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Re: Is there a lowest Eigenvalues function around?

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  • Subject: [mg11108] Re: Is there a lowest Eigenvalues function around?
  • From: (AES)
  • Date: Sun, 22 Feb 1998 14:55:34 -0500
  • Organization: Stanford University
  • References: <6c8q6s$> <6cg2u3$>

Paul Abbott,, notes that:

> The largest or smallest eigenvalue of a matrix (and the corresponding
> eigenvector) can be computed using the power method [G H Golub and C F
> van Loan, Matrix Computations, Johns Hopkins Press, Baltimore, 1989]. 

I'll just toss in that the power method is very widely used (or at least
used to be) in finding the lowest eigenmodes and eigenvalues of optical
resonators and lensguides, under the name of the "Fox and Li" method. 
It's very simple to program, as already noted, and so provides an
approach which, though not particularly elegant or sophisticated, makes
it easy to trade off programming complexities for burning up some
addition CPU cycles.

There are some practical things to watch out for. If the dominant
eigenvalues are near-degenerate, the method may converge very slowly. 
On the other hand, if the two dominant eigenvalues are complex, the
system vill eventually converge down to a sinusoidally oscillating
variation from bounce to bounce -- sorry, that's the optical version, I
mean, from iteration to iteration -- from which one can easily extract
the two dominant eigenvalues.

Finally, if the matrix has a smallish number N of dominant eigenvalues
and then a lot of smaller eigenvalues, there is a somewhat less
well-known procedure called the Prony method, where one can do just N
iterations -- or maybe it's 2N, been a long time -- and extract the N
dominant complex eigenvalues as the roots of an N-th order polynomial. 
For information on it, and refs to the Fox and Li method, have a look
at A. E. Siegman and H. Y. Miller, "Unstable optical resonator loss
calculations using the Prony method," Appl. Opt. 9, 2729--2763
(December 1970).

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