Abstract
In this paper, we propose a Shamanskii-like Levenberg-Marquardt method for nonlinear equations. At every iteration, not only a LM step but also m−1 approximate LM steps are computed, where m is a positive integer. Under the local error bound condition which is weaker than nonsingularity, we show the Shamanskii-like LM method converges with Q-order m+1. The trust region technique is also introduced to guarantee the global convergence of the method. Since the Jacobian evaluation and matrix factorization are done after every m computations of the step, the overall cost of the Shamanskii-like LM method is usually much less than that of the general LM method (the m=1 case).
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The author would appreciate the referees for the valuable suggestions which improve the presentation of the paper.
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Supported by Chinese NSF grants 10871127 and 11171217.
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Fan, J. A Shamanskii-like Levenberg-Marquardt method for nonlinear equations. Comput Optim Appl 56, 63–80 (2013). https://doi.org/10.1007/s10589-013-9549-4
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DOI: https://doi.org/10.1007/s10589-013-9549-4