link to my thesis

6 CHAPTER 1. INTRODUCTION
the quotient space is available, by means of an nD-systems approach?
To answer this question we need to address the following subquestions:
a) How to perform efficient computations for nD-systems?
b) How to exploit the Stetter structure of the involved eigenvectors?
c) How to tune the iterative eigenvalue solvers to gain efficiency?
2. How to find the global optimum to the H 2 model-order reduction problem for a
reduction of order N to order N − k, using the techniques of the Stetter-Möller
matrix method in combination with an nD-system and an iterative eigenvalue
solver?
The corresponding subquestions are:
a) How to improve the performance for co-order k = 1 reduction?
b) How to develop new techniques or extensions for co-order k = 2 reduction
and how to deal with the singular matrix pencils that occurs in this case?
c) How to develop new techniques or extensions for co-order k = 3 reduction
and how to deal with the structured and singular two-parameter polynomial
eigenvalue problem that occurs in this case?
1.4 Thesis outline
For readers unfamiliar with the algebraic background a brief introduction into the
most important concepts and definitions of algebraic geometry is given in Part I,
Chapter 2 of this thesis. Chapter 3 provides an overview of various methods which
are used to solve univariate and multivariate systems of polynomial equations.
The development and the efficiency improvements of the global polynomial optimization
method based on the Stetter-Möller matrix method are described in Part II
of this thesis.
In Chapter 4 we introduce the global optimization of a multivariate polynomial
and, in particular, the optimization of a Minkowski dominated polynomial. Subsequently,
we explain the techniques behind the Stetter-Möller matrix method.
In Chapter 5 the efficiency of the Stetter-Möller matrix method is improved by
associating the system of first-order conditions with an nD-system. The usage of
the nD-system is improved by setting up a corresponding shortest path problem as
in Section 5.3.2 and by implementing some heuristic procedures as in Section 5.3.3.
In Section 5.3.4 we try to improve the efficiency of the method by applying parallel
computing techniques.
The first section of Chapter 6 describes the main implementation aspects of
Jacobi–Davidson eigenvalue solvers including the pseudocode. Such a Jacobi–Davidson
method is able to focus on some specific eigenvalues of the involved matrix and,
therefore, in Section 6.3, a selection criterion is developed and embedded into the