A Curvilinear Search for Systems of Nonlinear Equations

We propose a curvilinear search for nonlinear systems of equations and path-following methods that are very nonlinear with the dominant part in the tangent space. The curvilinear search is very easy to implement and should be used with the (Gauss-)Newton method. At the cost of one function evaluation the curvature along the search direction can be reduced. For zero residual nonlinear least squares problems and nonlinear systems of equations the (Gauss-)Newton method with the curvilinear search method converges locally with q-cubic rate. For general use, well-deened Jacobians are required, which is the case for nonlinear Tikhonov regularization as well as for homotopy-continuation methods. Numerical results for artiicial problems and standard test problems are given.