Eigensystems

Date: Wed, 2 Nov 94 18:37:19 MST
From: jaime morales
Subject: Eigensystems

I am working on alternate methods to solve polynomial equations of 
degree greater than 3.  I have read in several books that the roots of 
polynomial equations can be found in by the eigenvalues of its 
companion matrix.

I have read of and applied Jacobi transformations to several matrices 
with limited success.  I was wondering if you could give me some 
insight on alternate methods of finding eigenvalues or give me the 
name of a GOOD book that covers this topic in great detail.

Thanks,
Jaime Morales

From: Dr. Ethan
Date: Thu, 3 Nov 1994 10:38:29 -0500 (EST)

        I must say I know very little about solving polynomials using 
their companion matrices.  I have had a little experience (very little) 
finding eigen values of matrices although as I understand it this is in 
general not an easy problem.  If you just want ideas about given a 
matrix what are its eigen values, I may be able to help; if not, I have 
passed on your question to a few who may know more than I do - and 
of course there may be a math Doctor lurking n the shadows who 
knows the answer.

     Either way we are on it; we have heard you cry for help and we 
are coming as fast as we can.

                Ethan - Doctor On Call
__________

Date: Sat, 19 Nov 1994 15:16:50 -0500
From: Gene Klotz
Subject: Re: Eigensystems

Dear Jaime:

Dr. Math hasn't forgotten your question, but this is at a higher level 
than we usually operate (pre-college).  I'll ask our local matrix algebra 
person and see what she suggests for a good reference.  Hang in there!

                Gene
__________

Date: Mon, 21 Nov 1994 15:36:51 -0500
From: Gene Klotz
Subject: Re: Eigensystems

Dear Jaime:

Here's more on eigensystems from our linear algebraist:

"This is a vast subject and I'm not really familiar with the literature on
it.  Wilkenson's book, "The Algebraic Eigenvalue Problem"  is one of 
the classics, but a lot has gone on in this subject in the past 20 years.
There is a recent book by Golub called "Matrix Computations"  or 
something like that.  There are a number of numerical methods for 
finding  eigenvalues of large matrices, but I think the choice of method 
depends a  lot on the particular sort of matrix you have.  For all I know, 
there could  be lots of papers out there about computing eigenvalues 
of companion  matrices.  You really need to get in touch with someone 
who knows more about numerical linear algebra."

I talked with her further about this and she said that a good beginning
book is Gilbert Strang, "Linear Algebra and its applications," Academic
Press.  Apparently there aren't any middle-level books.  I think
Wilkenson's is a very good advanced book.

She also didn't expect you to gain very much in translating from roots of
polynomials to eigenvalues of matrices, since finding eigenvalues is
usually "reduced" to finding roots of polynomials (which is done
numerically by Newton's method or whatever).  Alas.

        Hope this helped--      Gene Klotz for Dr.Math