link to my thesis

230 CHAPTER 12. CONCLUSIONS & DIRECTIONS FOR FURTHER RESEARCH
SPP too. A central role in this approach is played by the notion of stable patterns,
which are shifted along the 2D-grid. The same approach leads to partial results in
the 3D-case (and higher dimensions).
The results of Section 5.3.2 also underlie the design of the heuristic methods
discussed in Section 5.3.3. These heuristic procedures are developed to arrive cheaply
at suboptimal paths with acceptable performance. We have implemented five of these
heuristics and compared their performance.
Another way to improve the efficiency of an nD-system is to apply parallel computing
techniques as described in Section 5.3.4. Here the ‘least-increments’ method
is implemented in such a way that it computes each path in the nD-system on a
separate processor. The parallel nD-system is applied in combination with the JDQZ
and the JD methods. However, it turns out that parallelization is not useful in this
application of the nD-system.
The use of iterative eigenvalue solvers is described in Chapter 6. The iterative
eigenvalue solver implementations we used in this thesis are JDQR, JDQZ and JD. An
advantage of such an iterative eigenvalue solver, is that it is able to focus on a subset
of eigenvalues of the involved matrix. Here the most interesting eigenvalue is the
smallest real eigenvalue. Focusing on the smallest real eigenvalue of a matrix is not
a usual parameter setting for an iterative eigenvalue solver. Therefore this option
is developed and implemented as described in Section 6.3: the selection criterion is
adapted such that it is no longer necessary to compute all the eigenvalues of the
matrix but that it becomes possible to focus on the smallest real eigenvalues.
An important observation is that the eigenvectors of the matrix at hand exhibit
a large amount of internal structure, called the Stetter structure. In Section 6.4
some procedures are studied to project an approximate eigenvector to a close-by
vector with Stetter structure to speed up the convergence process of the iterative
solver. The main bottleneck was the existence of negative and complex entries in the
approximate eigenvectors in combination with the use of a logarithmic transformation.
Subsequently, two methods are given to embed such a projection method in the
Jacobi–Davidson implementation.
The development of a Jacobi–Davidson eigenvalue solver for commuting matrices
is described in Section 6.5. This Jacobi–Davidson method is called the JDCOMM method
and is an extended version of the JD method. Its most important newly implemented
feature is that it computes the eigenvalues of the matrix A p in the outer loop while
iterating with a much sparser (but commuting) matrix A xi in the inner loop. Most
of the computation time is spent in this inner loop, while the outer loop is only used
occasionally, which results in a speed up in computation time and a decrease in the
amount of required floating point operations.
Chapter 7 presents the results of the numerical experiments in which the global
minima of various Minkowski dominated polynomials are computed using the approaches
and techniques mentioned in Part II of this thesis.
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