A modification of the Delves-Lyness method for locating the zeros of analytic functions (Q1085564)

One of the variants of this general class of methods involves integrating the function \(z^ kf'(z)/f(z)\) round a closed contour. When this is done numerically one can avoid derivative evaluations by using integration by parts exploiting \(f'(z)/f(z)=d(\log f(z))/dz\). However, one is now left with the problem of keeping track of the argument of a multivalued analytic function as it winds round the origin (there being n winds when the number of zeros within the contour is n). This paper proposes a method which apparently reduces the chance of being misled. However, to use it one needs to know the numerical value of n. This reviewer's experience is that, when using these methods, the main problem is the determination of n. By the time enough information to be reasonably certain of this is available, locating the zeros is a simple ''spinoff'' calculation.