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Multipoint secant and interpolation methods are effective tools for solvingsystems of nonlinear equations. They use quasi-Newton updates for approximatingthe Jacobian matrix. Owing to their ability to more completely utilize theinformation about the Jacobian matrix gathered at the previous iterations,these methods are especially efficient in the case of expensive functions. Theyare known to be local and superlinearly convergent. We combine these methodswith the nonmonotone line search proposed by Li and Fukushima (2000), and studyglobal and superlinear convergence of this combination. Results of numericalexperiments are presented. They indicate that the multipoint secant andinterpolation methods tend to be more robust and efficient than Broyden'smethod globalized in the same way.

Multipoint secant and interpolation methods with nonmonotone line search for solving systems of nonlinear equations