The methods for solving k-eigenvalue problems for the multigroup neutron transport equation in one-dimensional slab geometry are presented. They are defined by means of multigroup and effective grey (one-group) low-order quasidiffusion (QD) equations. In this paper we formulate and study different variants of nonlinear QD iteration algorithms. These methods are analyzed on a set of test problems designed using C5G7 benchmark data. We present numerical results that demonstrate the performance of iteration schemes in different types of reactor physics problems. We consider tests that represent single-assembly and color-set calculations as well as a problem with elements of full-core computations involving a reflector zone.