Summary: The Arnoldi algorithm is applied to a linearization of a nonlinear path following problem. A shift and invert spectral transformation gives fast convergence to those eigenvalues that predict turning points or bifurcations along the path, in such points the matrix of partial derivatives becomes singular. This approach is of special utility for very large matrices coming from systems with many degrees of freedom. The technique is demonstrated on a discretization of a Taylor vortex flow problem.