Kantorovich theory and the Mysovskii theory for operator equations when the derivative is not necessarily invertible. The results reduce in the case of an invertible derivative to well-known theorems of Kantorovich and Mysovskii with no additional assumptions, unlike earlier theorems which impose strong conditions. The strategy of the analysis is based on Banach-type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses (to a given bounded linear operator) admits selections that behave like bounded linear inverses, in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator. We give two examples to illustrate our results and compare them with earlier results, and another numerical example to relate our results to computational issues.
Original languageEnglish
Pages (from-to)235-257
Number of pages23
JournalNumerische Mathematik
Issue number1
Publication statusPublished - 1 Dec 1993
Externally publishedYes


  • Mathematics Subject Classification (1991): 65J15

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • General Mathematics


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