Rapid and robust resolution of Underwood equations using convex transformations

In this work, a new method is proposed for solving Underwood's equations. Newton methods cannot be used without interval control, and may require many iterations or experience severe convergence problems if the roots are near poles and the initial guess is poor. It is shown that removing only one adjacent asymptote leads to convex functions, while removing both asymptotes leads to quasi convex functions which are close to linearity on wide intervals. Using a change of variable, a pair of convex functions is defined; at each point within the search interval one of the two functions is guaranteed to satisfy the monotonic convergence condition for Newton methods. The search interval is restricted to narrow solution windows (simple and costless) and a simple high quality initial guess can be obtained using their bounds. Two solution algorithms are presented: in the first one, Newton (including higher-order) methods can be safely applied without any interval control using the appropriate convex function; in the second one, Newton iterations are applied to a quasi-convex function, and convex functions are used only if an iterate goes out of its bounds. The algorithms are tested on several numerical examples, some of them recognized as very difficult in the literature. The proposed solution methods are simple, robust, very rapid (quadratic or super-quadratic convergence) and easy to implement. In most cases, convergence is obtained in 2–3 Newton iterations, even for roots extremely close to a pole.