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One of the most challenging and frequently arising problems in many areas ofscience is to find solutions of a system of multivariate nonlinear equations.There are several numerical methods that can find many (or all if the system issmall enough) solutions but they each exhibit characteristic problems.Moreover, traditional methods can break down if the system contains singularsolutions. Here, we propose an efficient implementation of Newton homotopies,which can sample a large number of the stationary points of complicatedmany-body potentials. We demonstrate how the procedure works by applying it tothe nearest-neighbor $\phi^4$ model and atomic clusters.

Communication: Newton homotopies for sampling stationary points of potential energy landscapes