link to my thesis

76 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
Figure 5.15: Points required by the Axis method for computing state vectors at time
instants (0, 500), (250, 250), and (125, 375) with a 3 × 3 initial state
are 7460, 127756, 113988 and 54597, respectively. Figure 5.17 shows the points of
Figure 5.16 in the neighbourhood of the requested point (250, 250). From this figure,
one can see that the thin paths showing up in the plots of the linear and equalizing
methods in Figure 5.16 are built up from separate points located on diagonals of
length 2m − 1=5.
Because the diagonal, equalizing and axis methods are not efficient in every situation
and because the linear method is only efficient for small values of n, but becomes
less efficient when the dimension n of the problem increases, a fifth method, the leastincrements
method, is implemented. It is a combination of the equalizing and axis
method and it applies that recurrence relation of the system (5.3) which requires a
minimal number of new points to be calculated. It also proceeds in a backwards
fashion (starting from a requested point and moving towards the initial state) but
when constructing a path it takes the points into account that have already been
included in the path. This method becomes more efficient than the linear method for
larger values of n. Figure 5.18 shows a plot of all the evaluated points needed for the
computation of the state vectors at the time instants (0, 500), (125, 375), (250, 250),
(375, 125), and (500, 0) using this least-increments method. Obviously, more sophisticated
methods can also be designed which take the global structure of the requested
points into account.
To get an idea of the complexity of the first four heuristic methods additional
computations are carried out using the values from Tables 5.1 and 5.2. The relative
growth in the number of stored points needed for the computation of a requested point
was investigated for two situations: when moving along the diagonal t 1 = t 2 and when
moving along a time axis. The increase in the number of stored points is displayed