From: israel@math.ubc.ca (Robert Israel)
Subject: Re: complex roots
Date: 25 Apr 2000 17:04:00 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8e2n6f$8i8$1@lacerta.tiscalinet.it>,
 "Roberto Diana" <rdiana@libero.it> writes:

> My name is Roberto Diana and I'm a Ph.D. student at Politecnico di Bari
> (Italy).
> In my research activity, I've a very complicated complex function. It's the
> determinant of a matrix, which is function of a complex variable x. The
> problem is to find a complex zero of this complex function. Every matrix
> element is a non linear function of argument; moreover the function contains
> a lot of poles in addition to zeros. The only informations are an initial
> estimate z0 of the zero.
> I've tried using Muller's method (starting from z0) but the function is too
> complicated, and the method seems to be not very fit.

If the initial approximation z0 is good enough, Newton's method starting
from  z0 should work.  Otherwise, you might try searching with a grid of starting
points near z0, using Newton's method with each of them.  
Note that if M is a matrix with elements m_{ij}(z), 
(d/dz) det M = sum_{j,k} m_{ij}' cofactor_{ij} 
where cofactor_{ij} = (-1)^(i+j) det(matrix obtained by deleting row i and
column j) 
This may be a better way to calculate the derivative than expanding it all out
first and then taking the derivative.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2