This paper presents a new look at Davidson's method for the calculation of the rightmost eigenvalue(s). The combination of time-stepping by the Krylov exponential propagator and the Davidson method leads to a method that builds a Krylov space of the matrix exponential. The method is well-suited when only a matrix-vector multiplication by A is possible. The paper presents some convergence results and numerical examples, including a comparison with the Jacobi-Davidson method. (C) 1999 Elsevier Science B.V. and IMACS. All rights reserved.