I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter $w$, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue of the corresponding matrix.

For small values of $w$, the corresponding matrix is small and I can use the so-called power method — start with some vector, and multiply it by the matrix over and over, and under certain conditions you'll get the eigenvector corresponding to the largest eigenvalue. However, for the values of $w$ I am interested in, the matrix becomes too large, and thus the vector becomes too large, say, $n > 10^7$ entries or so, and it can't be contained in the computer's memory anymore and I need extra programming tricks or a very powerful computer.

As for the matrix itself, I don't need to store it in memory — I can access it as a black box, i.e., given $i, j$ I can return $A_{ij}$ via a simple computation. Also, the matrix has only $0$ and $1$ entries, and I believe it to be sparse (i.e., only around $\log n$ of the entries are $1$'s, $n$ being the number of rows/columns). However, the matrix is **not** symmetric.

Is there some method more space-effective for computation of eigenvalues for a case like this?