Convergence of Newton-like methods for singular operator equations using outer inverses
For singular \(F\)-differentiable nonlinear operators between Banach spaces convergence theorems of (semilocal) Kantorovich-type are proved for Newton-like methods \(x_{k+1} = x_ k - A(x_ k)^ \#F(x_ k)\), \(k \geq 0\). Here \(A(x)\) is a bounded linear operator approximating \(F'(x)\) and \(A(x)^ \#\) a bounded outer inverse of \(A(x)\); that is, \(A(x)^ \# A(x) A(x)^ \# = A(x)^ \#\). The results reduce precisely to the well- known Kantorovich and Mysovskii theorems for the case of invertible derivatives. The analysis utilizes Banach-type lemmas and perturbation bounds for outer inverses which show that (for bounded linear operators) the set of outer inverses contains elements that behave similar to bounded linear inverses. Several examples compare the results with related known results and also addresses some computational issues. Various interesting remarks relevant to this setting conclude the paper.
semilocal Kantorovich-type convergence theorems
scientific article; zbMATH DE number 569183
scientific article