On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption

Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using mu(k)=parallel toF(x(k))parallel to(delta) where delta is an element of [1,2], instead of mu(k)=parallel toF(x(k))parallel to(2) as the Levenberg-Marquardt parameter. If parallel toF(x)parallel to provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x(k)} generated by the new method converges to a solution quadratically, which is stronger than dist(x(k),X*) --> 0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.