Our main contribution in this paper is to present an inexact Matrix-Newton algorithm that uses the tools for Newton's method in Banach spaces to solve a particular type of matrix valued problem: the nonlinear eigenvalue problem with eigenvector dependency (NEPv). We provide the conditions for our algorithm to be applicable to NEPv and show how to exploit the problem structure for an ef- ficient implementation. Various numerical experiments are provided that indicate the advantage of quadratic order of convergence over the linear order of the well- established SCF algorithm.
