For each in some domain D in the complex plane, let F() be a linear, compact operator on a Banach space X and let F be holomorphic in . Assuming that there is a so that I-F() is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue . In the Newton method the smallest eigenvalue of the operator pencil [I-F(), F()] 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(), I] with algebraic multiplicity m. For fixed let h() denote the arithmetic mean of the m eigenvalues of the pencil [I-F(), I] which are closest to 0. Then h is holomorphic in a neighborhood of and h()=0. Under suitable hypotheses the classical Muller's method applied to h converges locally with order approximately 1.84. 1988 Springer-Verlag.
