link to my thesis

Summary
The problem of computing the solutions of a system of multivariate polynomial equations
can be approached by the Stetter-Möller matrix method which casts the problem
into a large eigenvalue problem. This Stetter-Möller matrix method forms the starting
point for the development of computational procedures for the two main applications
addressed in this thesis:
• The global optimization of a multivariate polynomial, described in Part II of
this thesis, and
• the H 2 model-order reduction problem, described in Part III of this thesis.
Part I of this thesis provides an introduction in the background of algebraic geometry
and an overview of various methods to solve systems of polynomial equations.
In Chapter 4 a global optimization method is worked out which computes the
global minimum of a Minkowski dominated polynomial. The Stetter-Möller matrix
method transforms the problem of finding solutions to the system of first-order conditions
of this polynomial into an eigenvalue problem. This method is described in
[48], [50] and [81]. A drawback of this approach is that the matrices involved in this
eigenvalue problem are usually very large.
The research question which plays a central role in Part II of this thesis is formulated
as follows: How to improve the efficiency of the Stetter-Möller matrix method,
applied to the global optimization of a multivariate polynomial, by means of an nDsystems
approach? The efficiency of this method is improved in this thesis by using a
matrix-free implementation of the matrix-vector products and by using an iterative
eigenvalue solver instead of a direct eigenvalue solver.
The matrix-free implementation is achieved by the development and implementation
of an nD-system of difference equations as described in Chapter 5. This yields a
routine that computes the action of the involved large and sparse matrix on a given
257