  We present a method to linearize, without approximation, a specific class of
eigenvalue problems with eigenvector nonlinearities (NEPv), where the
nonlinearities are expressed by scalar functions that are defined by a quotient
of linear functions of the eigenvector. The exact linearization relies on an
equivalent multiparameter problem (MEP) that contains the exact solutions of
the NEPv. Due to the characterization of MEPs in terms of a generalized
eigenvalue problem this provides a direct way to compute all NEPv solutions for
small problems, and it opens up the possibility to develop locally convergent
iterative methods for larger problems. Moreover, the linear formulation allows
us to easily determine the number of solutions of the NEPv. We propose two
numerical schemes that exploit the structure of the linearization: inverse
iteration and residual inverse iteration. We show how symmetry in the MEP can
be used to improve reliability and reduce computational cost of both methods.
Two numerical examples verify the theoretical results and a third example shows
the potential of a hybrid scheme that is based on a combination of the two
proposed methods.
