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Sequential homotopy-based computation of multiple solutions to nonlinear equations

Homotopy methods have achieved significant success in solving systems of nonlinear equations for which the number of solutions are known and the homotopy paths are bounded. We present a two-stage homotopy process which does not require a-priori knowledge of the number of solutions to a system of nonlinear equations. This approach makes use of compact manifolds to find solutions sequentially along disconnected homotopy paths. The procedure is tested on two standard optimization and neural network benchmark problems.