A Levenberg–Marquardt algorithm with correction for singular system of nonlinear equations

In this paper we propose a new LM algorithm for the singular system of nonlinear equations F(x)=0F(x)=0, where F(x):Rn→RnF(x):Rn→Rn is continuously differentiable and F′(x)F′(x) is Lipschitz continuous. At every iteration, not only a LM step but also a correction step are computed. The LM parameter is taken as μk‖Fk‖δμk‖Fk‖δ, where δ∈(0,2]δ∈(0,2] and μkμk is updated by the trust region technique. Under the local error bound condition which is weaker than nonsingularity, we prove the convergence rate of the algorithm is min{2,1+2δ}min{2,1+2δ}. Numerical results show that the new LM algorithm performs very well.