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 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.