The authors present a new algorithm for numerically following solution paths and predicting singular points of large and sparse one-parameter problems \(F(x,\lambda)=0,\) where F: \({\mathbb{R}}^ n\times {\mathbb{R}}\to {\mathbb{R}}^ n\). The path following algorithm is the Euler-Newton continuation method. The involved linear systems are solved iteratively by a preconditioned Arnoldi algorithm. The accompanied prediction for singular values along the path is done by linearizing \(A(\lambda)=F_ x(x(\lambda),\lambda)\) with respect to \(\lambda\) and considering an appropriate generalized linear eigenvalue problem. It is shown how to exploit the Arnoldi algorithm for solving this eigenvalue problem. The case of turning points is dealt with by taking an augmented problem. The algorithm is applied to a Taylor problem of hydrodynamics: The steady axisymmetric flow of an incompressible viscous fluid between two rotating cylinders. As the parameter \(\lambda\) either the wavelength of the flow or the Reynolds number is considered.