I am looking for suitable algorithm to compute the eigenvalues and eigenvectors of a matrix. My matrix is sparse ( think of Finite Element Matrix), and it is very, very big ( think of hundreds of thousands or even million degrees of freedom).

The leading candidate for this task seems to be Lanczos algorithm.

The issue now is, how well Lanczos algorithm fare if the matrix is sparse? The reason I ask this is because I want to know if there are a lot of zero terms in a matrix, will Lanczos take advantage of this by storing only nonzero terms and operate on them? Since my matrix is big, I want to conserve as much memory space as possible.