Comparison of Krylov Subspace Methods on the PageRank Problem

PageRank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions. The problem incorporates a parameter $\alpha$ that determines the difficulty of the problem. In this paper, the effectiveness of stationary and non stationary methods are compared on some portion of real web matrices for different choices of $\alpha$. We see that stationary methods are very reliable and more competitive when the problem is well conditioned, that is for small values of $\alpha$. However, for large value of the parameter $\alpha$ the problem becomes more difficult and methods such as preconditioned BiCGStab or restarted preconditioned GMRES become competitive with stationary methods in terms of Mflops count as well as in number of iterations necessary to reach convergence.