SELECTIVE SOLUTIONS TO TRANSCENDENTAL EQUATIONS

This paper presents a combination of global iterative methods, based on the Fatou-Julia theory, and local methods to find selected roots of elementary transcendental equations, z = F(z, c), z and c complex, that occur in complex Sturm-Liouville eigenvalue problems, in dielectric spectroscopy and in orbit determination. Suitable starting values for the iteration function, Zn + 1 = F(z(n),c), and appropriate regions for each determination of the inverse iteration function, zn + 1 = F-1 (z(n),c), are presented. Convergence criteria are derived from the facts that F has very few attractive fixed points and that the attractive fixed points of F-1 have relatively large basins of attraction in the above-mentioned regions. Certain fixed points of F can be reached quickly by means of a local method, like Newton's method. It is shown by means of digital figures that, in general, Newton's method may lead to attractive cycles or to unpredictable roots when the starting values are near the Fatou-Julia set. Convergence to parasitic roots or even to strange attractors may occur with iterative methods of higher order.