Convergence and uniqueness properties of Gauss-Newton's method
The problem is to find a generalized solution for the equation \[ f(x)= 0,\tag{1} \] where \(f: \mathbb{R}^n\to \mathbb{R}^m\) is a nonlinear Fréchet-differentiable mapping and \(m\geq n\). The authors investigate the convergence of the Gauss-Newton method \[ x_{n+1}= x_n- [f'(x_n)^T f'(x_n)]^{-1} f'(x_n)^T f(x_n),\quad n= 0,1,2,\dots\tag{2} \] for finding the least squares solution to (2) under generalized Lipschitz conditions with \(L\) average. Some convergence theorems are proved as well as estimates for radius of the convergence ball and the uniqueness ball of the solution of (1) are derived. Some earlier and recent results are extended and improved sharply.
scientific article; zbMATH DE number 2142367