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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence


Author: Jinyan Fan
Journal: Math. Comp. 81 (2012), 447-466
MSC (2010): Primary 65K05, 90C30
Published electronically: June 23, 2011
MathSciNet review: 2833503
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Abstract: We propose a modified Levenberg-Marquardt method for nonlinear equations, in which not only a LM step but also an approximate LM step are computed at every iteration. To ensure the global convergence of the new method, a new kind of predicted reduction is introduced for the merit function when using the trust region technique. The cubic convergence of the modified LM method is proved under the local error bound condition which is weaker than nonsingularity. Numerical results show that the new method is very efficient and could save many calculations of the Jacobian.


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Additional Information

Jinyan Fan
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email: jyfan@sjtu.edu.cn

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02496-8
PII: S 0025-5718(2011)02496-8
Keywords: Nonlinear equations, Levenberg-Marquardt method, local error bound, cubic convergence
Received by editor(s): April 19, 2010
Received by editor(s) in revised form: September 20, 2010
Published electronically: June 23, 2011
Additional Notes: The author was supported by Chinese NSF grants 10871127, 10701056 and the Chenxing Program of SJTU
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.