I have a dense real symmetric square matrix. The dimension is about 1000x1000.I need to compute the first principal component and wonder what the best algorithm to do this might be.It seems that MATLAB uses the Arnoldi/Lanczos algorithms (for eigs). But from reading about them I'm not sure whether they have any advantages over simple power iteration, since my matrix is not sparse and I'm only interested in the first eigenvector.Any recommendations what's the fastest algorithm in this case?
What's the most efficient way to compute the eigenvector of a dense matrix corresponding to the eigenvalue of largest magnitude?
linear algebra;algorithms;eigensystem