  The estimation of weather forecast uncertainty with ensemble systems requires
a careful selection of perturbations to establish a reliable sampling of the
error growth potential in the phase space of the model. Usually, the singular
vectors of the tangent linear model propagator are used to identify the fastest
growing modes (classical singular vector perturbation (SV) method). In this
paper we present an efficient matrix-free block Krylov method for generating
fast growing perturbations in high dimensional dynamical systems. A specific
matrix containing the non-linear evolution of perturbations is introduced,
which we call Evolved Increment Matrix (EIM). Instead of solving an equivalent
eigenvalue problem, we use the Arnoldi method for a direct approximation of the
leading singular vectors of this matrix, which however is never computed
explicitly. This avoids linear and adjoint models but requires forecasts with
the full non-linear system. The performance of the approximated perturbations
is compared with singular vectors of a full EIM (not with the classical SV
method). We show promising results for the Lorenz96 differential equations and
a shallow water model, where we obtain good approximations of the fastest
growing perturbations by using only a small number of Arnoldi iterations.
