A note on the Levenberg–Marquardt parameter

In [C. Ma, L. Jiang, Some research on Levenberg–Marquardt method for the nonlinear equations, Appl. Math. Comput. 184 (2007) 1032–1040], the LM parameter at the k  th iteration is chosen as λk=θ‖Fk‖2+(1-θ)‖JkTFk‖2 where F is the residual function, J is the Jacobi of F  , and θ∈[0,1]θ∈[0,1] is a constant. In this note, we point out that the LM parameter can be any combination of ‖Fk‖2‖Fk‖2 and ‖JkTFk‖2 provided it is positive. Furthermore, we give a more general choice of the LM parameter, and show that the LM method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity.