A Levenberg-Marquardt Algorithm With Correction For Singular System Of Nonlinear Equations

The Levenberg-Marquardt method is a very important method for solving non-linear equations. The introduction of the positive parameter λkin the L-M method onone hand overcomes the difficulty when the Jacobian Matrix is singular or nearly sin-gular, on the other hand it makes the trial step away from the Moore-Penrose step. Inthis paper we consider introducing a correction step to make the new trial step closerto the Moore-Penrose step. We set the regularization parameter λδk=μkFkwithδ∈(0,2]. We show that the convergence rate of the L-M method with correctionis min{2,1+2δ} under the local error bound condition which is weaker than non-singularity. We use the inexact line search technique and trust region technique topresent the global convergence new L-M algorithms. Numerical results show that thealgorithm performs well.