Computing zeros of analytic functions in the complex plane without using derivatives
We present a package in Fortran 90 which solves f(z) = 0, where z is an element of W subset of C without requiring the evaluation of derivatives, f'(z). W is bounded by a simple closed curve and f (z) must be holomorphic within W. We have developed and tested the package to support our work in the modeling of high frequency and optical wave guiding and resonant structures. The respective eigenvalue problems are particularly challenging because they require the high precision computation of all multiple complex roots of f (z) confined to the specified finite domain. Generally f (z), despite being holomorphic, does not have explicit analytical form thereby inhibiting evaluation of its derivatives.
zeros analytic functions computation of zeros halley's method