An application of the finite element approximation method to find the complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑧)

Abstract

Using a finite element approximation, an iterative optimization scheme is described to find the z zeros of K n ( z ) {K_n}(z) for fixed order n. Two computer programs have been implemented to find the complex zeros with a computational accuracy of either 13 or 27 significant digits. The optimization scheme described in the paper may also be readily applied to find real and complex zeros of an arbitrary function with real and complex coefficients. Neither its accuracy nor its efficiency is affected by the number of the roots of the function.