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Accelerating the modified Levenberg-Marquardt method for nonlinear equations

Author: Jinyan Fan
Journal: Math. Comp. 83 (2014), 1173-1187
MSC (2010): Primary 65K05, 90C30
Published electronically: August 8, 2013
MathSciNet review: 3167454
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Abstract: In this paper we propose an accelerated version of the modified Levenberg-Marquardt method for nonlinear equations (see Jinyan Fan, Mathematics of Computation 81 (2012), no. 277, 447-466). The original version uses the addition of the LM step and the approximate LM step as the trial step at every iteration, and achieves the cubic convergence under the local error bound condition which is weaker than nonsingularity. The notable differences of the accelerated modified LM method from the modified LM method are that we introduce the line search for the approximate LM step and extend the LM parameter to more general cases. The convergence order of the new method is shown to be a continuous function with respect to the LM parameter. We compare it with both the LM method and the modified LM method; on the benchmark problems we observe competitive performance.

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Additional Information

Jinyan Fan
Affiliation: Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China

Keywords: Nonlinear equations, Levenberg-Marquardt method, local error bound
Received by editor(s): February 6, 2012
Received by editor(s) in revised form: August 8, 2012
Published electronically: August 8, 2013
Additional Notes: The author was supported by Chinese NSF grants 10871127 and 11171217
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.