The matrix eigenvalue problem is often encountered in scientific computing applications. Although it has an uncomplicated problem formulation, the best numerical algorithms devised to solve it are far from obvious. In my PhD we generalize methods of QR-type, the standard class of algorithms to solve the eigenvalue problem, to rational QR methods. This generalization uses a pole swapping technique. We show that the convergence behaviour of this new class of algorithms is governed by rational functions and that they are both more accurate and more efficient than existing methods.