TITLE: How to compute the smallest eigenvalue using the power iteration algorithm?
QUESTION [19 upvotes]: I need to write a program which computes the largest and the smallest (in terms of absolute value) eigenvalues using both power iteration and inverse iteration.
I can find them using the inverse iteration, and I can also find the largest one using the power method. But I have no idea how to find the smallest one using the power method.
I tried applying some kind of shift such as $A - \lambda_{max}I$, but to no avail.
How can I modify the power method so that it computes the smallest eigenvalue?

REPLY [2 votes]: If your matrix isn't too large, you could do linear regression on $\{I,A,A^2,\cdots,A^n\}$ to find the coefficients of the minimal polynomial. The minimal polynomial will have scalar roots the same as the characteristic polynomial but maybe smaller multiplicities. Thing is this polynomial will solve for the eigenvalues over the scalar field too (Caley-Hamilton theorem links these to each other). Then you can solve numerically over your scalar field for the solution closest to the origin, probably by setting 0 as the first guess to your root-finding method.
This does however deviate a bit from the requirement of using a power iterations for both, so maybe it does not answer your question - but on the other hand it is more general, not assuming any definiteness of the matrix.