Numerical algorithms for finding an eigenvector

[jostpuur]

All matrices $A\in\mathbb{C}^{n\times n}$ have at least one eigenvector $z\in\mathbb{C}^n$. I'm interested to know what kind of algorithms there are for the purpose of finding an eigenvector.

I noticed that

$$\frac{|z^{\dagger} A z|}{\|Az\|} = 1\quad\quad\quad\quad (1)$$

holds only when $z$ is an eigenvector, so I succeeded writing one algorithm using this fact. I let $z$ move on the sphere $\|z\|=1$ when the program runs iterations so that the quantity (1) is maximized. Unfortunately I run into some precision problems. For some reason my function seems to give only 3 or 4 decimals right for the components of the eigenvector, even though floating point numbers in C-language could contain more accuracy. I believe that the inaccuracy problem rises from the fact that the quantity (1) is approximately a paraboloid in the environment of the eigenvector. If the peak was sharp, then its location would be easier to find with iterations, but paraboloid is not so sharp peak, which makes its location less precise.

Anyway, are there other kind of algorithms out there too?

 

[CFDFEAGURU]

Yes there are,

see http://www.nrbook.com/a/bookcpdf.php

chapter 11

This is the free older 1992 C book but there are also algorithms in the new NR3 book.

 

[jostpuur]

Those pdf files are locked somehow. My pdf viewer software asks for a password.

update:

I succeeded getting the book as a pdf file in an alternative way, and it's working now. So thank's for mentioning the book, anyway.

 

[jostpuur]

Actually I did not succeed in learning how to find an eigenvector from that book. For my purposes finding an eigenvector is approximately the same as finding a Schur form, so tried to look for that. I see that the book explains how the Householder transformations can be used to obtain a Hessenberg form, but I don't see how to proceed from Hessenberg form to a Schur form.

 

[CFDFEAGURU]

For clarity,

The pdf files for the NR3 book are locked. However, the older editions are free to view.

Also, you might want to post the question in the NR forum.