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
我们开发了一款Fortran 90语言编写的程序包，可求解复平面区域$W$内的方程$f(z)=0$，其中$zin Wsubsetmathbb{C}$，且无需计算其导数$f'(z)$。$W$由一条简单闭合曲线界定，$f(z)$在$W$内为全纯函数（holomorphic）。该程序包已完成开发与测试，用于支撑我们在高频与光波导及谐振结构建模领域的研究工作。相关特征值问题极具挑战性，因其需要高精度计算所有多重复根……

## 程序名称：EZERO
## 目录编号：ADXY_v1_0

## 问题属性
求解方程$f(z)=0$的根，其中$z$为复平面内的变量，$f(z)$是其一阶导数表达式要么难以获取，要么即便表达式可获取但重复计算成本极高的函数。例如，若$f(z)$表示为一个大型矩阵的行列式，且该矩阵的每个元素均为含$z$的积分式。

## Mendeley Data中CPC知识库收录的本程序版本
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年）