A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain

Summary 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 1/2p times the change in Arg f(z) as z moves along the closed contour s D . Since the range of Arg f(z) is (-p, p] a critical point in the computational procedure is to assure that the discretization of s D , z i , i =1, ..., M , is such that $$|Δ _\text[z_i \text, \textz_i + 1 \text] Arg f(z)| ≤q π $$ . Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of s 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 s 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.