Nonlinear Eigenproblems

Abstract

Let $A(\lambda )$ be a holomorphic matrix-valued function defined on a domain $\Omega \subset {\Bbb C}.$ The nonlinear eigenproblem of finding generalized eigenpairs $\lambda ,\ v$ such that $A(\lambda )v=0$ is considered. The method which is exposed for solving this problem is based on the derivatives of the function $x(\lambda )=A(\lambda )^{-1}b$, where b is a given vector. Theoretical convergence results are established, and an algorithm is proposed for computing a few eigenvalues close to a given complex number together with some corresponding eigenvectors.