Abstract

We present a package in Fortran 90 which solves f(z)=0, where without requiring the evaluation of derivatives, f^′(z). is bounded by a simple closed curve and f(z) must be holomorphic within .

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 roots of f(z) confined to the specified finite domain. Generally f(z), despite being holomorphic, does not have explicit analytical form thereby inhibiting evaluation of its derivatives.

Program summary

Title of program:EZERO

Catalogue identifier:ADXY_v1_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXY_v1_0

Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland

Computer:IBM compatible desktop PC

Operating system:Fedora Core 2 Linux (with 2.6.5 kernel)

Programming languages used:Fortran 90

No. of bits in a word:32

No. of processors used:one

Has the code been vectorized:no

No. of lines in distributed program, including test data, etc.:21045

Number of bytes in distributed program including test data, etc.:223 756

Distribution format:tar.gz

Peripherals used:none

Method of solution:Our package uses the principle of the argument to count the number of zeros encompassed by a contour and then computes estimates for the zeros. Refined results for each zero are obtained by application of the derivative-free Halley method with or without Aitken acceleration, as the user wishes.

Keywords: Analytic functions; Zeros; Computation of zeros; Halley's method

PACS classification codes: 02.30.Dk; 02.60.Cb; 02.70.Pt; 02.70.-c

Article Outline

1. Introduction

2. The dispersion equation for a semiconductor layer

3. The method

3.1. Counting the roots

3.2. Localizing the roots

3.3. Refining the roots with Halley's method

3.4. Points to note

4. Implementation details

4.1. User interface

4.1.1. Namelist DIELECTRIC

4.1.2. Namelist INPUT

4.2. Code structure

5. Test runs

5.1. Test case 1

5.2. Test case 2: Modified Wilkinson polynomial

5.3. Gold film surrounded by glass substrate

6. Concluding remarks

Acknowledgements

References

This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).