Computing zeros of analytic functions in the complex plane without using derivatives

Abstract
We present a package in Fortran 90 which solves f (z) = 0, where z ∈ W ⊂ C without requiring the evaluation of derivatives, f^′(z). W is bounded by a simple closed curve and f (z) must be holomorphic within W. We have developed and tested the package to support our work in the modeling of high frequency and optical wave guiding and resonant structures. The respective eigenvalue problems are particularly challenging because they require the high precision computation of all multiple complex ...

Title of program: EZERO
Catalogue Id: ADXY_v1_0

Nature of problem
Finding solutions of the equation f(z)=0 where z is a variable in the complex plane and f(z) a function for which formulae for the first derivatives are either not easily obtainable or when such formulae are available are very expensive to compute repeatedly. For example suppose, f(z) is expressed as a determinant of a large matrix each element of which is an integral in which z is present in the integrand.

Versions of this program held in the CPC repository in Mendeley Data
ADXY_v1_0; EZERO; 10.1016/j.cpc.2006.04.007

This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2019)

**摘要**
本文介绍一款基于Fortran 90语言开发的程序包，可求解复平面区域$W subset mathbb{C}$内的方程$f(z)=0$，无需计算其导数$f'(z)$。其中区域$W$由简单闭合曲线界定，且函数$f(z)$在$W$内为全纯函数。该程序包的开发与测试旨在支撑我们在高频、光波导与谐振结构建模领域的研究工作。此类问题对应的特征值问题极具挑战性，因其需要对全部多重复根进行高精度计算……

程序名称：EZERO
目录编号：ADXY_v1_0

**问题类型**
求解方程$f(z)=0$的根，其中$z$为复平面变量，$f(z)$为一类一阶导数公式难以获取，或即便可获取但重复计算成本极高的函数。例如，若$f(z)$表示为大型矩阵的行列式，且该矩阵的每个元素均为含参变量$z$的积分式（被积函数包含$z$）。

存放在Mendeley数据平台CPC程序库中的该程序版本：ADXY_v1_0; EZERO; 10.1016/j.cpc.2006.04.007

本程序源自贝尔法斯特女王大学托管的CPC程序库（1969-2019）