A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain
The argument principle is a natural and simple method to determine the number of zeros of an analytic function f(z) in a bounded domain D. N, the number of zeros (counting multiplicities) of f(z), is 1/2\(\pi\) times the change in Arg f(z) as z moves along the closed contour \(\partial D\). Since the range of Arg f(z) is (-\(\pi\),\(\pi\) ] a critical point in the computational procedure is to assure that the discretization of \(\partial D\), \(\{z_ i,i=1,...,M\}\), is such that \(| \Delta _{[z_ i,z_{i+1}]}Arg f(z)| \leq \pi.\) Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of \(\partial D\) lead to a completely reliable algorithm to compute N. This algorithm also treats in an elementary way the case when a zero is on or near the contour \(\partial D\). Numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms.
scientific article; zbMATH DE number 4022465