Summary: This paper analyzes algorithms from the Broyden class of quasi-Newton methods for nonlinear unconstrained optimization. This class depends on a parameter \(\phi_ k\), for which the choices \(\phi_ k=0\) and \(\phi_ k=1\) give the well-known BFGS and DFP methods. This paper examines algorithms that allow for negative values of the parameter \(\phi_ k\). It shows that severe restrictions have to be imposed on the selection of \(\phi_ k\) to guarantee \(q\)-superlinear convergence. It is argued that negative values of \(\phi_ k\) are desirable, and conditions on \(\phi_ k\) that guarantee superlinear convergence are given. However, practical algorithms that preserve the excellent properties of the BFGS method are not easy to design.