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112 CHAPTER 7. NUMERICAL EXPERIMENTS
solutions of this system can be substituted into the polynomial (7.1) to find the global
minimum.
The results of the computations using these software packages (using the default
parameters) are collected in Table 7.3. Note that SOSTOOLS does not return the
coordinates of the global minimum.
Table 7.3: Results of SOSTOOLS, GloptiPoly, and SYNAPS
Method x 1 ,x 2 ,x 3 ,x 4 Global Minimum Error Time (s)
SOSTOOLS (−, −, −, −) 4.09516477401837 2.97 × 10 −8 10
GloptiPoly (0.876535, −0.903963, 0.862021, −0.835180) 4.09516476247764 1.81 × 10 −8 11
SYNAPS (0.876536, −0.903965, 0.862026, −0.835184) 4.09516474461324 2.50 × 10 −10 2
When comparing the results in Table 7.2 and Table 7.3, we see that the methods
SOSTOOLS, GloptiPoly and SYNAPS are faster than the methods using our nD-systems
approach. Moreover, the method Eigs performs poorly: it uses very many operator
actions and needs a lot of running time. But where the methods in Table 7.3 appear
to be faster, they are not as accurate as the iterative eigenvalue solvers in Table 7.2.
This may be due to the default tolerance settings used in these software packages:
a trade off is involved between computation time and accuracy/error. Although the
methods in Table 7.3 outperform the nD-systems approach in running time, the nDsystems
approach leaves us the possibility to increase its performance and accuracy
by making use of the internal structure and the sparsity of the problem. The results
of these attempts are described in the next sections.
7.2 Computing the global minimum using target selection
In [53] a set of 22 Minkowski dominated polynomials is generated. The number of
variables in these polynomials ranges from 2 to 4 and the total degree ranges from
4 to 24. The explicit polynomials are given in Appendix B of this thesis and some
characteristics are displayed in Table 7.4.
Table 7.4 shows the number n of variables of the polynomials, the total degree
2d, the matrix size of the involved matrices A T p which is computed as (2d − 1) n , the
value of the global minimum (also the smallest real eigenvalue λ min ) and the number
of terms of each of these polynomials.
The global minimum of all the polynomials in Table 7.4 is computed using the
nD-systems approach and an iterative solver. Each computation is repeated 20 times
because of the randomly chosen first approximate eigenvector these solvers start with.
The JD method, the implementation of the Jacobi–Davidson method by M. Hochstenbach
in MATLAB (see [58]), is used for the research presented here. Also, the original
Jacobi–Davidson method using QZ-decomposition, JDQZ [39], is used to compare the
influence of different implementations of the iterative eigenvalue solver. Both implementations
have various options that control their behavior. A preliminary analysis
showed that some of these options should be adjusted to make sure that the algorithms
are able to correctly converge to the smallest real eigenvalue. The set of parameters