I was assigned to make a program that finds the largest, the N-largest, the smallest and the N-smallest eigenvalues of a symmetric matrix, using the Power Method. So far, I've been able to succesfully calculate the largest eigenvalue using the traditional Power Method, the N-largest using the Power Method with Deflation, and the smallest using the Inverse Iteration (the Inverse Iteration as described here in section 3-2: Iterative Methods).
But, right now I have no idea how to determine the N-smallest. I tried using the Inverse Iteration with Deflation to calculate the N-smallest but it is not working. When I calculate the second smallest (and so on…) I don't get the expected results, as if it's not possible to simply apply the Inverse Iteration with deflation. What am I missing? What should be the right way to calculate de N-smallest?
Your help is deeply appreciated.
Best Answer
I assume by "largest" and "smallest" you mean in absolute value. The $n$'th smallest eigenvalue of $A$ is $\pm \sqrt{t - \mu}$ where $\mu$ is the $n$'th largest eigenvalue of $t I - A^2$ if $t$ is large enough. See how this moves if you replace $A$ by $A-\epsilon I$ for small $\epsilon$ and you can tell whether it's $+$ or $-$.