Applications
Although the power iteration method approximates only one eigenvalue of a matrix, it remains useful for certain computational problems. For instance, Google uses it to calculate the PageRank of documents in their search engine. For matrices that are well-conditioned and as sparse as the Web matrix, the power iteration method can be more efficient than other methods of finding the dominant eigenvector.
Some of the more advanced eigenvalue algorithms can be understood as variations of the power iteration. For instance, the inverse iteration method applies power iteration to the matrix . Other algorithms look at the whole subspace generated by the vectors . This subspace is known as the Krylov subspace. It can be computed by Arnoldi iteration or Lanczos iteration. Another variation of the power method that simultaneously gives n eigenvalues and eigenfunctions, as well as accelerated convergence as is "Multiple extremal eigenpairs by the power method" in the Journal of Computational Physics Volume 227 Issue 19, October, 2008, Pages 8508-8522 (Also see pdf below for Los Alamos National Laboratory report LA-UR-07-4046)
Read more about this topic: Power Iteration