  In this paper we take a quasi-Newton approach to nonlinear eigenvalue
problems (NEPs) of the type $M(\lambda)v=0$, where
$M:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ is a holomorphic function. We
investigate which types of approximations of the Jacobian matrix lead to
competitive algorithms, and provide convergence theory. The convergence
analysis is based on theory for quasi-Newton methods and Keldysh's theorem for
NEPs. We derive new algorithms and also show that several well-established
methods for NEPs can be interpreted as quasi-Newton methods, and thereby
provide insight to their convergence behavior. In particular, we establish
quasi-Newton interpretations of Neumaier's residual inverse iteration and
Ruhe's method of successive linear problems.
