In this manuscript a new method will be presented for performing a QR-iteration with (A − σ I)(A − κ I)−1 = QR without explicit inversion of the factor (A − κ I)−1. A QR-method driven by a rational function is attractive since convergence can occur at both sides of the matrix. Each step of this new iteration consists of two substeps. In the explicit version, first an RQ-factorization of the initial matrix A − κ I = RQ will be computed, followed by a QR-factorization of the matrix (A − σ I)Q H . The factorization of (A − σ I)Q H can be computed in an intelligent manner, exploiting properties of the already known RQ-factorization of A − κ I. The similarity transformation yielding the QR-step is defined by the unitary factor Q in the QR-factorization of the transformed matrix (A − σ I)Q H . Examples will be given, illustrating how to efficiently compute the factorization for some specific classes of matrices. The novelties of this approach with respect to these matrix classes will be discussed.