REGULARIZATION OF THE PROBLEM

A well-known solution for improving the convergence properties of the power algorithm by (Young and Doyle, 1990) is to regularize the real p problem, by adding a small amount e of complex uncertainty to each real uncertainty (Packard and Pandey, 1993). Let Ад and M(juj) the original data of the real ju problem, which thus consists in computing (a lower bound of) paR(M(ju>)). A model perturbation Ac is introduced, with the same structure as Ад, except that the real scalars become complex.

Let then the augmented model perturbation A = diag(An, Ac) and let:

The issue is now to compute a lower bound of /ід (H(ju)). The higher the value of e, the larger the amount of complex uncertainties, and the better the convergence properties of the power algorithm, which is applied to this regularized problem. e is classically chosen as 5 % or 10 %.

However, the lower bound obtained for this regularized p problem is not a lower bound for the original real p problem. The power algorithm provides indeed a critical model perturbation A* = diag(AR, A^), which renders the matrix I – H(juj)A* singular. No model perturbation Ад was found, which would render the matrix I — M{ju))A. R singular. It is especially worth emphasizing that the matrix I — M(ju})AR is not a priori singular.