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

Authors:
Jin-yan Fan;Ya-xiang Yuan
Affiliations:
State Key Laboratory of Scientific Engineering Computing, Institute of Computational Mathematics and Scientific Engineering Computing, The Academy of Mathematics and Systems Sciences, Chinese Acad ...;State Key Laboratory of Scientific Engineering Computing, Institute of Computational Mathematics and Scientific Engineering Computing, The Academy of Mathematics and Systems Sciences, Chinese Acad ...
Venue:
Computing
Year:
2005

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Abstract

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 µk = |F(xk|2-where δ ∈(1,2) instead of µk = F(xk)2 as the Levenberg-Marquardt parameter. If |F(x)| provides a local error bound for the system of nonlinear equations F(x) = 0, it is shown that the sequence {xk} generated by the new method converges to a solution quadratically, which is stronger than dist(xkċX∞) -- 0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.