link to my thesis

4 CHAPTER 1. INTRODUCTION
1.2 Approach
Finding the global minimum of a real-valued multivariate polynomial is a problem
which has many useful applications. To guarantee a finite number of solutions to
the system of first-order conditions of the polynomial under consideration, we focus
on Minkowski dominated polynomials. The system of first-order conditions of
a Minkowski dominated polynomial is directly in Gröbner basis form with a known
basis for the quotient space. However, in [63] and [64] it is shown that we could also
start from general polynomials or rational functions.
The Stetter-Möller matrix method reformulates the problem of finding the global
minimum of a Minkowski dominated polynomial as an eigenvalue problem. Actually,
it yields a set of large and sparse commuting matrices. To study and improve the
efficiency of the Stetter-Möller matrix method, it is worked out how the matrices that
show up in this method admit an interpretation in terms of nD-systems. Such an
nD-system is used to compute the action of a matrix more efficiently using recursions.
In this way it is possible to avoid the explicit construction of these matrices.
The eigenvalues of these matrices determine the solutions of the system of firstorder
conditions. To compute the global minimum, not all these eigenvalues and
eigenvectors are required, but we focus on a few selected solutions of the system of
first-order conditions. To achieve the computation of only a selected set of eigenvalues,
we use iterative eigenvalue solvers. This yields many possibilities to improve the
efficiency of the approach, and some of them have been investigated. First, the nDsystems
approach is incorporated in the Jacobi–Davidson eigenvalue solver, yielding
a matrix-free solver, and the efficient computation within the nD-system setting is
investigated. Second, the Jacobi–Davidson method is tuned in various ways, such
as changing the parameter settings, implementing a projection method that projects
an approximate eigenvector to a vector with Stetter structure and changing the operators
during the iteration process. In particular this last modification has lead to
the development of the JDCOMM iterative eigenvalue solver which has the following
properties: (i) it computes the eigenvalues of a matrix while iterating with a much
sparser matrix which results in a speed up in computation time and a decrease in
required floating point operations, and (ii) it targets on the smallest real eigenvalue of
the matrix first. Moreover, this solver is compatible with the matrix-free nD-system
approach. Finally, all the methods mentioned here have been tested on a set of polynomials,
varying in degree and number of variables. It turns out that in some cases
the approach described here has a superior performance in terms of time and accuracy
over other more conventional methods.
Preliminary and partial results described in Part II of this thesis have been communicated
in [13], [14], [15], [16], [17], [18], [20], [53], and [88].
The same techniques are applied to address the H 2 model-order reduction problem
of a system of order N to a reduced system of order N − k. The case where the order
of the given system of order N is reduced to order N − k is called the co-order k case.