Higher-order iterative methods for approximating zeros of analytic functions

AbstractIterative methods with extremely rapid convergence in floating-point arithmetic and circular arithmetic for simultaneously approximating simple zeros of analytic functions (inside a simple smooth closed contour in the complex plane) are presented. The R-order of convergence of the basic total-step and single-step methods, as well as their improvements which use Newton's and Halley's corrections, is given. Some hybrid algorithms that combine the efficiency of ordinary floating-point iterative methods with the accuracy control provided by interval arithmetic are also considered

Convergence of the accelerated overrelaxation method

summary:The convergence of the Accelerated Overrelaxation (AOR) method is discussed. It is shown that the intervals of convergence for the parameters $\sigma$ and $\omega$ are not always of the following form: $0\leq \omega \leq \omega_1, -\sigma_1\leq\sigma\leq\sigma_2, \sigma_1, \sigma_2\geq 0$