Local convergence theorems for Newton methods (Q2731126)

Inexact Newton-like methods are discussed for solving the operator equation \(F(x)=0\), where \(F\) is a mapping defined on an open convex subset \(D\) of a Banach space \(E_1\) with values in a Banach space \(E_2\). These are iterative methods of the generic form NEWLINE\[NEWLINE x_{n+1}=x_n+s_n,\quad n=0, 1,2 \cdots NEWLINE\]NEWLINE and \(s_n\) is solution of the linear problem NEWLINE\[NEWLINE A(x_n) s_n=-F(x_n)+r_n. NEWLINE\]NEWLINE Here \(A(x)\in L(E_1,E_2)\) for \(x\in D\) is an approximate of \(F'(x)\) and \(r_n\in D,~ n=0,1,2,\cdots \) are small perturbations. If \(r_n\equiv 0, ~n=0, 1, 2,\cdots\), then these are the Newton-like methods. If \(A(x)=F(x)\), they reduce to the inexact Newton methods. Convergence and radius of convergence of inexact Newton-like methods are analyzed and compared with other results in the literature.