Re: Eigenvalues and eigenvectors of an ill-conditioned matrix

From: The Phantom (phantom_at_aol.com)
Date: 12/31/04 

Date: Fri, 31 Dec 2004 14:41:54 -0800

On 31 Dec 2004 10:03:39 -0800, jcooper@ucalgary.ca wrote:

>I want to find the eigenvalues and eigenvectors for a matrix which is
>extremely ill-conditioned (true eigenvalues span perhaps 20-30 orders
>of magnitude). I trust the values in the matrix. When I try to obtain
>the eigenvalues and eigenvectors using LAPACK's DSYEV, however, the
>values are good only down to about 1.0E-16 of the highest eigenvalue.

   What's wrong with using a mathematical software that is capable of arbitrary precision
arithmetic, such as Mathematica or Maple? Using Mathematica, I found the eigenvalues of
the order 30 Hilbert matrix to better than 40 digit accuracy in substantially less than a
second, using 80 digit arithmetic. Finding the eigenvectors as well only took about twice
as long. The same problem (eigenvalues only) using 500 digit arithmetic took about 5.7
seconds on a 600 MHz laptop, but I didn't bother to find out how many digits were
accurate. I would think this procedure should meet your needs.

>Below that, there's a pile-up of random values including some negative
>(whereas I know the matrix to be positive semidefinite).
>
>The cutoff seems suspiciously close to the machine error for a
>double-precision floating point value, and given the negative
>eigenvalues I suspect that's what I'm running up against.
>
>Is there a standard way to obtain the eigenvalues and eigenvectors of
>such a matrix?
>
>Thank you.

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