A robust homotopy continuation method for seeking all real roots of unconstrained systems of nonlinear algebraic and transcendental equations

A new homotopy developed for finding all real roots to a single nonlinear equation is extended to a system of nonlinear algebraic and transcendental equations written as f\x\ = 0 to find all real roots, including those on isolas. To reach roots that may lie on isolas, the functions are squared. This causes all roots to be bifurcation points that are connected to each other through stemming branches. As a result, a new system of homotopy functions, including numerous bifurcation points, is formed as H\x,t\ = (x - x0)(1 + f2 - t), where x0 is the starting point. Because the functions are squared, many systems of equations must be solved to find a starting point on a reduced system of homotopy functions written as H\x,t\ = 1 + f2 - t. Therefore, robustness is achieved at the expense of increased computation time. To improve the efficiency of the algorithm, the Levenberg-Marquardt method is used to find the starting point for the reduced homotopy system by solving a system of nonlinear equations with the degree of freedom equal to one. Then, a continuation method is used to track the paths from the resulting starting point to seek at least one root. Because all roots are bifurcation points, tracking the stemming branches from each subsequent root is the final step. The new algorithm was able to find successfully all the reported roots for 20 test problems that included a variety of algebraic and transcendental terms. In some cases additional roots were obtained. © 2011 American Chemical Society.