On The Complexity Of The Modified Levenberg-Marquardt Algorithm For Nonlinear Equations

We propose a new modified Levenberg-Marquardt algorithm for the system ofnonlinear equationsF(x) = 0.The Levenberg-Marquardt parameter is chosen as ||Fk||δwhereδis a positive constant. The new algorithm is globally convergent under somesuitable conditions.Under the local error bound condition which is weaker than thenonsingularity,we show that the new algorithm converges superlinearly to the solutionforδ∈(0, 1),while quadratically for anyδ∈[1, 2].We also prove that the globalcomplexity bound of the new algorithm is O(-2).Numerical results show that thenew algorithm performs well for some singular problems.