For each \(\lambda\) in some domain D in the complex plane, let F(\(\lambda)\) be a linear, compact operator on a Banach space X and let F be holomorphic in \(\lambda\). Assuming that there is a \(\xi\) so that I- F(\(\xi)\) is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue \(\xi\). In the Newton method the smallest eigenvalue of the operator pencil [I-F(\(\lambda)\),F'(\(\lambda)\)] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I-F(\(\xi)\),I] with algebraic multiplicity m. For fixed \(\lambda\) let h(\(\lambda)\) denote the arithmetic mean of the m eigenvalues of the pencil [I-F(\(\lambda)\),I] which are closest to 0. Then h is holomorphic in a neighborhood of \(\xi\) and \(h(\xi)=0\). Under suitable hypotheses the classical Muller's method applied to h converges locally with order approximately 1.84.