(warning, I am not a numerical computation specialist) Krylov subspaces are a (the ?) common framework to compute eigenvalues and singular values, and it works

(warning, I am not a numerical computation specialist) Krylov subspaces are a (the ?) common framework to compute eigenvalues and singular values, and it works for sparse matrices as well. IOW, it is extremely useful for most problems that can be expressed as matrix factorization (e.g. recommendation using collaborative filtering, initial page rank-like algos).