  A new iterative method for solving large scale symmetric nonlinear eigenvalue
problems is presented. We firstly derive an infinite dimensional symmetric
linearization of the nonlinear eigenvalue problem, then we apply the indefinite
Lanczos method to this specific linearization, resulting in a short-term
recurrence. We show how, under specific assumption on the starting vector, this
method can be carried out in finite arithmetic and how the exploitation of the
problem structure leads to improvements in terms of computation time. The
eigenpair approximations are extracted with the nonlinear Rayleigh-Ritz
procedure combined with a specific choice of the projection space. We
illustrate how this extraction technique resolves the instability issues that
may occur due to the loss of orthogonality in many standard Lanczos-type
methods.
