From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math.research
Subject: Re: Criteria for the roots of a polynomial to be outside |z| <= 1
Date: 7 Mar 1996 01:29:25 GMT

In article <4hk4is$17r@ramsey.cs.laurentian.ca>, sawyer@ramsey.cs.laurentian.ca (P. Sawyer) writes:
 
|> I would be interested in any criteria for the roots of a polynomial to be 
|> outside the unit disk.  

Well, of course a necessary and sufficient condition is int_C f'(z)/f(z) dz = 0
where C is the unit circle.
 
|> I have a more specific question: what conditions on the matrices $A_k$ of 
|> dimension $n\times n$  will ensure that the roots of
|> 
|> $\det(I_n - \sum_{k=1}^p A_k z^k) = 0 
|> 
|> are outside the unit disk. 
|> 
|> One condition which is sufficient, but too strict, is that $\sum_{k=1}^p 
|> \|A_k\| < 1$  (where $\| \|$ is a matrix norm). 

Unless you know something about the A_k, you're unlikely to get anything very
much better than that.  In special cases (e.g. if the A_k commute), more
can be said.  Well, here's one possibility: writing 
F(z) = I - sum_{k=1}^p A_k z^k (a matrix-valued polynomial), consider the
Maclaurin series for F(z)^(-1):
F(z)^(-1) = sum_{k=0}^infinity B_k z^k
where B_0 = I and B_k = sum_{j=1}^p A_j B_{k-j} for k >= 1 
(taking B_{k-j} as 0 for j>k).  
Then all roots of det(F(z)) are outside the closed unit disk
iff F(z) is invertible in a neighbourhood of the closed unit disk
iff \|B_k\| < C exp(-epsilon k) for some constants C and k > 0.

-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Y4

==============================================================================

From: pmt6jrp@gps.leeds.ac.uk (J R Partington)
Newsgroups: sci.math.research
Subject: Re: Criteria for the roots of a polynomial to be outside |z| <= 1
Date: Fri, 8 Mar 1996 09:52:59 +0000 (GMT)

In article <4hk4is$17r@ramsey.cs.laurentian.ca> sawyer@ramsey.cs.laurentian.ca (P. Sawyer) writes:
>
>I would be interested in any criteria for the roots of a polynomial to be 
>outside the unit disk.  

Look in a book on Control Theory (e.g. "Introduction to Control Theory"
by O.L.R. Jacobs) for the Schur-Cohn test - this checks whether all
the zeroes are inside the disc, but taking z^np(1/z) you can check whether
they are outside instead.

It's a bit long for me to write down here but involves checking the
signs of n determinants for a polynomial of degree n, the jth being of
order 2j.

Alternatively use a bilinear map to move the critical region to
the left half plane and use the Routh-Hurwitz criterion.

-- 
Dr Jonathan R. Partington,      Tel: UK: (0113) 2335123. Int: +44 113 2335123
School of Mathematics,          Fax: UK: (0113) 2335145. Int: +44 113 2335145
University of Leeds, Leeds LS2 9JT, U.K.    Email: J.R.Partington@leeds.ac.uk