A Shamanskii-like self-adaptive Levenberg-Marquardt method for nonlinear equations

Abstract In this paper, we are concerned with the Shamanskii-like self-adaptive Levenberg–Marquardt methods for nonlinear equations. We consider two choices of Levenberg–Marquardt parameter. One of them is the standard self-adaptive Levenberg–Marquardt parameter, the other is nonmonotone self-adaptive Levenberg–Marquardt parameter by using the nonmonotone technique of Grippo, Lampariello and Lucidi. Under the error bound condition which is weaker than nonsingularity, we show that the Shamanskii-like self-adaptive Levenberg–Marquardt methods converge with Q -order ( m + 1 ) . Numerical experiments show the new algorithms are efficient and promising.

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