# How do I estimate complex eigenvalues?

1. May 13, 2007

### photis

Let A be a matrix with real elements. The problem is to estimate eigenvalues of A, real and complex. QR algorithm is fine for real eigenvalues, but obviously fails to converge on complex eigenvalues... So, I'm looking for an alternative that could provide an estimate for complex eigenvalues of A. Can anybody help?

Thanks,
photis

2. May 13, 2007

### pkleinod

Go to http://www.netlib.org/lapack/double/. The routine dgeev.f (a fortran subprogramme) is good and does not require complex arithmetic because you have a real matrix, so that all complex eigenvalues (if there are any) come in pairs of complex conjugates. I have used this routine successfully many many times in the past but I don't remember the details of the numerical method. Usually, the matrix has to be reduced to Hessenberg form first; then, I think there may be a variant of the QR algorithm that works. I found the discussions in "Numerical Recipes" by Press et al. and "Numerical methods that work" by F.S. Acton very imformative. Good luck---the world of non-symmetric matrices is not a pleasant place!

For more about the numerical procedure, look in
http://www.netlib.org/lapack/lug/node50.html

Last edited: May 13, 2007
3. May 13, 2007

### photis

Thanks, your help was really valueable:!!) . Following the links, I found this: http://www.acm.caltech.edu/~mlatini/research/qr_alg-feb04.pdf

As eigenvalues come in conjugate pairs, QR apparently fails (no dominant eigenvalue exists). However, instead of generating a single eigenvalue estimation, QR produces a 2x2 matrix "containing" the conjugate pair. One can either (a) calculate the eigenvalues of the 2x2 matrix directly and proceed with next eigenvalue(s) or (b) use "Wilkinson shift" to move QR on the complex plain.

(a) may affect estimates of the remaining eigenvalues, but (b) introduces complex arithmetic, so (a) seems preferable. After all, I use QR to get reasonable initial approximations for inverse power algorithm.

Does anybody know if there is an alternative method (not QR) to ger Schur quasitriangular form of a real matrix?

4. Jun 29, 2007

### Reverie

5. Sep 27, 2010

### shasa

i have a matrix that its elements are complex ,ineed its eignvalue , did any onehave any subroutine for that?

(not double percision)
tnx