A family of secant methods based on general rank-1 updates has been revisited in view of the construction of iterative solvers for large non-Hermitian linear systems. As it turns out, both Broyden's “good” and “bad” update techniques play a special role—but should be associated with two different line search principles. For Broyden's bad update technique, a minimum residual principle is natural—thus making it theoretically comparable with a series of well-known alogrithms like GMRES. Broyden's good update technique, however, is shoen to be naturally linked with a minimum “next correction” principle—which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized PDEs of convection diffusion type in 2-D with internal layers give a first impression of the possible power of the derived good Broyden variant.