On the Gauss-Newton method

We provide a new semilocal convergence analysis of the Gauss-Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243-252, 1966; Chen and Nashed, Numer. Math. 66:235-257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1-10, 1979; Guo, J. Comput. Math. 25:231-242, 2007; Häuler, Numer. Math. 48:119-125, 1986; Hu et al., J. Comput. Appl. Math. 219:110-122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243-252, 1966; Chen and Nashed, Numer. Math. 66:235-257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1-10, 1979; Guo, J. Comput. Math. 25:231-242, 2007; Häuler, Numer. Math. 48:119-125, 1986; Hu et al., J. Comput. Appl. Math. 219:110-122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982). © 2010 Korean Society for Computational and Applied Mathematics.