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

TL;DR

This paper enhances multipoint secant and interpolation methods for solving nonlinear systems by integrating a nonmonotone line search, demonstrating improved robustness and efficiency over Broyden's method through theoretical analysis and numerical experiments.

Contribution

It introduces a combination of multipoint secant and interpolation methods with nonmonotone line search, analyzing their global and superlinear convergence properties.

Findings

Methods are more robust than Broyden's method.

Numerical experiments show improved efficiency.

Theoretical results confirm superlinear convergence.

Abstract

Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the information about the Jacobian matrix gathered at the previous iterations, these methods are especially efficient in the case of expensive functions. They are known to be local and superlinearly convergent. We combine these methods with the nonmonotone line search proposed by Li and Fukushima (2000), and study global and superlinear convergence of this combination. Results of numerical experiments are presented. They indicate that the multipoint secant and interpolation methods tend to be more robust and efficient than Broyden's method globalized in the same way.