A generalized refined Arnoldi method based on the weighted inner product is presented for computing PageRank. The properties of the generalized refined Arnoldi method were studied. To speed up the convergence performance for computing PageRank, we propose to change the weights adaptively where the weights are calculated based on the current residual corresponding to the approximate PageRank vector. Numerical results show that the proposed Arnoldi method converges faster than existing methods, in particular when the damping factor is large. Copyright © 2011 John Wiley & Sons, Ltd.