We introduce a new Krylov subspace iteration for large scale eigenvalue problems that is able to accelerate the convergence through an inexact (iterative) solution to a shift-invert equation. The new method can take also full advantage of an exact solution when it is possible to apply a sparse direct method to solve the shift-invert equations. We call this new iteration the Truncated RQ Iteration (TRQ). It is based upon a recursion that develops in the leading kcolumns of the implicitly shifted RQ-Iteration for dense matrices. The main advantage in the large scale setting is that inverse-iteration like convergence occurs in the leading column of the updated basis vectors. The leading k-terms of a Schur decomposition rapidly emerge with desired eigenvalues appearing on the leading diagonal elements of the triangular matrix of the Schur decomposition. The updating equations for TRQ have a great deal in common with the update equations that define the Rational Krylov Method of Ruhe, and also the projected correction equations that define the Jacobi-Davidson Method of Van der Vorst et. al. The TRQ Iteration is quite competitive with the Rational Krylov Method when the shift-invert equations can be solved directly and with the Jacobi-Davidson Method when these equations are solved inexactly with a preconditioned iterative method. The TRQ Iteration is derived directly from the RQ-Iteration and thus inherits the convergence properties of that method. Existing RQ deflation strategies may be employed when necessary.