We consider a class of nonlinear eigenvalue problems including those arising from stability
analysis of linear systems of delay differential equations.
Our aim is to compute the pseudospectral abscissa, i.e. the real part of the rightmost point in the
pseudospectrum, and detect - for a stable system - when the pseudospectral abscissa vanishes, which
determines a nearby system which is not anymore stable. If this occurs for a small perturbation of
the considered stable system, this is a sign of lack of robustness.
In analogy to the linear eigenvalue problem we have that it is sufficient to restrict the analysis
to rank-1 perturbations of the matrices of the system.
Using this main result we present a gradient system approach which only requires the computation of
the spectral abscissa of a sequence of problems obtained by adding rank one updates to the matrices.
In order to be applied these methods simply require a procedure to compute the rightmost eigenvalue
and the corresponding left and right eigenvectors.
In addition, if the matrices are large and sparse then the computation of the rightmost eigenvalue
can for many classes of nonlinear eigenvalue problems be performed in an efficient way by iterative
algorithms which only rely on matrix vector multiplication and on solving systems of linear equations,
where the structure of the matrices (sparse plus rank one updates) can be exploited.