EigensystemsDate: 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 |
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