Some research on Levenberg–Marquardt method for the nonlinear equations

Levenberg–Marquardt method is one of the most important methods for solving systems of nonlinear equations. In this paper, we consider the convergence of a new Levenberg–Marquardt method (i.e. λk=θ‖Fk‖+(1-θ)‖JkTFk‖, where θ ∈ [0, 1] is a real parameter) for solving a system of singular nonlinear equations F(x) = 0, where F is a mapping from Rn into Rm. We will show that if ∥F(x)∥ provides a local error bound which is weaker than the condition of nonsingular for the system of nonsingular for the system of nonlinear equations, the sequence generated by the new Levenberg–Marquardt method convergence to a point of the solution set X∗ quadratically. Numerical experiments and comparisons are reported.