Theoretical Background

This section of the documentation contains the theoretical underpinning of the algorithms currently available as part of PyZEAL. A list of original research publications providing significantly more details can be found in the Bibliography below.

Important

The bibliography provided below is necessarily incomplete. It can only be a starting point for anyone wanting to go deeper into the theory of the numerical calculation of divisors of meromorphic functions.

Newton Grid Algorithm

• The Grid-Based Newton Algorithm

Simple Argument Algorithm

• The Simple Argument Algorithm

Simple Argument Newton Algorithm

• The Simple Argument with Newton Algorithm

Associated Polynomial Algorithm

• The Associated Polynomial Algorithm

Future Directions

• Future Directions

Bibliography

[Che22]

Haotian Chen. On locating the zeros and poles of a meromorphic function. Journal of Computational and Applied Mathematics, 402:113796, 2022. doi:https://doi.org/10.1016/j.cam.2021.113796.

[DSchutzZ02]

Michael Dellnitz, Oliver Schütz, and Qinghua Zheng. Locating all the zeros of an analytic function in one complex variable. Journal of Computational and Applied Mathematics, 138():325–333, 2002.

[DL67]

L. M. Delves and J. N. Lyness. A Numerical Method for Locating the Zeros of an Analytic Function. Mathematics of Computation, 21(100):543–60, 1967. doi:https://doi.org/10.2307/2004999.

[KVanBarelR+00]

P. Kravanja, M. Van Barel, O. Ragos, M.N. Vrahatis, and F.A. Zafiropoulos. ZEAL: A mathematical software package for computing zeros of analytic functions. Computer Physics Communications, 124(2):212–232, 2000. doi:https://doi.org/10.1016/S0010-4655(99)00429-4.

[KVanBarel99]

Peter Kravanja and Marc Van Barel. A derivative-free algorithm for computing zeros of analytic functions. Computing, 63(1):69–91, 1999.

[KVanBarel00]

Peter Kravanja and Marc Van Barel. Computing the Zeros of Analytic Functions. Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 1st edition, 2000. ISBN 978-3-540-67162-6.

[KVanBarelH99]

Peter Kravanja, Marc Van Barel, and Ann Haegemans. On computing zeros and poles of meromorphic functions. Computational Methods and Function Theory 1997, pages 359–369, 1999.

[LD67]

J.N. Lyness and L.M. Delves. On numerical contour integration round a closed contour. Mathematics of Computation, 21(100):561–577, 1967. doi:https://doi.org/10.2307/2005000.

[RS02]

Reinhold Remmert and Georg Schumacher. Funktionentheorie 1. Springer-Lehrbuch. Springer Berlin, Heidelberg, 5th edition, 2002. ISBN 978-3-540-41855-9.

[YNK88]

Xingren Ying and I Norman Katz. A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain. Numerische Mathematik, 53(1):143–163, 1988.