The thesis is divided into two main parts. In the first part we present the background material and current state of the art in greater depth and review the existing litera-ture. The major topics of relevance are: the theory and techniques of the solution of systems of equations, inequalities, and optimisation problems; topological degree theory;
Bernstein expansion; and interval arithmetic. The second part of the thesis consists of contributions in four areas, each of which is given in a separate chapter: a major in-vestigation of a recursive topological degree computation algorithm, a branch-and-bound method for systems of polynomial equations, an improved Bernstein expansion, and affine bounding functions. This second part includes results which appear in the publications [GS01a, GS01b, GJS03a, GJS03b, GS04, GS05, GS07, GS08, Smi09]. Here follows a brief overview of each chapter:
Chapter 2: An introduction to interval analysis and arithmetic. Beginning with a brief discussion on uncertainty, we proceed to review the fundamental theory of interval arith-metic and its properties. We introduce interval enclosures and give an overview of the main categories of interval algorithms. The algorithm in Chapter 7 deals centrally with intervals and the Bernstein expansion (utilised in Chapters 8–10) is closely related to interval analysis.
Chapter 3: A treatment of the Bernstein expansion. Firstly we deal with the Bern-stein basis, the BernBern-stein basis polynomials, and conversions to and from the BernBern-stein and power bases. The major properties of the Bernstein coefficients are then discussed, before proceeding to a number of important algorithms for the computation of the Bernstein coeffi-cients. Finally, there is a brief discussion of the mean value Bernstein form and Bézier curves.
Chapter 4: An introduction to topological degree. We review the foundations of topo-logical degree theory, in particular the Brouwer degree and its properties. There follows a review of existing known algorithms for its calculation, which are relatively few and mostly grounded upon similar ideas. The recursive method of Aberth [Abe94] is explained at some length, with an illustrative worked example, serving as the starting point for Chapter 7.
Chapter 5: A discussion on systems of polynomial equations. We firstly mention sev-eral application areas where such systems arise and give a simple example. There follows a discussion on the categorisation of solution methodology and an overview of five known techniques.
1 Introduction
Chapter 6: A review of problems involving polynomial inequalities. We briefly cover systems of polynomial inequalities, including stability regions, and global optimisation prob-lems, including the use of relaxations.
Chapter 7: A detailed study of a recursive algorithm for the computation of Brouwer degree. The behavioural attributes of the recursive method in practice (especially the com-plexity) were not previously well-understood and so we begin with some open questions. A detailed abstract analysis is followed by a number of computational examples. An estimate of the complexity, with a focus on the face subdivision process, is obtained through an unorthodox abstract complexity study, a probabilistic analysis based on a simplified geo-metric model. The software is exercised with a catalogue of examples, and data are obtained through the large-scale repetition of randomly-generated test problems, which support the conclusions of the complexity study. A further abstract analysis concerning an optimal face subdivision strategy is undertaken, introducing some new terminology for strategy at-tributes and identifying a crucial sub-problem. This material motivates the conception of a proposed new subdivision heuristic.
Chapter 8: A branch-and-bound method for the solution of systems of polynomial equa-tions. We describe a new algorithm, which is based on the Bernstein expansion, coupled with a simple existence test which utilises interval arithmetic. The method is tested with several examples, and a couple of simple strategies for the choice of subdivision direction selection are compared. The method is then improved by the addition of a preconditioning scheme which results in a reduced complexity.
Chapter 9: A new representation for the Bernstein coefficients which offers a major computational advantage. We firstly derive some key results concerning the Bernstein coef-ficients of monomials, and their monotonicity. This motivates the formulation of an implicit Bernstein representation. Coupled with three tests which are proposed, this form poten-tially allows a much faster computation of the Bernstein enclosure. An illustrative example is given, and the scheme is tested with several example polynomials from the literature. It is seen that the complexity is much reduced for many types of sparse polynomial.
Chapter 10: Several different types of affine bounding functions for polynomials, based upon the Bernstein expansion. After a brief discussion on convex envelopes, we consider the advantageous construction of affine bounding functions, which might be utilised in a relaxation scheme within a branch-and-bound framework. Six different algorithms, some simple and some elaborate, are described. They are compared with a large catalogue of randomly-generated polynomials. Finally, there is a brief discussion on the potential use of the implicit Bernstein form, as well as the application of such affine bounding functions for polynomials.
1 Introduction
Chapter 11: Summary and conclusions. In the final chapter we review the main results of the thesis, placing them in context, and outline possible directions for further related work.
Appendix A: Software. In the appendix we present an overview of the developed soft-ware, both for the computation of topological degree and for the computation of the Bern-stein expansion and associated bounding functions.