(75d) Development of a New Homotopy Method for Finding All Real Roots of Systems of Unconstrained Nonlinear Algebraic Equations

Although several methods are available for seeking all real roots of systems of algebraic equations, only homotopy-continuation methods can solve unconstrained systems. For a single nonlinear equation, a new homotopy was recently developed that can be embedded into a very efficient continuation method that avoids formation of an isola and locates all roots on a path that consists of just two continuous branches that stem from a single bifurcation point that is easily found. The new homotopy is a linear combination of the fixed-point and Newton homotopies, and is referred to as the FPN homotopy. This paper describes the extension of the new homotopy continuation method for a single equation to a system of unconstrained nonlinear algebraic and transcendental equations. The development was an evolutionary three-step process, whose goal was to find all real roots robustly and efficiently. All three steps were initiated from one or more bifurcation points. The first step was a simple extension of the single equation method. Although it was efficient, unreachable isolas with one or more roots sometimes existed; thus, the method was not robust. To eliminate unreachable isolas containing roots and, thereby, achieve robustness, the method was considerably improved in the second step by squaring the functions before forming the FPN homotopy. This caused all roots to be bifurcation points. From one starting point, all roots could be traced. However, for some problems, many paths had to be traced, and thus robustness was achieved at greatly reduced efficiency. Robustness was maintained and efficiency was significantly increased in the third step, which involved locating at least one optimal root from an arbitrary starting guess using just the squared functions. To obtain an optimal root, first the Levenberg-Marquardt algorithm is used to solve the (N-1) equations obtained by equalizing the N original equations to find a starting point, which most often is not a root. Next, that point is used in the continuation process to locate at least one optimal root, which is also a bifurcation points from which continuation finds all real roots. The development of the method is illustrated by solving a system of 7 nonlinear equations for all 10 real roots. The system describes a process consisting of a sequence of two stirred-tank reactors, both of which produce liquid as well as equilibrium vapor effluents.