Geodesic
Local convergence theorems of Newton’s method for nonlinear equations using outer or generalized inverses
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 603-614.

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We provide local convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations.
Classification : 47H17, 47J25, 49D15, 65J15
Keywords: Newton’s method; Banach space; Fréchet-derivative; local convergence; outer inverse; generalized inverse 
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     author = {Argyros, Ioannis K.},
     title = {Local convergence theorems of {Newton{\textquoteright}s} method for nonlinear equations using outer or generalized inverses},
     journal = {Czechoslovak Mathematical Journal},
     pages = {603--614},
     publisher = {mathdoc},
     volume = {50},
     number = {3},
     year = {2000},
     mrnumber = {1777480},
     zbl = {1079.65528},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2000__50_3_a12/}
}
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Argyros, Ioannis K. Local convergence theorems of Newton’s method for nonlinear equations using outer or generalized inverses. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 603-614. http://geodesic.mathdoc.fr/item/CMJ_2000__50_3_a12/