Complex Roots of Meromorphic Functions. 1. How to Find One Zero

摘要

A number of numerical methods proposed to compute complex roots of analytic, or more generally meromorphic, functions f(z) are compared critically and their basic limitations are pointed out. It is shown that these can be overcome via a suitable steepest descent algorithm for /f(z)/ exp 2 . The strategy proposed allows to find a zero (as well as its multiplicity) with a predetermined precision provided that the precision with which f(z) is computed is known. Some peculiar features of the algorithm (which does not require an explicit knowledge of f'(z)) are illustrated in a number of non trivial examples. (Atomindex citation 11:560805)