Finding Zeros of Analytic Functions and Local Eigenvalue Analysis Using Contour Integral Method in Examples
Advances in Electrical and Electronic Engineering
A numerical method for computing zeros of analytic complex functions is presented. It relies on Cauchy's residue theorem and the method of Newton's identities, which translates the problem to finding zeros of a polynomial. In order to stabilize the numerical algorithm, formal orthogonal polynomials are employed. At the end the method is adapted to finding eigenvalues of a matrix pencil in a bounded domain in the complex plane. This work is based on a series of papers of Professor Sakurai and
... ssor Sakurai and collaborators. Our aim is to make their work available by means of a systematic study of properly chosen examples. Keywords Contour integral method, formal orthogonal polynomials, generalized eigenvalue problem, zeros of analytic functions. Clearly, z 1 = 1, α 1 = 10, z 2 = 5, α 2 = 5. However, the method of Newton's identities and ε int := 0.01 now give very poor results: z 1 = 0.638482651241363 + 0.1123392797585549i, z 2 = 0.638587425742418 − 0.1126503196732164i, z 3 = 0.763900657783945 + 0.3022377533042421i, z 4 = 0.764166341345210 − 0.3024211941351311i, z 5 = 0.981330371128332 + 0.3925444707253891i, z 6 = 0.981647031382095 − 0.3925384311440422i, z 7 = 1.223616331736505 + 0.3378606328576522i, z 8 = 1.223870848730167 − 0.3376746496432903i, z 9 = 1.392149341950558 + 0.1355311955012755i, z 10 = 1.392248774201433 − 0.1352287380346158i, z 11 = 4.969215063333387 + 0.0229039229183072i, z 12 = 4.969232020549779 − 0.0229259755221052i, z 13 = 5.012225738075729 + 0.0354938759143295i, z 14 = 5.012250530712597 − 0.0354846773469500i, z 15 = 5.037076872119689 + 0.0000128545195791i. The poor results of the last example are caused by the instability of the mapping of the polynomial coefficients to the roots. In the next section we introduce the concept of formal orthogonal polynomials that will allow to separate the problem into two subtasks: • first determine all mutually distinct zeros of f , • and then determine their multiplicities. By this approach the last example gets well-posed.