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The closest unstable equilibrium point (UEP) method is a well-known direct method of the Lyapunov type for optimally estimating stability regions of nonlinear dynamical systems. One key step involved in the closest UEP methodology is the computation of the closest unstable equilibrium point that has the lowest Lyapunov function value on the stability boundary. In this paper, a new computational algorithm to compute the closest UEP is presented. The proposed algorithm is based on a homotopy-continuation method combined with the singular fixed-point strategy. Numerical simulation results show that the algorithm outperforms previously reported existing techniques.