From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Newsgroups: sci.math.num-analysis
Subject: Re: Complex root finder
Date: 3 Nov 1998 17:42:54 GMT

In article <363E08FB.6BBB20FF@arctic.mit.edu>,
 Jaiyong Lee <jai@arctic.mit.edu> writes:
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|> Hi.
|> 
|> I am looking for a complex root finder for a complex function
|> 
|> ( non polynomial function ).   I've tried Muller's algorithm,
|> 
|> but it has some problems finding zeros for some cases.
|>
snip
Laguerres method is the method of choice for determining
zeroes of complex analytic functions.
see p.henrici: computational complex analysis (I believe it is in volume 2,
have it not handy)
hope this helps
peter
==============================================================================

From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Newsgroups: sci.math.num-analysis
Subject: Re: root finder in complex plane
Date: 11 Dec 1998 18:14:10 GMT

In article <Pine.SOL.3.96.981211094652.15441A-100000@ux8.cso.uiuc.edu>,
 bin hu <binhu@students.uiuc.edu> writes:
|> Hi
|> 
|> Is there any subroutine that can perform the root finding in complex
|> plane? I have a complex function and need to find out all of its zeros in
|> a specific region(not a big region) in the complex plane. I hope there is
|> a routine existing that can let me input the region information and finds
|> all the zeros in that region. Does anyone has some experience on that?

not that easy.
by separating real and imaginary parts you can use any nonlinear solver for 
several variables. looking at these, you will find that most of them 
find _one_ zero (maybe near your initial guess, maybe not). but finding all
zeroes in a region is hard to do. for zeroes in a box there exists the
code INTBIS in netlib/toms (code toms/681)
that your problem is not so easy can be seen if you formulate it as 
min_z |f(z)|^2. you want _all_ _global_ minimizers in a region 
(hopefully a box). this can be done only using local methods combined with
estimation and branch and bound.
normally, for finding _a_ zero of a complex (analytic) function, it is not 
necessary and even not advisable to transform the problem to one in real 2-space.
The method of Laguerre is especially well suited for that case.
see henrici: computational complex analysis (it is in volume 2, if i remember
right). you may split out the zero , dealing then with f(z)/(z-z0), and try 
again. but I know of no ready to use code which combines this with 
constraints on the region. a last resort would be to use 
minimization of |f(z)|^2 as a function of Re z and Im z with the 
constraints on the region given by some functions from R^2->R again, then use
nonlinear constrained global optimization. (this an area of active research).
for more information see
http://plato.la.asu.edu/guide.html
hope this helps
peter