link to my thesis

Chapter 7
Numerical experiments
This chapter presents numerical experiments in which the global minima of various
Minkowski dominated polynomials are computed using the approaches of the previous
chapters. Section 7.1 describes the computations on a polynomial of order
eight in four variables. To compute the global minimum of this polynomial conventional
methods are used first. Then the nD-systems approach is used in combination
with the iterative eigenproblem solvers JDQR, JDQZ, and Eigs. Finally, the outcomes
of these computations are compared with those of the software packages SOSTOOLS,
GloptiPoly and SYNAPS. In Section 7.2 the nD-systems approach is used to compute
the global minima of 22 distinct polynomials. These 22 polynomials vary in the number
of variables and in the total degree and therefore they also vary in the size of
the involved linear operators and in their numerical conditioning. In this section the
target selection, described in Section 6.3, is used to let the iterative solvers JD and
JDQZ focus on the smallest real eigenvalues first. To put the performance of these
methods into perspective it is compared with the performance of the SOSTOOLS package.
The Sections 7.3 and 7.4 show the results of applying the projection method of
Section 6.4 and the parallel approach of Section 5.3.4 on the same set of 22 polynomials.
Finally, Section 7.5 describes four experiments in which the JDCOMM method is
used to compute the global minima. The JDCOMM method is an extended version of
the JD method for commuting matrices and is described in Section 6.5 of this thesis.
The global minima in this section are also computed by the JD method and by the
SOSTOOLS and GloptiPoly packages to be able to compare the performance and the
efficiency of the various methods used.
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