AN ALGORITHM THE BASIC ALGORITHM

The issue is to compute the maximal s. s.v. n(M(ju>)) over a large frequency interval [camjn, штах- Let:

Emax = max u{M{ju)) (10.37)

^€[^тгп» Mjnax]

The basic algorithm consists in the following steps:

is split into a set of small frequency intervals [щ, uij+i]

• The method of subsection 3.1 is applied to each interval [tup Wj+і]. The validity of the computed D, G scaling matrices on the interval is checked with the method of subsection 4.3. If the scaling matrices are indeed valid on [wj, а Ц upper bound Pi is found on this interval, and nothing

more is to be done inside [tap tuj+і]. Otherwise, the interval is split into two smaller intervals [tap (taj + u>i+i)/2] and [(w* + ші+і)/2, u>i+i],and the method of subsection 3.1 is applied to each of these two intervals. This process of branching on the frequency is repeated until а ц upper bound Pj is found on each frequency interval [tup Uj+i], whose union gives the large initial frequency interval [camjn, wmM].

• A reliable upper bound of Umax is the maximal value of the Pi’s. Either the state-space method by (Magni and Doll, 1997), or the frequency – domain method of chapter 5 (section 3.) provides a lower bound of цтах- If the gap between the bounds is not sufficiently small, it is possible to further branch on the frequency, in order to decrease the value of the upper bound of Umax-