| The asymptotic behavior of fixed-point methods in the complex domain is studied. It is shown that both direct substitution and Newton's method exhibit stable periodic and aperiodic behavior from real- or complex-valued starting points. Moreover, multiple stable periodic orbits can exist for direct substitution. Traditional trust region (or dogleg) methods, on the other hand, often terminate at singular points, which correspond to nonzero-valued saddle points in the least squares function that can be arbitrarily far from a solution. Furthermore, the basins of attraction of these singular points are usually dispersed throughout the basin boundaries in the complex domain, clearly illustrating that singular points (via the dogleg strategy) also attract either real- or complex-valued starting points. In light of this, an extension of the dogleg strategy to the complex domain, based on a simple norm-reducing, singular point perturbation, is proposed. It is shown that this extended trust region method removes all forms of nonconvergent behavior and always terminates at a fixed-point, even from critical point (i.e., worst-case) initial values. Many numerical results and geometric illustrations using chemical process simulation examples are presented. |
