Implicit schemes for solving large-scale linear dierential-algebraic systems with constant coef-cients necessitate at each integration step the solution of a linear system, typically obtainedby a Krylov subspace method such as GMRES. To accelerate the convergence, an approach isproposed that computes good initial guesses for each linear system to be solved in the implicitscheme. This approach requires, at each integration step, a small dimensional subspace where agood initial guess is found using the Petrov-Galerkin process. It is shown that the residual asso-ciated with the computed initial guess depends on the dimension of the subspace, the order ofthe implicit scheme, and the discretization stepsize. Several numerical illustrations are reported.