  The pseudospectrum of a linear time-invariant system is the set in the
complex plane consisting of all the roots of the characteristic equation when
the system matrices are subjected to all possible perturbations with a given
upper bound. The pseudospectral abscissa is defined as the maximum real part of
the characteristic roots in the pseudospectrum and, therefore, it is for
instance important from a robust stability point of view. In this paper we
present an accurate method for the computation of the pseudospectral abscissa
of retarded delay differential equations with discrete pointwise delays. Our
approach is based on the connections between the pseudospectrum and the level
sets of an appropriately defined complex function. The computation is done in
two steps. In the prediction step, an approximation of the pseudospectral
abscissa is obtained based on a rational approximation of the characteristic
matrix and the application of a bisection algorithm. Each step in this
bisection algorithm relies on checking the presence of the imaginary axis
eigenvalues of a complex matrix, similar to the delay free case. In the
corrector step, the approximate pseudospectral abscissa is corrected to any
given accuracy, by solving a set of nonlinear equations that characterize
extreme points in the pseudospectrum contours.
