17. March 2026, Tuesday, 14:15, in seminar room J2.138
Title:
Analysis of Adaptive Damping Strategies in Newton Methods for the Numerical Computation of Complex Zeros
Abstract:
This master thesis investigates numerical methods for the computation of complex zeros of analytic functions. The focus lies on the comparison of Newton-based methods and argument principle–based approaches with respect to correctness and runtime behavior. The algorithms Simple-Argument, Simple-Argument-Newton, and Newton-Grid from the software library PyZeal are analyzed in detail. In addition, damped variants of the Newton method are implemented and compared to their undamped counterparts. Furthermore, a hybrid algorithm is developed that combines global root localization based on the argument principle with local damped Newton iterations.
The performance of the algorithms is evaluated through numerical experiments using complex polynomials with different geometric configurations of zeros, including uniformly distributed and strongly clustered arrangements. The results indicate that adaptive damping strategies can improve the robustness of Newton methods in certain scenarios. The proposed hybrid algorithm shows particularly strong reliability in cases with clustered zeros, although at the cost of increased computational effort.