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5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 79
Table 5.4: Increase in the number of stored points along the axis
Increase factor Diagonal method Linear/Axis method Equalizing method
n =2/ n =3 n =2and n =3 n =2/ n =3
2 3.7 / 7.0 2.0 3.5 / 6.2
5 15.7 / 49.5 4.5 11.9 / 30.3
10 58.9 / 344.5 8.8 41.8 / 187.5
5.3.4 Parallel computations in the nD-systems approach
Suppose one wants to compute the state vectors at the time instants (0, 500), (125, 375),
(250, 250), (375, 125), and (500, 0). Figure 5.18 shows that the least-increments
method determines five paths for this purpose. In every iteration of the eigenvalue
solver the values of w 0,500 , w 125,375 , w 250,250 ,w 375,125 , and w 500,0 have to be computed
following these paths given the values of the initial state w 0,0 . Because these
five paths are well separated from each other (except in the very beginning near
the initial state), one could think of determining and computing these paths of the
least-increments method in parallel, i.e., each path is considered by one processor.
In such a situation a parallel approach is reasonable, but it has no effect when using
an nD-systems approach for global polynomial optimization. This is a conclusion
from the parallel numerical examples described in Section 7.4. The same results
are described in [53]. The reason that parallelization does not give any benefit in
an nD-systems approach for global polynomial optimization is the absence of long
paths. Such long paths as displayed in Figure 5.18 can not occur: the smallest
real eigenvalue of the operator A T p λ (x 1,...,x n)
is of importance as it yields the value of
the global minimum of p λ . The total time of the state vectors w t1,...,t n
, needed to
compute the action of A T p λ (x 1,...,x n)
is smaller than or equal to 2d, since the order of
the polynomial p λ is 2d (see the Equations (4.1) and (5.10)). The maximum total
time of a time instant y t1,...,t n
in the initial state w 0,...,0 is 2d − 2. As a result the
occurring paths contain time instants with a total time increasing from 2d − 2to2d
and therefore these paths are clearly shorter as the paths denoted in Figure 5.18.
The state vectors required to compute the action of the operator A T p λ (x ,
1,...,x n)
among which also the initial state w 0,...,0 , have the dimensions (2d − 1) × (2d − 1) ×
···×(2d − 1) and they all lie inside the region of points with maximum total time
2d, thereby forming a ‘closed’ region. These ’closed’ regions are displayed for the 2-
dimensional and 3-dimensional case in the Figures 5.19(a) and 5.19(b), respectively,
by a solid line.
The number of paths is obviously given by the number of monomial terms in the
polynomial p λ . Because of the fact that all these short paths start from the initial
state w 0,...,0 , there is much overlap especially near this initial state.
Remark 5.1. Note that even when the degree of p λ increases the paths remain short
because only the initial state, containing the given values, becomes larger and still
only a few steps in the nD-system are required to follow one path.